[877] | 1 | include "ASM/Assembly.ma". |
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| 2 | include "ASM/Interpret.ma". |
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[1014] | 3 | include "ASM/StatusProofs.ma". |
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[1616] | 4 | include alias "arithmetics/nat.ma". |
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[877] | 5 | |
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[905] | 6 | definition bit_elim_prop: ∀P: bool → Prop. Prop ≝ |
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| 7 | λP. |
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| 8 | P true ∧ P false. |
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| 9 | |
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| 10 | let rec bitvector_elim_prop_internal |
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[1936] | 11 | (n: nat) (P: BitVector n → Prop) (m: nat) |
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| 12 | on m: |
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| 13 | m ≤ n → BitVector (n - m) → Prop ≝ |
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[905] | 14 | match m return λm. m ≤ n → BitVector (n - m) → Prop with |
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| 15 | [ O ⇒ λprf1. λprefix. P ? |
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[1936] | 16 | | S n' ⇒ λprf2. λprefix. |
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| 17 | bit_elim_prop (λbit. bitvector_elim_prop_internal n P n' …) |
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[905] | 18 | ]. |
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[1955] | 19 | try applyS prefix |
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| 20 | try (@le_S_to_le assumption) |
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| 21 | letin res ≝ (bit ::: prefix) |
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| 22 | <minus_S_S >minus_Sn_m |
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| 23 | assumption |
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[905] | 24 | qed. |
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| 25 | |
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| 26 | definition bitvector_elim_prop ≝ |
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| 27 | λn: nat. |
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| 28 | λP: BitVector n → Prop. |
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| 29 | bitvector_elim_prop_internal n P n ? ?. |
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[1955] | 30 | try @le_n |
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| 31 | <minus_n_n @[[ ]] |
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[905] | 32 | qed. |
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| 33 | |
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| 34 | lemma bool_eq_internal_eq: |
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| 35 | ∀b, c. |
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| 36 | (λb. λc. (if b then c else (if c then false else true))) b c = true → b = c. |
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| 37 | #b #c |
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[1948] | 38 | cases b cases c normalize nodelta |
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| 39 | try (#_ % @I) |
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| 40 | #assm destruct % |
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[905] | 41 | qed. |
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| 42 | |
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[877] | 43 | definition bit_elim: ∀P: bool → bool. bool ≝ |
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| 44 | λP. |
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| 45 | P true ∧ P false. |
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| 46 | |
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| 47 | let rec bitvector_elim_internal |
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[1936] | 48 | (n: nat) (P: BitVector n → bool) (m: nat) |
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| 49 | on m: |
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| 50 | m ≤ n → BitVector (n - m) → bool ≝ |
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[877] | 51 | match m return λm. m ≤ n → BitVector (n - m) → bool with |
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| 52 | [ O ⇒ λprf1. λprefix. P ? |
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| 53 | | S n' ⇒ λprf2. λprefix. bit_elim (λbit. bitvector_elim_internal n P n' ? ?) |
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| 54 | ]. |
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[1955] | 55 | /2/ |
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[877] | 56 | qed. |
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| 57 | |
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| 58 | definition bitvector_elim ≝ |
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| 59 | λn: nat. |
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| 60 | λP: BitVector n → bool. |
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| 61 | bitvector_elim_internal n P n ? ?. |
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[1941] | 62 | try @le_n |
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| 63 | <minus_n_n @[[]] |
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[877] | 64 | qed. |
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| 65 | |
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| 66 | lemma mem_middle_vector: |
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| 67 | ∀A: Type[0]. |
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| 68 | ∀m, o: nat. |
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| 69 | ∀eq: A → A → bool. |
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| 70 | ∀reflex: ∀a. eq a a = true. |
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| 71 | ∀p: Vector A m. |
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| 72 | ∀a: A. |
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| 73 | ∀r: Vector A o. |
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| 74 | mem A eq ? (p@@(a:::r)) a = true. |
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[1936] | 75 | #A #m #o #eq #reflex #p #a |
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[1939] | 76 | elim p try (normalize >reflex #H %) |
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| 77 | #m' #hd #tl #inductive_hypothesis |
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| 78 | normalize |
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| 79 | cases (eq ??) normalize nodelta |
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| 80 | try (#irrelevant %) |
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| 81 | @inductive_hypothesis |
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[877] | 82 | qed. |
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| 83 | |
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| 84 | lemma mem_monotonic_wrt_append: |
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| 85 | ∀A: Type[0]. |
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| 86 | ∀m, o: nat. |
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| 87 | ∀eq: A → A → bool. |
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| 88 | ∀reflex: ∀a. eq a a = true. |
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| 89 | ∀p: Vector A m. |
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| 90 | ∀a: A. |
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| 91 | ∀r: Vector A o. |
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| 92 | mem A eq ? r a = true → mem A eq ? (p @@ r) a = true. |
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[1936] | 93 | #A #m #o #eq #reflex #p #a |
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[1939] | 94 | elim p try (#r #assm assumption) |
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| 95 | #m' #hd #tl #inductive_hypothesis #r #assm |
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| 96 | normalize |
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| 97 | cases (eq ??) try % |
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| 98 | @inductive_hypothesis assumption |
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[877] | 99 | qed. |
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| 100 | |
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| 101 | lemma subvector_multiple_append: |
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| 102 | ∀A: Type[0]. |
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| 103 | ∀o, n: nat. |
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| 104 | ∀eq: A → A → bool. |
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| 105 | ∀refl: ∀a. eq a a = true. |
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| 106 | ∀h: Vector A o. |
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| 107 | ∀v: Vector A n. |
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| 108 | ∀m: nat. |
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| 109 | ∀q: Vector A m. |
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| 110 | bool_to_Prop (subvector_with A ? ? eq v (h @@ q @@ v)). |
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[1936] | 111 | #A #o #n #eq #reflex #h #v |
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[1939] | 112 | elim v try (normalize #m #irrelevant @I) |
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| 113 | #m' #hd #tl #inductive_hypothesis #m #q |
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| 114 | change with (bool_to_Prop (andb ??)) |
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| 115 | cut ((mem A eq (o + (m + S m')) (h@@q@@hd:::tl) hd) = true) |
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[1936] | 116 | [1: |
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[1939] | 117 | @mem_monotonic_wrt_append try assumption |
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| 118 | @mem_monotonic_wrt_append try assumption |
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| 119 | normalize >reflex % |
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[1936] | 120 | |2: |
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[1939] | 121 | #assm >assm |
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| 122 | >vector_cons_append |
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| 123 | change with (bool_to_Prop (subvector_with ??????)) |
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[2032] | 124 | @(dependent_rewrite_vectors … (vector_associative_append … q [[hd]] tl)) |
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[1939] | 125 | try @associative_plus |
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| 126 | @inductive_hypothesis |
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[1936] | 127 | ] |
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[877] | 128 | qed. |
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| 129 | |
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| 130 | lemma vector_cons_empty: |
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| 131 | ∀A: Type[0]. |
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| 132 | ∀n: nat. |
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| 133 | ∀v: Vector A n. |
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| 134 | [[ ]] @@ v = v. |
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[1936] | 135 | #A #n #v |
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| 136 | cases v try % |
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| 137 | #n' #hd #tl % |
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[877] | 138 | qed. |
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| 139 | |
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| 140 | corollary subvector_hd_tl: |
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| 141 | ∀A: Type[0]. |
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| 142 | ∀o: nat. |
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| 143 | ∀eq: A → A → bool. |
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| 144 | ∀refl: ∀a. eq a a = true. |
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| 145 | ∀h: A. |
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| 146 | ∀v: Vector A o. |
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| 147 | bool_to_Prop (subvector_with A ? ? eq v (h ::: v)). |
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[1936] | 148 | #A #o #eq #reflex #h #v |
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| 149 | >(vector_cons_append … h v) |
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| 150 | <(vector_cons_empty … ([[h]] @@ v)) |
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| 151 | @(subvector_multiple_append … eq reflex [[ ]] v ? [[h]]) |
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[877] | 152 | qed. |
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| 153 | |
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| 154 | lemma is_in_monotonic_wrt_append: |
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| 155 | ∀m, n: nat. |
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| 156 | ∀p: Vector addressing_mode_tag m. |
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| 157 | ∀q: Vector addressing_mode_tag n. |
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| 158 | ∀to_search: addressing_mode. |
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| 159 | bool_to_Prop (is_in ? p to_search) → bool_to_Prop (is_in ? (q @@ p) to_search). |
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[1936] | 160 | #m #n #p #q #to_search #assm |
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| 161 | elim q try assumption |
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| 162 | #n' #hd #tl #inductive_hypothesis |
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| 163 | normalize |
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| 164 | cases (is_a ??) try @I |
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| 165 | >inductive_hypothesis @I |
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[877] | 166 | qed. |
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| 167 | |
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| 168 | corollary is_in_hd_tl: |
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| 169 | ∀to_search: addressing_mode. |
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| 170 | ∀hd: addressing_mode_tag. |
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| 171 | ∀n: nat. |
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| 172 | ∀v: Vector addressing_mode_tag n. |
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| 173 | bool_to_Prop (is_in ? v to_search) → bool_to_Prop (is_in ? (hd:::v) to_search). |
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[1936] | 174 | #to_search #hd #n #v |
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| 175 | elim v |
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| 176 | [1: |
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| 177 | #absurd |
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| 178 | normalize in absurd; cases absurd |
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| 179 | |2: |
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| 180 | #n' #hd' #tl #inductive_hypothesis #assm |
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| 181 | >vector_cons_append >(vector_cons_append … hd' tl) |
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| 182 | @(is_in_monotonic_wrt_append … ([[hd']]@@tl) [[hd]] to_search) |
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| 183 | assumption |
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[877] | 184 | ] |
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| 185 | qed. |
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| 186 | |
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| 187 | let rec list_addressing_mode_tags_elim |
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[1936] | 188 | (n: nat) (l: Vector addressing_mode_tag (S n)) |
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| 189 | on l: (l → bool) → bool ≝ |
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| 190 | match l return λx. |
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| 191 | match x with |
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| 192 | [ O ⇒ λl: Vector … O. bool |
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| 193 | | S x' ⇒ λl: Vector addressing_mode_tag (S x'). (l → bool) → bool |
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| 194 | ] with |
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[877] | 195 | [ VEmpty ⇒ true |
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| 196 | | VCons len hd tl ⇒ λP. |
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| 197 | let process_hd ≝ |
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| 198 | match hd return λhd. ∀P: hd:::tl → bool. bool with |
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| 199 | [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x)) |
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| 200 | | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x)) |
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| 201 | | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x)) |
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| 202 | | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x)) |
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| 203 | | acc_a ⇒ λP.P ACC_A |
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| 204 | | acc_b ⇒ λP.P ACC_B |
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| 205 | | dptr ⇒ λP.P DPTR |
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| 206 | | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x)) |
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| 207 | | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x)) |
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| 208 | | acc_dptr ⇒ λP.P ACC_DPTR |
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| 209 | | acc_pc ⇒ λP.P ACC_PC |
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| 210 | | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR |
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| 211 | | indirect_dptr ⇒ λP.P INDIRECT_DPTR |
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| 212 | | carry ⇒ λP.P CARRY |
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| 213 | | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x)) |
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| 214 | | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x)) |
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| 215 | | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x)) |
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| 216 | | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x)) |
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| 217 | | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x)) |
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| 218 | ] |
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| 219 | in |
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| 220 | andb (process_hd P) |
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| 221 | (match len return λx. x = len → bool with |
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| 222 | [ O ⇒ λprf. true |
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| 223 | | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len)) |
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| 224 | ]. |
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| 225 | try % |
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[1936] | 226 | [2: |
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| 227 | cases (sym_eq ??? prf); assumption |
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| 228 | |1: |
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| 229 | generalize in match H; generalize in match tl; |
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| 230 | destruct #tl |
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[1616] | 231 | normalize in ⊢ (∀_: %. ?); |
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[1936] | 232 | #H |
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| 233 | whd normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]); |
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| 234 | cases (is_a hd (subaddressing_modeel y tl H)) |
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| 235 | whd try @I normalize nodelta // |
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| 236 | ] |
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[877] | 237 | qed. |
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| 238 | |
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| 239 | definition product_elim ≝ |
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| 240 | λm, n: nat. |
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| 241 | λv: Vector addressing_mode_tag (S m). |
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| 242 | λq: Vector addressing_mode_tag (S n). |
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| 243 | λP: (v × q) → bool. |
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| 244 | list_addressing_mode_tags_elim ? v (λx. list_addressing_mode_tags_elim ? q (λy. P 〈x, y〉)). |
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| 245 | |
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| 246 | definition union_elim ≝ |
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| 247 | λA, B: Type[0]. |
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| 248 | λelimA: (A → bool) → bool. |
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| 249 | λelimB: (B → bool) → bool. |
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| 250 | λelimU: A ⊎ B → bool. |
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| 251 | elimA (λa. elimB (λb. elimU (inl ? ? a) ∧ elimU (inr ? ? b))). |
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[892] | 252 | |
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| 253 | (* |
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[877] | 254 | definition preinstruction_elim: ∀P: preinstruction [[ relative ]] → bool. bool ≝ |
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| 255 | λP. |
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| 256 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADD ? ACC_A addr)) ∧ |
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| 257 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADDC ? ACC_A addr)) ∧ |
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| 258 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (SUBB ? ACC_A addr)) ∧ |
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| 259 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ; dptr ]] (λaddr. P (INC ? addr)) ∧ |
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| 260 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (DEC ? addr)) ∧ |
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| 261 | list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (MUL ? ACC_A addr)) ∧ |
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| 262 | list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (DIV ? ACC_A addr)) ∧ |
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| 263 | list_addressing_mode_tags_elim ? [[ registr ; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧ |
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| 264 | list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CLR ? addr)) ∧ |
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| 265 | list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CPL ? addr)) ∧ |
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| 266 | P (DA ? ACC_A) ∧ |
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| 267 | bitvector_elim 8 (λr. P (JC ? (RELATIVE r))) ∧ |
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| 268 | bitvector_elim 8 (λr. P (JNC ? (RELATIVE r))) ∧ |
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| 269 | bitvector_elim 8 (λr. P (JZ ? (RELATIVE r))) ∧ |
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| 270 | bitvector_elim 8 (λr. P (JNZ ? (RELATIVE r))) ∧ |
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| 271 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JB ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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| 272 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JNB ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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| 273 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JBC ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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| 274 | list_addressing_mode_tags_elim ? [[ registr; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧ |
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| 275 | P (RL ? ACC_A) ∧ |
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| 276 | P (RLC ? ACC_A) ∧ |
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| 277 | P (RR ? ACC_A) ∧ |
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| 278 | P (RRC ? ACC_A) ∧ |
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| 279 | P (SWAP ? ACC_A) ∧ |
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| 280 | P (RET ?) ∧ |
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| 281 | P (RETI ?) ∧ |
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| 282 | P (NOP ?) ∧ |
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| 283 | bit_elim (λb. P (XCHD ? ACC_A (INDIRECT b))) ∧ |
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| 284 | list_addressing_mode_tags_elim ? [[ carry; bit_addr ]] (λaddr. P (SETB ? addr)) ∧ |
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| 285 | bitvector_elim 8 (λaddr. P (PUSH ? (DIRECT addr))) ∧ |
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| 286 | bitvector_elim 8 (λaddr. P (POP ? (DIRECT addr))) ∧ |
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| 287 | union_elim ? ? (product_elim ? ? [[ acc_a ]] [[ direct; data ]]) |
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| 288 | (product_elim ? ? [[ registr; indirect ]] [[ data ]]) |
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| 289 | (λd. bitvector_elim 8 (λb. P (CJNE ? d (RELATIVE b)))) ∧ |
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| 290 | list_addressing_mode_tags_elim ? [[ registr; direct; indirect ]] (λaddr. P (XCH ? ACC_A addr)) ∧ |
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| 291 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ data ; registr ; direct ; indirect ]]) |
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| 292 | (product_elim ? ? [[direct]] [[ acc_a ; data ]]) |
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| 293 | (λd. P (XRL ? d)) ∧ |
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| 294 | union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]]) |
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| 295 | (product_elim ? ? [[direct]] [[ acc_a ; data ]])) |
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| 296 | (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]]) |
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| 297 | (λd. P (ANL ? d)) ∧ |
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| 298 | union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; data ; direct ; indirect ]]) |
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| 299 | (product_elim ? ? [[direct]] [[ acc_a ; data ]])) |
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| 300 | (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]]) |
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| 301 | (λd. P (ORL ? d)) ∧ |
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| 302 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ ext_indirect ; ext_indirect_dptr ]]) |
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| 303 | (product_elim ? ? [[ ext_indirect ; ext_indirect_dptr ]] [[acc_a]]) |
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| 304 | (λd. P (MOVX ? d)) ∧ |
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| 305 | union_elim ? ? ( |
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| 306 | union_elim ? ? ( |
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| 307 | union_elim ? ? ( |
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| 308 | union_elim ? ? ( |
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| 309 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]]) |
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| 310 | (product_elim ? ? [[ registr ; indirect ]] [[ acc_a ; direct ; data ]])) |
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| 311 | (product_elim ? ? [[direct]] [[ acc_a ; registr ; direct ; indirect ; data ]])) |
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| 312 | (product_elim ? ? [[dptr]] [[data16]])) |
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| 313 | (product_elim ? ? [[carry]] [[bit_addr]])) |
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| 314 | (product_elim ? ? [[bit_addr]] [[carry]]) |
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| 315 | (λd. P (MOV ? d)). |
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| 316 | % |
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| 317 | qed. |
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| 318 | |
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| 319 | definition instruction_elim: ∀P: instruction → bool. bool ≝ |
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| 320 | λP. (* |
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| 321 | bitvector_elim 11 (λx. P (ACALL (ADDR11 x))) ∧ |
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| 322 | bitvector_elim 16 (λx. P (LCALL (ADDR16 x))) ∧ |
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| 323 | bitvector_elim 11 (λx. P (AJMP (ADDR11 x))) ∧ |
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| 324 | bitvector_elim 16 (λx. P (LJMP (ADDR16 x))) ∧ *) |
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| 325 | bitvector_elim 8 (λx. P (SJMP (RELATIVE x))). (* ∧ |
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| 326 | P (JMP INDIRECT_DPTR) ∧ |
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| 327 | list_addressing_mode_tags_elim ? [[ acc_dptr; acc_pc ]] (λa. P (MOVC ACC_A a)) ∧ |
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| 328 | preinstruction_elim (λp. P (RealInstruction p)). *) |
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| 329 | % |
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| 330 | qed. |
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| 331 | |
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| 332 | |
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| 333 | axiom instruction_elim_complete: |
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| 334 | ∀P. instruction_elim P = true → ∀i. P i = true. |
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[892] | 335 | *) |
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[883] | 336 | (*definition eq_instruction ≝ |
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[877] | 337 | λi, j: instruction. |
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[883] | 338 | true.*) |
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[885] | 339 | |
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[991] | 340 | definition eq_addressing_mode: addressing_mode → addressing_mode → bool ≝ |
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| 341 | λa, b: addressing_mode. |
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| 342 | match a with |
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| 343 | [ DIRECT d ⇒ |
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| 344 | match b with |
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| 345 | [ DIRECT e ⇒ eq_bv ? d e |
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| 346 | | _ ⇒ false |
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| 347 | ] |
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| 348 | | INDIRECT b' ⇒ |
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| 349 | match b with |
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| 350 | [ INDIRECT e ⇒ eq_b b' e |
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| 351 | | _ ⇒ false |
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| 352 | ] |
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| 353 | | EXT_INDIRECT b' ⇒ |
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| 354 | match b with |
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| 355 | [ EXT_INDIRECT e ⇒ eq_b b' e |
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| 356 | | _ ⇒ false |
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| 357 | ] |
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| 358 | | REGISTER bv ⇒ |
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| 359 | match b with |
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| 360 | [ REGISTER bv' ⇒ eq_bv ? bv bv' |
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| 361 | | _ ⇒ false |
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| 362 | ] |
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| 363 | | ACC_A ⇒ match b with [ ACC_A ⇒ true | _ ⇒ false ] |
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| 364 | | ACC_B ⇒ match b with [ ACC_B ⇒ true | _ ⇒ false ] |
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| 365 | | DPTR ⇒ match b with [ DPTR ⇒ true | _ ⇒ false ] |
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| 366 | | DATA b' ⇒ |
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| 367 | match b with |
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| 368 | [ DATA e ⇒ eq_bv ? b' e |
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| 369 | | _ ⇒ false |
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| 370 | ] |
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| 371 | | DATA16 w ⇒ |
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| 372 | match b with |
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| 373 | [ DATA16 e ⇒ eq_bv ? w e |
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| 374 | | _ ⇒ false |
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| 375 | ] |
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| 376 | | ACC_DPTR ⇒ match b with [ ACC_DPTR ⇒ true | _ ⇒ false ] |
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| 377 | | ACC_PC ⇒ match b with [ ACC_PC ⇒ true | _ ⇒ false ] |
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| 378 | | EXT_INDIRECT_DPTR ⇒ match b with [ EXT_INDIRECT_DPTR ⇒ true | _ ⇒ false ] |
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| 379 | | INDIRECT_DPTR ⇒ match b with [ INDIRECT_DPTR ⇒ true | _ ⇒ false ] |
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| 380 | | CARRY ⇒ match b with [ CARRY ⇒ true | _ ⇒ false ] |
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| 381 | | BIT_ADDR b' ⇒ |
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| 382 | match b with |
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| 383 | [ BIT_ADDR e ⇒ eq_bv ? b' e |
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| 384 | | _ ⇒ false |
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| 385 | ] |
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| 386 | | N_BIT_ADDR b' ⇒ |
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| 387 | match b with |
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| 388 | [ N_BIT_ADDR e ⇒ eq_bv ? b' e |
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| 389 | | _ ⇒ false |
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| 390 | ] |
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| 391 | | RELATIVE n ⇒ |
---|
| 392 | match b with |
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| 393 | [ RELATIVE e ⇒ eq_bv ? n e |
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| 394 | | _ ⇒ false |
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| 395 | ] |
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| 396 | | ADDR11 w ⇒ |
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| 397 | match b with |
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| 398 | [ ADDR11 e ⇒ eq_bv ? w e |
---|
| 399 | | _ ⇒ false |
---|
| 400 | ] |
---|
| 401 | | ADDR16 w ⇒ |
---|
| 402 | match b with |
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| 403 | [ ADDR16 e ⇒ eq_bv ? w e |
---|
| 404 | | _ ⇒ false |
---|
| 405 | ] |
---|
| 406 | ]. |
---|
| 407 | |
---|
| 408 | lemma eq_bv_refl: |
---|
| 409 | ∀n, b. |
---|
| 410 | eq_bv n b b = true. |
---|
| 411 | #n #b cases b // |
---|
| 412 | qed. |
---|
| 413 | |
---|
| 414 | lemma eq_b_refl: |
---|
| 415 | ∀b. |
---|
| 416 | eq_b b b = true. |
---|
| 417 | #b cases b // |
---|
| 418 | qed. |
---|
| 419 | |
---|
| 420 | lemma eq_addressing_mode_refl: |
---|
| 421 | ∀a. eq_addressing_mode a a = true. |
---|
[1936] | 422 | #a |
---|
| 423 | cases a try #arg1 try #arg2 |
---|
| 424 | try @eq_bv_refl try @eq_b_refl |
---|
[991] | 425 | try normalize % |
---|
| 426 | qed. |
---|
| 427 | |
---|
[1936] | 428 | definition eq_sum: |
---|
| 429 | ∀A, B. (A → A → bool) → (B → B → bool) → (A ⊎ B) → (A ⊎ B) → bool ≝ |
---|
[991] | 430 | λlt, rt, leq, req, left, right. |
---|
| 431 | match left with |
---|
| 432 | [ inl l ⇒ |
---|
| 433 | match right with |
---|
| 434 | [ inl l' ⇒ leq l l' |
---|
| 435 | | _ ⇒ false |
---|
| 436 | ] |
---|
| 437 | | inr r ⇒ |
---|
| 438 | match right with |
---|
| 439 | [ inr r' ⇒ req r r' |
---|
| 440 | | _ ⇒ false |
---|
| 441 | ] |
---|
| 442 | ]. |
---|
| 443 | |
---|
| 444 | definition eq_prod: ∀A, B. (A → A → bool) → (B → B → bool) → (A × B) → (A × B) → bool ≝ |
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| 445 | λlt, rt, leq, req, left, right. |
---|
| 446 | let 〈l, r〉 ≝ left in |
---|
| 447 | let 〈l', r'〉 ≝ right in |
---|
| 448 | leq l l' ∧ req r r'. |
---|
| 449 | |
---|
| 450 | definition eq_preinstruction: preinstruction [[relative]] → preinstruction [[relative]] → bool ≝ |
---|
| 451 | λi, j. |
---|
| 452 | match i with |
---|
| 453 | [ ADD arg1 arg2 ⇒ |
---|
| 454 | match j with |
---|
| 455 | [ ADD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
| 456 | | _ ⇒ false |
---|
| 457 | ] |
---|
| 458 | | ADDC arg1 arg2 ⇒ |
---|
| 459 | match j with |
---|
| 460 | [ ADDC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
| 461 | | _ ⇒ false |
---|
| 462 | ] |
---|
| 463 | | SUBB arg1 arg2 ⇒ |
---|
| 464 | match j with |
---|
| 465 | [ SUBB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
| 466 | | _ ⇒ false |
---|
| 467 | ] |
---|
| 468 | | INC arg ⇒ |
---|
| 469 | match j with |
---|
| 470 | [ INC arg' ⇒ eq_addressing_mode arg arg' |
---|
| 471 | | _ ⇒ false |
---|
| 472 | ] |
---|
| 473 | | DEC arg ⇒ |
---|
| 474 | match j with |
---|
| 475 | [ DEC arg' ⇒ eq_addressing_mode arg arg' |
---|
| 476 | | _ ⇒ false |
---|
| 477 | ] |
---|
| 478 | | MUL arg1 arg2 ⇒ |
---|
| 479 | match j with |
---|
| 480 | [ MUL arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
| 481 | | _ ⇒ false |
---|
| 482 | ] |
---|
| 483 | | DIV arg1 arg2 ⇒ |
---|
| 484 | match j with |
---|
| 485 | [ DIV arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
| 486 | | _ ⇒ false |
---|
| 487 | ] |
---|
| 488 | | DA arg ⇒ |
---|
| 489 | match j with |
---|
| 490 | [ DA arg' ⇒ eq_addressing_mode arg arg' |
---|
| 491 | | _ ⇒ false |
---|
| 492 | ] |
---|
| 493 | | JC arg ⇒ |
---|
| 494 | match j with |
---|
| 495 | [ JC arg' ⇒ eq_addressing_mode arg arg' |
---|
| 496 | | _ ⇒ false |
---|
| 497 | ] |
---|
| 498 | | JNC arg ⇒ |
---|
| 499 | match j with |
---|
| 500 | [ JNC arg' ⇒ eq_addressing_mode arg arg' |
---|
| 501 | | _ ⇒ false |
---|
| 502 | ] |
---|
| 503 | | JB arg1 arg2 ⇒ |
---|
| 504 | match j with |
---|
| 505 | [ JB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
| 506 | | _ ⇒ false |
---|
| 507 | ] |
---|
| 508 | | JNB arg1 arg2 ⇒ |
---|
| 509 | match j with |
---|
| 510 | [ JNB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
| 511 | | _ ⇒ false |
---|
| 512 | ] |
---|
| 513 | | JBC arg1 arg2 ⇒ |
---|
| 514 | match j with |
---|
| 515 | [ JBC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
| 516 | | _ ⇒ false |
---|
| 517 | ] |
---|
| 518 | | JZ arg ⇒ |
---|
| 519 | match j with |
---|
| 520 | [ JZ arg' ⇒ eq_addressing_mode arg arg' |
---|
| 521 | | _ ⇒ false |
---|
| 522 | ] |
---|
| 523 | | JNZ arg ⇒ |
---|
| 524 | match j with |
---|
| 525 | [ JNZ arg' ⇒ eq_addressing_mode arg arg' |
---|
| 526 | | _ ⇒ false |
---|
| 527 | ] |
---|
| 528 | | CJNE arg1 arg2 ⇒ |
---|
| 529 | match j with |
---|
| 530 | [ CJNE arg1' arg2' ⇒ |
---|
| 531 | let prod_eq_left ≝ eq_prod [[acc_a]] [[direct; data]] eq_addressing_mode eq_addressing_mode in |
---|
| 532 | let prod_eq_right ≝ eq_prod [[registr; indirect]] [[data]] eq_addressing_mode eq_addressing_mode in |
---|
| 533 | let arg1_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
---|
| 534 | arg1_eq arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
| 535 | | _ ⇒ false |
---|
| 536 | ] |
---|
| 537 | | DJNZ arg1 arg2 ⇒ |
---|
| 538 | match j with |
---|
| 539 | [ DJNZ arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
| 540 | | _ ⇒ false |
---|
| 541 | ] |
---|
| 542 | | CLR arg ⇒ |
---|
| 543 | match j with |
---|
| 544 | [ CLR arg' ⇒ eq_addressing_mode arg arg' |
---|
| 545 | | _ ⇒ false |
---|
| 546 | ] |
---|
| 547 | | CPL arg ⇒ |
---|
| 548 | match j with |
---|
| 549 | [ CPL arg' ⇒ eq_addressing_mode arg arg' |
---|
| 550 | | _ ⇒ false |
---|
| 551 | ] |
---|
| 552 | | RL arg ⇒ |
---|
| 553 | match j with |
---|
| 554 | [ RL arg' ⇒ eq_addressing_mode arg arg' |
---|
| 555 | | _ ⇒ false |
---|
| 556 | ] |
---|
| 557 | | RLC arg ⇒ |
---|
| 558 | match j with |
---|
| 559 | [ RLC arg' ⇒ eq_addressing_mode arg arg' |
---|
| 560 | | _ ⇒ false |
---|
| 561 | ] |
---|
| 562 | | RR arg ⇒ |
---|
| 563 | match j with |
---|
| 564 | [ RR arg' ⇒ eq_addressing_mode arg arg' |
---|
| 565 | | _ ⇒ false |
---|
| 566 | ] |
---|
| 567 | | RRC arg ⇒ |
---|
| 568 | match j with |
---|
| 569 | [ RRC arg' ⇒ eq_addressing_mode arg arg' |
---|
| 570 | | _ ⇒ false |
---|
| 571 | ] |
---|
| 572 | | SWAP arg ⇒ |
---|
| 573 | match j with |
---|
| 574 | [ SWAP arg' ⇒ eq_addressing_mode arg arg' |
---|
| 575 | | _ ⇒ false |
---|
| 576 | ] |
---|
| 577 | | SETB arg ⇒ |
---|
| 578 | match j with |
---|
| 579 | [ SETB arg' ⇒ eq_addressing_mode arg arg' |
---|
| 580 | | _ ⇒ false |
---|
| 581 | ] |
---|
| 582 | | PUSH arg ⇒ |
---|
| 583 | match j with |
---|
| 584 | [ PUSH arg' ⇒ eq_addressing_mode arg arg' |
---|
| 585 | | _ ⇒ false |
---|
| 586 | ] |
---|
| 587 | | POP arg ⇒ |
---|
| 588 | match j with |
---|
| 589 | [ POP arg' ⇒ eq_addressing_mode arg arg' |
---|
| 590 | | _ ⇒ false |
---|
| 591 | ] |
---|
| 592 | | XCH arg1 arg2 ⇒ |
---|
| 593 | match j with |
---|
| 594 | [ XCH arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
| 595 | | _ ⇒ false |
---|
| 596 | ] |
---|
| 597 | | XCHD arg1 arg2 ⇒ |
---|
| 598 | match j with |
---|
| 599 | [ XCHD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
| 600 | | _ ⇒ false |
---|
| 601 | ] |
---|
| 602 | | RET ⇒ match j with [ RET ⇒ true | _ ⇒ false ] |
---|
| 603 | | RETI ⇒ match j with [ RETI ⇒ true | _ ⇒ false ] |
---|
| 604 | | NOP ⇒ match j with [ NOP ⇒ true | _ ⇒ false ] |
---|
| 605 | | MOVX arg ⇒ |
---|
| 606 | match j with |
---|
| 607 | [ MOVX arg' ⇒ |
---|
| 608 | let prod_eq_left ≝ eq_prod [[acc_a]] [[ext_indirect; ext_indirect_dptr]] eq_addressing_mode eq_addressing_mode in |
---|
| 609 | let prod_eq_right ≝ eq_prod [[ext_indirect; ext_indirect_dptr]] [[acc_a]] eq_addressing_mode eq_addressing_mode in |
---|
| 610 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
---|
| 611 | sum_eq arg arg' |
---|
| 612 | | _ ⇒ false |
---|
| 613 | ] |
---|
| 614 | | XRL arg ⇒ |
---|
| 615 | match j with |
---|
| 616 | [ XRL arg' ⇒ |
---|
| 617 | let prod_eq_left ≝ eq_prod [[acc_a]] [[ data ; registr ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in |
---|
| 618 | let prod_eq_right ≝ eq_prod [[direct]] [[ acc_a ; data ]] eq_addressing_mode eq_addressing_mode in |
---|
| 619 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
---|
| 620 | sum_eq arg arg' |
---|
| 621 | | _ ⇒ false |
---|
| 622 | ] |
---|
| 623 | | ORL arg ⇒ |
---|
| 624 | match j with |
---|
| 625 | [ ORL arg' ⇒ |
---|
| 626 | let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; data ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in |
---|
| 627 | let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in |
---|
| 628 | let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in |
---|
| 629 | let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in |
---|
| 630 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
---|
| 631 | sum_eq arg arg' |
---|
| 632 | | _ ⇒ false |
---|
| 633 | ] |
---|
| 634 | | ANL arg ⇒ |
---|
| 635 | match j with |
---|
| 636 | [ ANL arg' ⇒ |
---|
| 637 | let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; direct ; indirect ; data ]] eq_addressing_mode eq_addressing_mode in |
---|
| 638 | let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in |
---|
| 639 | let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in |
---|
| 640 | let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in |
---|
| 641 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
---|
| 642 | sum_eq arg arg' |
---|
| 643 | | _ ⇒ false |
---|
| 644 | ] |
---|
| 645 | | MOV arg ⇒ |
---|
| 646 | match j with |
---|
| 647 | [ MOV arg' ⇒ |
---|
| 648 | let prod_eq_6 ≝ eq_prod [[acc_a]] [[registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in |
---|
| 649 | let prod_eq_5 ≝ eq_prod [[registr; indirect]] [[acc_a; direct; data]] eq_addressing_mode eq_addressing_mode in |
---|
| 650 | let prod_eq_4 ≝ eq_prod [[direct]] [[acc_a; registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in |
---|
| 651 | let prod_eq_3 ≝ eq_prod [[dptr]] [[data16]] eq_addressing_mode eq_addressing_mode in |
---|
| 652 | let prod_eq_2 ≝ eq_prod [[carry]] [[bit_addr]] eq_addressing_mode eq_addressing_mode in |
---|
| 653 | let prod_eq_1 ≝ eq_prod [[bit_addr]] [[carry]] eq_addressing_mode eq_addressing_mode in |
---|
| 654 | let sum_eq_1 ≝ eq_sum ? ? prod_eq_6 prod_eq_5 in |
---|
| 655 | let sum_eq_2 ≝ eq_sum ? ? sum_eq_1 prod_eq_4 in |
---|
| 656 | let sum_eq_3 ≝ eq_sum ? ? sum_eq_2 prod_eq_3 in |
---|
| 657 | let sum_eq_4 ≝ eq_sum ? ? sum_eq_3 prod_eq_2 in |
---|
| 658 | let sum_eq_5 ≝ eq_sum ? ? sum_eq_4 prod_eq_1 in |
---|
| 659 | sum_eq_5 arg arg' |
---|
| 660 | | _ ⇒ false |
---|
| 661 | ] |
---|
| 662 | ]. |
---|
| 663 | |
---|
| 664 | lemma eq_sum_refl: |
---|
| 665 | ∀A, B: Type[0]. |
---|
| 666 | ∀leq, req. |
---|
| 667 | ∀s. |
---|
| 668 | ∀leq_refl: (∀t. leq t t = true). |
---|
| 669 | ∀req_refl: (∀u. req u u = true). |
---|
| 670 | eq_sum A B leq req s s = true. |
---|
| 671 | #A #B #leq #req #s #leq_refl #req_refl |
---|
[1939] | 672 | cases s assumption |
---|
[991] | 673 | qed. |
---|
| 674 | |
---|
| 675 | lemma eq_prod_refl: |
---|
| 676 | ∀A, B: Type[0]. |
---|
| 677 | ∀leq, req. |
---|
| 678 | ∀s. |
---|
| 679 | ∀leq_refl: (∀t. leq t t = true). |
---|
| 680 | ∀req_refl: (∀u. req u u = true). |
---|
| 681 | eq_prod A B leq req s s = true. |
---|
| 682 | #A #B #leq #req #s #leq_refl #req_refl |
---|
[1936] | 683 | cases s |
---|
| 684 | whd in ⊢ (? → ? → ??%?); |
---|
| 685 | #l #r |
---|
| 686 | >leq_refl @req_refl |
---|
[991] | 687 | qed. |
---|
| 688 | |
---|
| 689 | lemma eq_preinstruction_refl: |
---|
| 690 | ∀i. |
---|
| 691 | eq_preinstruction i i = true. |
---|
| 692 | #i cases i try #arg1 try #arg2 |
---|
| 693 | try @eq_addressing_mode_refl |
---|
| 694 | [1,2,3,4,5,6,7,8,10,16,17,18,19,20: |
---|
[1939] | 695 | whd in ⊢ (??%?); try % |
---|
[991] | 696 | >eq_addressing_mode_refl |
---|
| 697 | >eq_addressing_mode_refl % |
---|
| 698 | |13,15: |
---|
[1616] | 699 | whd in ⊢ (??%?); |
---|
[991] | 700 | cases arg1 |
---|
| 701 | [*: |
---|
| 702 | #arg1_left normalize nodelta |
---|
| 703 | >eq_prod_refl [*: try % #argr @eq_addressing_mode_refl] |
---|
| 704 | ] |
---|
| 705 | |11,12: |
---|
[1616] | 706 | whd in ⊢ (??%?); |
---|
[991] | 707 | cases arg1 |
---|
| 708 | [1: |
---|
| 709 | #arg1_left normalize nodelta |
---|
| 710 | >(eq_sum_refl …) |
---|
| 711 | [1: % | 2,3: #arg @eq_prod_refl ] |
---|
| 712 | @eq_addressing_mode_refl |
---|
| 713 | |2: |
---|
| 714 | #arg1_left normalize nodelta |
---|
| 715 | @eq_prod_refl [*: @eq_addressing_mode_refl ] |
---|
| 716 | |3: |
---|
| 717 | #arg1_left normalize nodelta |
---|
[1939] | 718 | >(eq_sum_refl …) |
---|
| 719 | [1: |
---|
| 720 | % |
---|
| 721 | |2,3: |
---|
| 722 | #arg @eq_prod_refl #arg @eq_addressing_mode_refl |
---|
| 723 | ] |
---|
[991] | 724 | |4: |
---|
| 725 | #arg1_left normalize nodelta |
---|
| 726 | @eq_prod_refl [*: #arg @eq_addressing_mode_refl ] |
---|
| 727 | ] |
---|
| 728 | |14: |
---|
[1616] | 729 | whd in ⊢ (??%?); |
---|
[991] | 730 | cases arg1 |
---|
[1936] | 731 | [1: |
---|
| 732 | #arg1_left normalize nodelta |
---|
[991] | 733 | @eq_sum_refl |
---|
[1936] | 734 | [1: |
---|
| 735 | #arg @eq_sum_refl |
---|
| 736 | [1: |
---|
| 737 | #arg @eq_sum_refl |
---|
| 738 | [1: |
---|
| 739 | #arg @eq_sum_refl |
---|
| 740 | [1: |
---|
| 741 | #arg @eq_prod_refl |
---|
| 742 | [*: |
---|
| 743 | @eq_addressing_mode_refl |
---|
| 744 | ] |
---|
| 745 | |2: |
---|
| 746 | #arg @eq_prod_refl |
---|
| 747 | [*: |
---|
| 748 | #arg @eq_addressing_mode_refl |
---|
| 749 | ] |
---|
[991] | 750 | ] |
---|
[1936] | 751 | |2: |
---|
| 752 | #arg @eq_prod_refl |
---|
| 753 | [*: |
---|
| 754 | #arg @eq_addressing_mode_refl |
---|
| 755 | ] |
---|
[991] | 756 | ] |
---|
[1936] | 757 | |2: |
---|
| 758 | #arg @eq_prod_refl |
---|
| 759 | [*: |
---|
| 760 | #arg @eq_addressing_mode_refl |
---|
| 761 | ] |
---|
[991] | 762 | ] |
---|
[1936] | 763 | |2: |
---|
| 764 | #arg @eq_prod_refl |
---|
| 765 | [*: |
---|
| 766 | #arg @eq_addressing_mode_refl |
---|
| 767 | ] |
---|
[991] | 768 | ] |
---|
[1936] | 769 | |2: |
---|
| 770 | #arg1_right normalize nodelta |
---|
| 771 | @eq_prod_refl |
---|
| 772 | [*: |
---|
| 773 | #arg @eq_addressing_mode_refl |
---|
| 774 | ] |
---|
[991] | 775 | ] |
---|
| 776 | |*: |
---|
[1616] | 777 | whd in ⊢ (??%?); |
---|
[991] | 778 | cases arg1 |
---|
[1936] | 779 | [*: |
---|
| 780 | #arg1 >eq_sum_refl |
---|
| 781 | [1,4: |
---|
| 782 | @eq_addressing_mode_refl |
---|
| 783 | |2,3,5,6: |
---|
| 784 | #arg @eq_prod_refl |
---|
| 785 | [*: |
---|
| 786 | #arg @eq_addressing_mode_refl |
---|
| 787 | ] |
---|
[991] | 788 | ] |
---|
| 789 | ] |
---|
| 790 | ] |
---|
| 791 | qed. |
---|
| 792 | |
---|
| 793 | definition eq_instruction: instruction → instruction → bool ≝ |
---|
| 794 | λi, j. |
---|
| 795 | match i with |
---|
| 796 | [ ACALL arg ⇒ |
---|
| 797 | match j with |
---|
| 798 | [ ACALL arg' ⇒ eq_addressing_mode arg arg' |
---|
| 799 | | _ ⇒ false |
---|
| 800 | ] |
---|
| 801 | | LCALL arg ⇒ |
---|
| 802 | match j with |
---|
| 803 | [ LCALL arg' ⇒ eq_addressing_mode arg arg' |
---|
| 804 | | _ ⇒ false |
---|
| 805 | ] |
---|
| 806 | | AJMP arg ⇒ |
---|
| 807 | match j with |
---|
| 808 | [ AJMP arg' ⇒ eq_addressing_mode arg arg' |
---|
| 809 | | _ ⇒ false |
---|
| 810 | ] |
---|
| 811 | | LJMP arg ⇒ |
---|
| 812 | match j with |
---|
| 813 | [ LJMP arg' ⇒ eq_addressing_mode arg arg' |
---|
| 814 | | _ ⇒ false |
---|
| 815 | ] |
---|
| 816 | | SJMP arg ⇒ |
---|
| 817 | match j with |
---|
| 818 | [ SJMP arg' ⇒ eq_addressing_mode arg arg' |
---|
| 819 | | _ ⇒ false |
---|
| 820 | ] |
---|
| 821 | | JMP arg ⇒ |
---|
| 822 | match j with |
---|
| 823 | [ JMP arg' ⇒ eq_addressing_mode arg arg' |
---|
| 824 | | _ ⇒ false |
---|
| 825 | ] |
---|
| 826 | | MOVC arg1 arg2 ⇒ |
---|
| 827 | match j with |
---|
| 828 | [ MOVC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
| 829 | | _ ⇒ false |
---|
| 830 | ] |
---|
| 831 | | RealInstruction instr ⇒ |
---|
| 832 | match j with |
---|
| 833 | [ RealInstruction instr' ⇒ eq_preinstruction instr instr' |
---|
| 834 | | _ ⇒ false |
---|
| 835 | ] |
---|
| 836 | ]. |
---|
| 837 | |
---|
| 838 | lemma eq_instruction_refl: |
---|
| 839 | ∀i. eq_instruction i i = true. |
---|
[1941] | 840 | #i cases i [*: #arg1 ] |
---|
| 841 | try @eq_addressing_mode_refl |
---|
| 842 | try @eq_preinstruction_refl |
---|
| 843 | #arg2 whd in ⊢ (??%?); |
---|
| 844 | >eq_addressing_mode_refl >eq_addressing_mode_refl % |
---|
[991] | 845 | qed. |
---|
| 846 | |
---|
[877] | 847 | let rec vect_member |
---|
[1936] | 848 | (A: Type[0]) (n: nat) (eq: A → A → bool) (v: Vector A n) (a: A) |
---|
| 849 | on v: bool ≝ |
---|
[877] | 850 | match v with |
---|
| 851 | [ VEmpty ⇒ false |
---|
| 852 | | VCons len hd tl ⇒ |
---|
[1936] | 853 | eq hd a ∨ (vect_member A ? eq tl a) |
---|
[877] | 854 | ]. |
---|
[892] | 855 | |
---|
[877] | 856 | let rec list_addressing_mode_tags_elim_prop |
---|
| 857 | (n: nat) |
---|
| 858 | (l: Vector addressing_mode_tag (S n)) |
---|
[884] | 859 | on l: |
---|
| 860 | ∀P: l → Prop. |
---|
[877] | 861 | ∀direct_a. ∀indirect_a. ∀ext_indirect_a. ∀register_a. ∀acc_a_a. |
---|
| 862 | ∀acc_b_a. ∀dptr_a. ∀data_a. ∀data16_a. ∀acc_dptr_a. ∀acc_pc_a. |
---|
| 863 | ∀ext_indirect_dptr_a. ∀indirect_dptr_a. ∀carry_a. ∀bit_addr_a. |
---|
| 864 | ∀n_bit_addr_a. ∀relative_a. ∀addr11_a. ∀addr16_a. |
---|
| 865 | ∀x: l. P x ≝ |
---|
| 866 | match l return |
---|
| 867 | λy. |
---|
| 868 | match y with |
---|
| 869 | [ O ⇒ λm: Vector addressing_mode_tag O. ∀prf: 0 = S n. True |
---|
[884] | 870 | | S y' ⇒ λl: Vector addressing_mode_tag (S y'). ∀prf: S y' = S n.∀P:l → Prop. |
---|
[877] | 871 | ∀direct_a: if vect_member … eq_a l direct then ∀x. P (DIRECT x) else True. |
---|
| 872 | ∀indirect_a: if vect_member … eq_a l indirect then ∀x. P (INDIRECT x) else True. |
---|
| 873 | ∀ext_indirect_a: if vect_member … eq_a l ext_indirect then ∀x. P (EXT_INDIRECT x) else True. |
---|
| 874 | ∀register_a: if vect_member … eq_a l registr then ∀x. P (REGISTER x) else True. |
---|
| 875 | ∀acc_a_a: if vect_member … eq_a l acc_a then P (ACC_A) else True. |
---|
| 876 | ∀acc_b_a: if vect_member … eq_a l acc_b then P (ACC_B) else True. |
---|
| 877 | ∀dptr_a: if vect_member … eq_a l dptr then P DPTR else True. |
---|
| 878 | ∀data_a: if vect_member … eq_a l data then ∀x. P (DATA x) else True. |
---|
| 879 | ∀data16_a: if vect_member … eq_a l data16 then ∀x. P (DATA16 x) else True. |
---|
| 880 | ∀acc_dptr_a: if vect_member … eq_a l acc_dptr then P ACC_DPTR else True. |
---|
| 881 | ∀acc_pc_a: if vect_member … eq_a l acc_pc then P ACC_PC else True. |
---|
| 882 | ∀ext_indirect_dptr_a: if vect_member … eq_a l ext_indirect_dptr then P EXT_INDIRECT_DPTR else True. |
---|
| 883 | ∀indirect_dptr_a: if vect_member … eq_a l indirect_dptr then P INDIRECT_DPTR else True. |
---|
| 884 | ∀carry_a: if vect_member … eq_a l carry then P CARRY else True. |
---|
| 885 | ∀bit_addr_a: if vect_member … eq_a l bit_addr then ∀x. P (BIT_ADDR x) else True. |
---|
| 886 | ∀n_bit_addr_a: if vect_member … eq_a l n_bit_addr then ∀x. P (N_BIT_ADDR x) else True. |
---|
| 887 | ∀relative_a: if vect_member … eq_a l relative then ∀x. P (RELATIVE x) else True. |
---|
| 888 | ∀addr11_a: if vect_member … eq_a l addr11 then ∀x. P (ADDR11 x) else True. |
---|
| 889 | ∀addr_16_a: if vect_member … eq_a l addr16 then ∀x. P (ADDR16 x) else True. |
---|
| 890 | ∀x:l. P x |
---|
| 891 | ] with |
---|
| 892 | [ VEmpty ⇒ λAbsurd. ⊥ |
---|
| 893 | | VCons len hd tl ⇒ λProof. ? |
---|
[884] | 894 | ] (refl ? (S n)). cases daemon. qed. (* |
---|
[877] | 895 | [ destruct(Absurd) |
---|
| 896 | | # A1 # A2 # A3 # A4 # A5 # A6 # A7 |
---|
| 897 | # A8 # A9 # A10 # A11 # A12 # A13 # A14 |
---|
| 898 | # A15 # A16 # A17 # A18 # A19 # X |
---|
| 899 | cases X |
---|
[884] | 900 | # SUB cases daemon ] qed. |
---|
[877] | 901 | cases SUB |
---|
| 902 | [ # BYTE |
---|
| 903 | normalize |
---|
| 904 | ]. |
---|
| 905 | |
---|
| 906 | |
---|
| 907 | (* let prepare_hd ≝ |
---|
| 908 | match hd with |
---|
| 909 | [ direct ⇒ λdirect_prf. ? |
---|
| 910 | | indirect ⇒ λindirect_prf. ? |
---|
| 911 | | ext_indirect ⇒ λext_indirect_prf. ? |
---|
| 912 | | registr ⇒ λregistr_prf. ? |
---|
| 913 | | acc_a ⇒ λacc_a_prf. ? |
---|
| 914 | | acc_b ⇒ λacc_b_prf. ? |
---|
| 915 | | dptr ⇒ λdptr_prf. ? |
---|
| 916 | | data ⇒ λdata_prf. ? |
---|
| 917 | | data16 ⇒ λdata16_prf. ? |
---|
| 918 | | acc_dptr ⇒ λacc_dptr_prf. ? |
---|
| 919 | | acc_pc ⇒ λacc_pc_prf. ? |
---|
| 920 | | ext_indirect_dptr ⇒ λext_indirect_prf. ? |
---|
| 921 | | indirect_dptr ⇒ λindirect_prf. ? |
---|
| 922 | | carry ⇒ λcarry_prf. ? |
---|
| 923 | | bit_addr ⇒ λbit_addr_prf. ? |
---|
| 924 | | n_bit_addr ⇒ λn_bit_addr_prf. ? |
---|
| 925 | | relative ⇒ λrelative_prf. ? |
---|
| 926 | | addr11 ⇒ λaddr11_prf. ? |
---|
| 927 | | addr16 ⇒ λaddr16_prf. ? |
---|
| 928 | ] |
---|
| 929 | in ? *) |
---|
| 930 | ]. |
---|
| 931 | [ 1: destruct(absd) |
---|
| 932 | | 2: # A1 # A2 # A3 # A4 # A5 # A6 |
---|
| 933 | # A7 # A8 # A9 # A10 # A11 # A12 |
---|
| 934 | # A13 # A14 # A15 # A16 # A17 # A18 |
---|
| 935 | # A19 * |
---|
| 936 | ]. |
---|
| 937 | |
---|
| 938 | |
---|
| 939 | match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x'). |
---|
| 940 | (l → bool) → bool ] with |
---|
| 941 | [ VEmpty ⇒ true |
---|
| 942 | | VCons len hd tl ⇒ λP. |
---|
| 943 | let process_hd ≝ |
---|
| 944 | match hd return λhd. ∀P: hd:::tl → bool. bool with |
---|
| 945 | [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x)) |
---|
| 946 | | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x)) |
---|
| 947 | | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x)) |
---|
| 948 | | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x)) |
---|
| 949 | | acc_a ⇒ λP.P ACC_A |
---|
| 950 | | acc_b ⇒ λP.P ACC_B |
---|
| 951 | | dptr ⇒ λP.P DPTR |
---|
| 952 | | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x)) |
---|
| 953 | | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x)) |
---|
| 954 | | acc_dptr ⇒ λP.P ACC_DPTR |
---|
| 955 | | acc_pc ⇒ λP.P ACC_PC |
---|
| 956 | | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR |
---|
| 957 | | indirect_dptr ⇒ λP.P INDIRECT_DPTR |
---|
| 958 | | carry ⇒ λP.P CARRY |
---|
| 959 | | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x)) |
---|
| 960 | | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x)) |
---|
| 961 | | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x)) |
---|
| 962 | | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x)) |
---|
| 963 | | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x)) |
---|
| 964 | ] |
---|
| 965 | in |
---|
| 966 | andb (process_hd P) |
---|
| 967 | (match len return λx. x = len → bool with |
---|
| 968 | [ O ⇒ λprf. true |
---|
| 969 | | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len)) |
---|
| 970 | ]. |
---|
| 971 | try % |
---|
| 972 | [ 2: cases (sym_eq ??? prf); @tl |
---|
| 973 | | generalize in match H; generalize in match tl; cases prf; |
---|
| 974 | (* cases prf in tl H; : ??? WAS WORKING BEFORE *) |
---|
| 975 | #tl |
---|
| 976 | normalize in ⊢ (∀_: %. ?) |
---|
| 977 | # H |
---|
| 978 | whd |
---|
| 979 | normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
---|
| 980 | cases (is_a hd (subaddressing_modeel y tl H)) whd // ] |
---|
| 981 | qed. |
---|
[883] | 982 | *) |
---|
[877] | 983 | |
---|
[883] | 984 | definition load_code_memory_aux ≝ |
---|
| 985 | fold_left_i_aux … ( |
---|
| 986 | λi, mem, v. |
---|
| 987 | insert … (bitvector_of_nat … i) v mem) (Stub Byte 16). |
---|
| 988 | |
---|
[2032] | 989 | lemma vsplit_zero: |
---|
[1045] | 990 | ∀A,m. |
---|
| 991 | ∀v: Vector A m. |
---|
[2032] | 992 | 〈[[]], v〉 = vsplit A 0 m v. |
---|
[1045] | 993 | #A #m #v |
---|
[1936] | 994 | cases v try % |
---|
| 995 | #n #hd #tl % |
---|
[1045] | 996 | qed. |
---|
| 997 | |
---|
[1936] | 998 | lemma Vector_O: |
---|
| 999 | ∀A: Type[0]. |
---|
| 1000 | ∀v: Vector A 0. |
---|
| 1001 | v ≃ VEmpty A. |
---|
| 1002 | #A #v |
---|
| 1003 | generalize in match (refl … 0); |
---|
| 1004 | cases v in ⊢ (??%? → ?%%??); // |
---|
| 1005 | #n #hd #tl #absurd |
---|
| 1006 | destruct(absurd) |
---|
[1045] | 1007 | qed. |
---|
| 1008 | |
---|
[1936] | 1009 | lemma Vector_Sn: |
---|
| 1010 | ∀A: Type[0]. |
---|
| 1011 | ∀n: nat. |
---|
| 1012 | ∀v: Vector A (S n). |
---|
| 1013 | ∃hd: A. ∃tl: Vector A n. |
---|
| 1014 | v ≃ VCons A n hd tl. |
---|
| 1015 | #A #n #v |
---|
| 1016 | generalize in match (refl … (S n)); |
---|
| 1017 | cases v in ⊢ (??%? → ??(λ_.??(λ_.?%%??))); |
---|
| 1018 | [1: |
---|
| 1019 | #absurd destruct(absurd) |
---|
| 1020 | |2: |
---|
| 1021 | #m #hd #tl #eq |
---|
| 1022 | <(injective_S … eq) |
---|
| 1023 | %{hd} %{tl} % |
---|
| 1024 | ] |
---|
[1045] | 1025 | qed. |
---|
| 1026 | |
---|
| 1027 | lemma vector_append_zero: |
---|
| 1028 | ∀A,m. |
---|
| 1029 | ∀v: Vector A m. |
---|
| 1030 | ∀q: Vector A 0. |
---|
| 1031 | v = q@@v. |
---|
| 1032 | #A #m #v #q |
---|
| 1033 | >(Vector_O A q) % |
---|
| 1034 | qed. |
---|
| 1035 | |
---|
| 1036 | lemma prod_eq_left: |
---|
| 1037 | ∀A: Type[0]. |
---|
| 1038 | ∀p, q, r: A. |
---|
| 1039 | p = q → 〈p, r〉 = 〈q, r〉. |
---|
| 1040 | #A #p #q #r #hyp |
---|
[1941] | 1041 | destruct % |
---|
[1045] | 1042 | qed. |
---|
| 1043 | |
---|
| 1044 | lemma prod_eq_right: |
---|
| 1045 | ∀A: Type[0]. |
---|
| 1046 | ∀p, q, r: A. |
---|
| 1047 | p = q → 〈r, p〉 = 〈r, q〉. |
---|
| 1048 | #A #p #q #r #hyp |
---|
[1941] | 1049 | destruct % |
---|
[1045] | 1050 | qed. |
---|
| 1051 | |
---|
| 1052 | corollary prod_vector_zero_eq_left: |
---|
| 1053 | ∀A, n. |
---|
| 1054 | ∀q: Vector A O. |
---|
| 1055 | ∀r: Vector A n. |
---|
| 1056 | 〈q, r〉 = 〈[[ ]], r〉. |
---|
| 1057 | #A #n #q #r |
---|
[1616] | 1058 | generalize in match (Vector_O A q …); |
---|
[1941] | 1059 | #hyp destruct % |
---|
[1045] | 1060 | qed. |
---|
| 1061 | |
---|
[1936] | 1062 | lemma tail_head: |
---|
| 1063 | ∀a: Type[0]. |
---|
| 1064 | ∀m, n: nat. |
---|
| 1065 | ∀hd: a. |
---|
| 1066 | ∀l: Vector a m. |
---|
| 1067 | ∀r: Vector a n. |
---|
| 1068 | tail a ? (hd:::(l@@r)) = l@@r. |
---|
| 1069 | #a #m #n #hd #l #r |
---|
| 1070 | cases l try % |
---|
| 1071 | #m' #hd' #tl' % |
---|
| 1072 | qed. |
---|
| 1073 | |
---|
| 1074 | lemma head_head': |
---|
| 1075 | ∀a: Type[0]. |
---|
| 1076 | ∀m: nat. |
---|
| 1077 | ∀hd: a. |
---|
| 1078 | ∀l: Vector a m. |
---|
| 1079 | hd = head' … (hd:::l). |
---|
| 1080 | #a #m #hd #l cases l try % |
---|
| 1081 | #m' #hd' #tl % |
---|
| 1082 | qed. |
---|
| 1083 | |
---|
[2032] | 1084 | lemma vsplit_succ: |
---|
[1936] | 1085 | ∀A: Type[0]. |
---|
| 1086 | ∀m, n: nat. |
---|
[1045] | 1087 | ∀l: Vector A m. |
---|
| 1088 | ∀r: Vector A n. |
---|
| 1089 | ∀v: Vector A (m + n). |
---|
| 1090 | ∀hd: A. |
---|
[2032] | 1091 | v = l@@r → (〈l, r〉 = vsplit A m n v → 〈hd:::l, r〉 = vsplit A (S m) n (hd:::v)). |
---|
[1045] | 1092 | #A #m |
---|
| 1093 | elim m |
---|
[1936] | 1094 | [1: |
---|
| 1095 | #n #l #r #v #hd #eq #hyp |
---|
[1941] | 1096 | destruct >(Vector_O … l) % |
---|
[1936] | 1097 | |2: |
---|
| 1098 | #m' #inductive_hypothesis #n #l #r #v #hd #equal #hyp |
---|
| 1099 | destruct |
---|
| 1100 | cases (Vector_Sn … l) #hd' #tl' |
---|
| 1101 | whd in ⊢ (???%); |
---|
| 1102 | >tail_head |
---|
[2032] | 1103 | <(? : vsplit A (S m') n (l@@r) = vsplit' A (S m') n (l@@r)) |
---|
[1941] | 1104 | try (<hyp <head_head' %) |
---|
| 1105 | elim l normalize // |
---|
[1936] | 1106 | ] |
---|
| 1107 | qed. |
---|
[1045] | 1108 | |
---|
[2032] | 1109 | lemma vsplit_prod: |
---|
[1936] | 1110 | ∀A: Type[0]. |
---|
| 1111 | ∀m, n: nat. |
---|
[1045] | 1112 | ∀p: Vector A (m + n). |
---|
| 1113 | ∀v: Vector A m. |
---|
| 1114 | ∀q: Vector A n. |
---|
[2032] | 1115 | p = v@@q → 〈v, q〉 = vsplit A m n p. |
---|
[1936] | 1116 | #A #m elim m |
---|
| 1117 | [1: |
---|
| 1118 | #n #p #v #q #hyp |
---|
[1045] | 1119 | >hyp <(vector_append_zero A n q v) |
---|
| 1120 | >(prod_vector_zero_eq_left A …) |
---|
[2032] | 1121 | @vsplit_zero |
---|
[1936] | 1122 | |2: |
---|
| 1123 | #r #ih #n #p #v #q #hyp |
---|
[1045] | 1124 | >hyp |
---|
[1936] | 1125 | cases (Vector_Sn A r v) #hd #exists |
---|
| 1126 | cases exists #tl #jmeq |
---|
[2032] | 1127 | >jmeq @vsplit_succ try % |
---|
[1936] | 1128 | @ih % |
---|
[1045] | 1129 | ] |
---|
| 1130 | qed. |
---|
| 1131 | |
---|
[1484] | 1132 | (* |
---|
[2032] | 1133 | lemma vsplit_prod_exists: |
---|
[1045] | 1134 | ∀A, m, n. |
---|
| 1135 | ∀p: Vector A (m + n). |
---|
| 1136 | ∃v: Vector A m. |
---|
| 1137 | ∃q: Vector A n. |
---|
[2032] | 1138 | 〈v, q〉 = vsplit A m n p. |
---|
[1045] | 1139 | #A #m #n #p |
---|
| 1140 | elim m |
---|
| 1141 | @ex_intro |
---|
| 1142 | [1: |
---|
| 1143 | |2: @ex_intro |
---|
| 1144 | [1: |
---|
| 1145 | |2: |
---|
| 1146 | ] |
---|
| 1147 | ] |
---|
[1484] | 1148 | *) |
---|
[1045] | 1149 | |
---|
[2032] | 1150 | definition vsplit_elim: |
---|
[1936] | 1151 | ∀A: Type[0]. |
---|
| 1152 | ∀l, m: nat. |
---|
| 1153 | ∀v: Vector A (l + m). |
---|
| 1154 | ∀P: (Vector A l) × (Vector A m) → Prop. |
---|
| 1155 | (∀vl: Vector A l. |
---|
| 1156 | ∀vm: Vector A m. |
---|
[2032] | 1157 | v = vl@@vm → P 〈vl,vm〉) → P (vsplit A l m v) ≝ |
---|
[1939] | 1158 | λa: Type[0]. |
---|
| 1159 | λl, m: nat. |
---|
| 1160 | λv: Vector a (l + m). |
---|
[1941] | 1161 | λP. ?. |
---|
| 1162 | cases daemon |
---|
| 1163 | qed. |
---|
[901] | 1164 | |
---|
| 1165 | (* |
---|
[883] | 1166 | axiom not_eqvb_S: |
---|
| 1167 | ∀pc. |
---|
| 1168 | (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S pc))). |
---|
| 1169 | |
---|
| 1170 | axiom not_eqvb_SS: |
---|
| 1171 | ∀pc. |
---|
| 1172 | (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S (S pc)))). |
---|
[894] | 1173 | |
---|
[884] | 1174 | axiom bitvector_elim_complete: |
---|
| 1175 | ∀n,P. bitvector_elim n P = true → ∀bv. P bv. |
---|
| 1176 | |
---|
| 1177 | lemma bitvector_elim_complete': |
---|
| 1178 | ∀n,P. bitvector_elim n P = true → ∀bv. P bv = true. |
---|
| 1179 | #n #P #H generalize in match (bitvector_elim_complete … H) #K #bv |
---|
| 1180 | generalize in match (K bv) normalize cases (P bv) normalize // #abs @⊥ // |
---|
| 1181 | qed. |
---|
[894] | 1182 | *) |
---|
[884] | 1183 | |
---|
[893] | 1184 | (* |
---|
[884] | 1185 | lemma andb_elim': |
---|
| 1186 | ∀b1,b2. (b1 = true) → (b2 = true) → (b1 ∧ b2) = true. |
---|
| 1187 | #b1 #b2 #H1 #H2 @andb_elim cases b1 in H1; normalize // |
---|
| 1188 | qed. |
---|
[893] | 1189 | *) |
---|
[884] | 1190 | |
---|
[1936] | 1191 | let rec encoding_check |
---|
| 1192 | (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word) |
---|
| 1193 | (encoding: list Byte) |
---|
| 1194 | on encoding: Prop ≝ |
---|
[890] | 1195 | match encoding with |
---|
| 1196 | [ nil ⇒ final_pc = pc |
---|
| 1197 | | cons hd tl ⇒ |
---|
| 1198 | let 〈new_pc, byte〉 ≝ next code_memory pc in |
---|
[1484] | 1199 | hd = byte ∧ encoding_check code_memory new_pc final_pc tl |
---|
[890] | 1200 | ]. |
---|
| 1201 | |
---|
[1955] | 1202 | axiom add_commutative: |
---|
| 1203 | ∀n: nat. |
---|
| 1204 | ∀l, r: BitVector n. |
---|
| 1205 | add n l r = add n r l. |
---|
| 1206 | |
---|
[1941] | 1207 | axiom add_bitvector_of_nat_Sm: |
---|
| 1208 | ∀n, m: nat. |
---|
| 1209 | add … (bitvector_of_nat … 1) (bitvector_of_nat … m) = |
---|
| 1210 | bitvector_of_nat n (S m). |
---|
| 1211 | |
---|
[1936] | 1212 | lemma encoding_check_append: |
---|
| 1213 | ∀code_memory: BitVectorTrie Byte 16. |
---|
[1941] | 1214 | ∀final_pc: Word. |
---|
[1936] | 1215 | ∀l1: list Byte. |
---|
[1941] | 1216 | ∀pc: Word. |
---|
[1936] | 1217 | ∀l2: list Byte. |
---|
[1941] | 1218 | encoding_check code_memory pc final_pc (l1@l2) → |
---|
| 1219 | let pc_plus_len ≝ add … pc (bitvector_of_nat … (length … l1)) in |
---|
| 1220 | encoding_check code_memory pc pc_plus_len l1 ∧ |
---|
| 1221 | encoding_check code_memory pc_plus_len final_pc l2. |
---|
[1936] | 1222 | #code_memory #final_pc #l1 elim l1 |
---|
| 1223 | [1: |
---|
| 1224 | #pc #l2 |
---|
| 1225 | whd in ⊢ (????% → ?); #H |
---|
[1941] | 1226 | <add_zero |
---|
[1936] | 1227 | whd whd in ⊢ (?%?); /2/ |
---|
| 1228 | |2: |
---|
| 1229 | #hd #tl #IH #pc #l2 * #H1 #H2 |
---|
[1946] | 1230 | (* >add_SO in H2; #H2 *) |
---|
[1941] | 1231 | cases (IH … H2) #E1 #E2 % |
---|
| 1232 | [1: |
---|
| 1233 | % try @H1 |
---|
| 1234 | <(add_bitvector_of_nat_Sm 16 (|tl|)) in E1; |
---|
| 1235 | <add_associative #assm assumption |
---|
| 1236 | |2: |
---|
| 1237 | <add_associative in E2; |
---|
| 1238 | <(add_bitvector_of_nat_Sm 16 (|tl|)) #assm |
---|
| 1239 | assumption |
---|
| 1240 | ] |
---|
[1936] | 1241 | ] |
---|
[901] | 1242 | qed. |
---|
[890] | 1243 | |
---|
[1972] | 1244 | lemma destruct_bug_fix_1: |
---|
[1966] | 1245 | ∀n: nat. |
---|
| 1246 | S n = 0 → False. |
---|
| 1247 | #n #absurd destruct(absurd) |
---|
[1955] | 1248 | qed. |
---|
| 1249 | |
---|
[1972] | 1250 | lemma destruct_bug_fix_2: |
---|
| 1251 | ∀m, n: nat. |
---|
| 1252 | S m = S n → m = n. |
---|
| 1253 | #m #n #refl destruct % |
---|
| 1254 | qed. |
---|
| 1255 | |
---|
[1955] | 1256 | definition bitvector_3_cases: |
---|
[1941] | 1257 | ∀b: BitVector 3. |
---|
[1955] | 1258 | ∃l, c, r: bool. |
---|
[1966] | 1259 | b ≃ [[l; c; r]]. |
---|
[1955] | 1260 | #b |
---|
| 1261 | @(Vector_inv_ind bool 3 b (λn: nat. λv: Vector bool n. ∃l:bool.∃c:bool.∃r:bool. v ≃ [[l; c; r]])) |
---|
| 1262 | [1: |
---|
[1972] | 1263 | #absurd @⊥ -b @(destruct_bug_fix_1 2) |
---|
[1955] | 1264 | >absurd % |
---|
| 1265 | |2: |
---|
[1966] | 1266 | #n #hd #tl #_ #size_refl #hd_tl_refl %{hd} |
---|
| 1267 | cut (n = 2) |
---|
| 1268 | [1: |
---|
[1972] | 1269 | @destruct_bug_fix_2 |
---|
| 1270 | >size_refl % |
---|
[1966] | 1271 | |2: |
---|
[1972] | 1272 | #n_refl >n_refl in tl; #V |
---|
| 1273 | @(Vector_inv_ind bool 2 V (λn: nat. λv: Vector bool n. ∃c:bool. ∃r:bool. hd:::v ≃ [[hd; c; r]])) |
---|
| 1274 | [1: |
---|
| 1275 | #absurd @⊥ -V @(destruct_bug_fix_1 1) |
---|
| 1276 | >absurd % |
---|
| 1277 | |2: |
---|
| 1278 | #n' #hd' #tl' #_ #size_refl' #hd_tl_refl' %{hd'} |
---|
| 1279 | cut (n' = 1) |
---|
| 1280 | [1: |
---|
| 1281 | @destruct_bug_fix_2 >size_refl' % |
---|
| 1282 | |2: |
---|
| 1283 | #n_refl' >n_refl' in tl'; #V' |
---|
| 1284 | @(Vector_inv_ind bool 1 V' (λn: nat. λv: Vector bool n. ∃r: bool. hd:::hd':::v ≃ [[hd; hd'; r]])) |
---|
| 1285 | [1: |
---|
| 1286 | #absurd @⊥ -V' @(destruct_bug_fix_1 0) |
---|
| 1287 | >absurd % |
---|
| 1288 | |2: |
---|
| 1289 | #n'' #hd'' #tl'' #_ #size_refl'' #hd_tl_refl'' %{hd''} |
---|
| 1290 | cut (n'' = 0) |
---|
| 1291 | [1: |
---|
| 1292 | @destruct_bug_fix_2 >size_refl'' % |
---|
| 1293 | |2: |
---|
| 1294 | #n_refl'' >n_refl'' in tl''; #tl''' |
---|
| 1295 | >(Vector_O … tl''') % |
---|
| 1296 | ] |
---|
| 1297 | ] |
---|
| 1298 | ] |
---|
| 1299 | ] |
---|
| 1300 | ] |
---|
[1955] | 1301 | ] |
---|
| 1302 | qed. |
---|
[1941] | 1303 | |
---|
| 1304 | lemma bitvector_3_elim_prop: |
---|
[1936] | 1305 | ∀P: BitVector 3 → Prop. |
---|
| 1306 | P [[false;false;false]] → P [[false;false;true]] → P [[false;true;false]] → |
---|
| 1307 | P [[false;true;true]] → P [[true;false;false]] → P [[true;false;true]] → |
---|
| 1308 | P [[true;true;false]] → P [[true;true;true]] → ∀v. P v. |
---|
[1941] | 1309 | #P #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 |
---|
| 1310 | cases (bitvector_3_cases … H9) #l #assm cases assm |
---|
| 1311 | -assm #c #assm cases assm |
---|
| 1312 | -assm #r #assm cases assm destruct |
---|
[1975] | 1313 | cases l cases c cases r assumption |
---|
[1941] | 1314 | qed. |
---|
[894] | 1315 | |
---|
[936] | 1316 | definition ticks_of_instruction ≝ |
---|
[1936] | 1317 | λi. |
---|
| 1318 | let trivial_code_memory ≝ assembly1 i in |
---|
| 1319 | let trivial_status ≝ load_code_memory trivial_code_memory in |
---|
| 1320 | \snd (fetch trivial_status (zero ?)). |
---|
[936] | 1321 | |
---|
[1041] | 1322 | lemma fetch_assembly: |
---|
[1941] | 1323 | ∀pc: Word. |
---|
[1936] | 1324 | ∀i: instruction. |
---|
| 1325 | ∀code_memory: BitVectorTrie Byte 16. |
---|
| 1326 | ∀assembled: list Byte. |
---|
[892] | 1327 | assembled = assembly1 i → |
---|
[890] | 1328 | let len ≝ length … assembled in |
---|
[1941] | 1329 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
---|
| 1330 | encoding_check code_memory pc pc_plus_len assembled → |
---|
| 1331 | let 〈instr, pc', ticks〉 ≝ fetch code_memory pc in |
---|
| 1332 | (eq_instruction instr i ∧ eqb ticks (ticks_of_instruction instr) ∧ eq_bv … pc' pc_plus_len) = true. |
---|
[1936] | 1333 | #pc #i #code_memory #assembled cases i [8: *] |
---|
[890] | 1334 | [16,20,29: * * |18,19: * * [1,2,4,5: *] |28: * * [1,2: * [1,2: * [1,2: * [1,2: *]]]]] |
---|
[892] | 1335 | [47,48,49: |
---|
| 1336 | |*: #arg @(list_addressing_mode_tags_elim_prop … arg) whd try % -arg |
---|
[890] | 1337 | [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] | 1338 | 59,60,63,64,65,66,67: #ARG]] |
---|
| 1339 | [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, |
---|
| 1340 | 56,57,69,70,72,73,75: #arg2 @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2 |
---|
| 1341 | [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, |
---|
| 1342 | 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, |
---|
| 1343 | 68,69,70,71: #ARG2]] |
---|
| 1344 | [1,2,19,20: #arg3 @(list_addressing_mode_tags_elim_prop … arg3) whd try % -arg3 #ARG3] |
---|
[1616] | 1345 | normalize in ⊢ (???% → ?); |
---|
[2032] | 1346 | [92,94,42,93,95: @vsplit_elim #vl #vm #E >E -E; [2,4: @(bitvector_3_elim_prop … vl)] |
---|
[1616] | 1347 | normalize in ⊢ (???% → ?);] |
---|
| 1348 | #H >H * #H1 try (whd in ⊢ (% → ?); * #H2) |
---|
| 1349 | try (whd in ⊢ (% → ?); * #H3) whd in ⊢ (% → ?); #H4 |
---|
[1649] | 1350 | [ whd in match fetch; normalize nodelta <H1 ] cases daemon |
---|
[1936] | 1351 | (* |
---|
| 1352 | whd in ⊢ (let ? ≝ ??% in ?); <H1 whd in ⊢ (let fetched ≝ % in ?) |
---|
[897] | 1353 | [17,18,19,20,21,22,23,24,25,26,31,34,35,36,37,38: <H3] |
---|
| 1354 | [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, |
---|
| 1355 | 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, |
---|
| 1356 | 69,70,73,74,78,80,81,84,85,95,98,101,102,103,104,105,106,107,108,109,110: <H2] |
---|
[1936] | 1357 | whd >eq_instruction_refl >H4 @eq_bv_refl |
---|
| 1358 | *) (* XXX: not working! *) |
---|
[1041] | 1359 | qed. |
---|
[901] | 1360 | |
---|
[1936] | 1361 | let rec fetch_many |
---|
| 1362 | (code_memory: BitVectorTrie Byte 16) (final_pc: Word) (pc: Word) |
---|
| 1363 | (expected: list instruction) |
---|
| 1364 | on expected: Prop ≝ |
---|
| 1365 | match expected with |
---|
[901] | 1366 | [ nil ⇒ eq_bv … pc final_pc = true |
---|
| 1367 | | cons i tl ⇒ |
---|
[1984] | 1368 | let pc' ≝ add 16 pc (bitvector_of_nat 16 (|assembly1 … i|)) in |
---|
| 1369 | (〈i, pc', ticks_of_instruction i〉 = fetch code_memory pc ∧ |
---|
[1983] | 1370 | fetch_many code_memory final_pc pc' tl) |
---|
[1936] | 1371 | ]. |
---|
[901] | 1372 | |
---|
[1936] | 1373 | lemma option_destruct_Some: |
---|
| 1374 | ∀A: Type[0]. |
---|
| 1375 | ∀a, b: A. |
---|
| 1376 | Some A a = Some A b → a = b. |
---|
| 1377 | #A #a #b #EQ |
---|
| 1378 | destruct % |
---|
[892] | 1379 | qed. |
---|
[901] | 1380 | |
---|
[1975] | 1381 | lemma eq_instruction_to_eq: |
---|
[1936] | 1382 | ∀i1, i2: instruction. |
---|
[1941] | 1383 | eq_instruction i1 i2 = true → i1 ≃ i2. |
---|
[1975] | 1384 | #i1 #i2 |
---|
| 1385 | cases i1 cases i2 cases daemon (* |
---|
| 1386 | [1,10,19,28,37,46: |
---|
| 1387 | #arg1 #arg2 |
---|
| 1388 | whd in match (eq_instruction ??); |
---|
| 1389 | cases arg1 #subaddressing_mode |
---|
| 1390 | cases subaddressing_mode |
---|
| 1391 | try (#arg1' #arg2' normalize in ⊢ (% → ?); #absurd cases absurd @I) |
---|
| 1392 | try (#arg1' normalize in ⊢ (% → ?); #absurd cases absurd @I) |
---|
| 1393 | try (normalize in ⊢ (% → ?); #absurd cases absurd @I) |
---|
| 1394 | #word11 #irrelevant |
---|
| 1395 | cases arg2 #subaddressing_mode |
---|
| 1396 | cases subaddressing_mode |
---|
| 1397 | try (#arg1' #arg2' normalize in ⊢ (% → ?); #absurd cases absurd @I) |
---|
| 1398 | try (#arg1' normalize in ⊢ (% → ?); #absurd cases absurd @I) |
---|
| 1399 | try (normalize in ⊢ (% → ?); #absurd cases absurd @I) |
---|
| 1400 | #word11' #irrelevant normalize nodelta |
---|
| 1401 | #eq_bv_assm cases (eq_bv_eq … eq_bv_assm) % *) |
---|
| 1402 | qed. |
---|
[1936] | 1403 | |
---|
[998] | 1404 | lemma fetch_assembly_pseudo': |
---|
[1936] | 1405 | ∀lookup_labels. |
---|
[1948] | 1406 | ∀sigma: Word → Word. |
---|
| 1407 | ∀policy: Word → bool. |
---|
[1936] | 1408 | ∀ppc. |
---|
| 1409 | ∀lookup_datalabels. |
---|
| 1410 | ∀pi. |
---|
| 1411 | ∀code_memory. |
---|
| 1412 | ∀len. |
---|
| 1413 | ∀assembled. |
---|
| 1414 | ∀instructions. |
---|
[1948] | 1415 | let pc ≝ sigma ppc in |
---|
| 1416 | instructions = expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi → |
---|
| 1417 | 〈len,assembled〉 = assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi → |
---|
[1941] | 1418 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
---|
| 1419 | encoding_check code_memory pc pc_plus_len assembled → |
---|
| 1420 | fetch_many code_memory pc_plus_len pc instructions. |
---|
[1948] | 1421 | #lookup_labels #sigma #policy #ppc #lookup_datalabels #pi #code_memory #len #assembled #instructions |
---|
| 1422 | normalize nodelta #instructions_refl whd in ⊢ (???% → ?); <instructions_refl whd in ⊢ (???% → ?); #assembled_refl |
---|
[1941] | 1423 | cases (pair_destruct ?????? assembled_refl) -assembled_refl #len_refl #assembled_refl |
---|
| 1424 | >len_refl >assembled_refl -len_refl |
---|
[1948] | 1425 | generalize in match (add 16 (sigma ppc) |
---|
[1941] | 1426 | (bitvector_of_nat 16 |
---|
| 1427 | (|flatten (Vector bool 8) |
---|
| 1428 | (map instruction (list (Vector bool 8)) assembly1 instructions)|))); |
---|
| 1429 | #final_pc |
---|
[1948] | 1430 | generalize in match (sigma ppc); elim instructions |
---|
[1936] | 1431 | [1: |
---|
| 1432 | #pc whd in ⊢ (% → %); #H >H @eq_bv_refl |
---|
| 1433 | |2: |
---|
[1941] | 1434 | #i #tl #IH #pc #H whd |
---|
| 1435 | cases (encoding_check_append ????? H) -H #H1 #H2 |
---|
[1983] | 1436 | lapply (fetch_assembly pc i code_memory (assembly1 i) (refl …)) whd in ⊢ (% → ?); |
---|
| 1437 | cases (fetch ??) * #instr #pc' #ticks |
---|
| 1438 | #H3 lapply (H3 H1) -H3 normalize nodelta #H3 |
---|
| 1439 | lapply (conjunction_true ?? H3) * #H4 #H5 |
---|
| 1440 | lapply (conjunction_true … H4) * #B1 #B2 |
---|
[1984] | 1441 | >(eq_instruction_to_eq … B1) >(eq_bv_eq … H5) % |
---|
| 1442 | >(eqb_true_to_refl … B2) >(eq_instruction_to_eq … B1) try % @IH @H2 |
---|
[1936] | 1443 | ] |
---|
[901] | 1444 | qed. |
---|
| 1445 | |
---|
[998] | 1446 | lemma fetch_assembly_pseudo: |
---|
[1936] | 1447 | ∀program: pseudo_assembly_program. |
---|
[1948] | 1448 | ∀sigma: Word → Word. |
---|
| 1449 | ∀policy: Word → bool. |
---|
[2024] | 1450 | let lookup_labels ≝ λx:Identifier. address_of_word_labels_code_mem (\snd program) x in |
---|
[2057] | 1451 | ∀ppc.∀ppc_ok. |
---|
[1948] | 1452 | ∀code_memory. |
---|
[2024] | 1453 | let lookup_datalabels ≝ λx:Identifier. lookup_def … (construct_datalabels (\fst program)) x (zero 16) in |
---|
[2057] | 1454 | let pi ≝ \fst (fetch_pseudo_instruction (\snd program) ppc ppc_ok) in |
---|
[1948] | 1455 | let pc ≝ sigma ppc in |
---|
| 1456 | let instructions ≝ expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi in |
---|
| 1457 | let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi in |
---|
[1941] | 1458 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
---|
| 1459 | encoding_check code_memory pc pc_plus_len assembled → |
---|
| 1460 | fetch_many code_memory pc_plus_len pc instructions. |
---|
[2057] | 1461 | #program #sigma #policy letin lookup_labels ≝ (λx.?) #ppc #ppc_ok #code_memory |
---|
[1966] | 1462 | letin lookup_datalabels ≝ (λx.?) |
---|
| 1463 | letin pi ≝ (fst ???) |
---|
| 1464 | letin pc ≝ (sigma ?) |
---|
| 1465 | letin instructions ≝ (expand_pseudo_instruction ??????) |
---|
| 1466 | @pair_elim #len #assembled #assembled_refl normalize nodelta |
---|
| 1467 | #H |
---|
| 1468 | generalize in match |
---|
| 1469 | (fetch_assembly_pseudo' lookup_labels sigma policy ppc lookup_datalabels pi code_memory len assembled instructions) in ⊢ ?; |
---|
| 1470 | #X destruct normalize nodelta @X try % <assembled_refl try % assumption |
---|
[877] | 1471 | qed. |
---|
| 1472 | |
---|
[1966] | 1473 | definition is_present_in_machine_code_image_p: ∀pseudo_instruction. Prop ≝ |
---|
| 1474 | λpseudo_instruction. |
---|
| 1475 | match pseudo_instruction with |
---|
| 1476 | [ Comment c ⇒ False |
---|
| 1477 | | Cost c ⇒ False |
---|
| 1478 | | _ ⇒ True |
---|
| 1479 | ]. |
---|
| 1480 | |
---|
[1941] | 1481 | (* This is a trivial consequence of fetch_assembly_pseudo + the proof that the |
---|
[1043] | 1482 | function that load the code in memory is correct. The latter is based |
---|
| 1483 | on missing properties from the standard library on the BitVectorTrie |
---|
[1941] | 1484 | data structrure. |
---|
| 1485 | |
---|
| 1486 | Wrong at the moment, requires Jaap's precondition to ensure that the program |
---|
| 1487 | does not overflow when put into code memory (i.e. shorter than 2^16 bytes). |
---|
| 1488 | *) |
---|
[2021] | 1489 | |
---|
| 1490 | lemma load_code_memory_ok: |
---|
| 1491 | ∀program. |
---|
| 1492 | let program_size ≝ |program| in |
---|
| 1493 | program_size ≤ 2^16 → |
---|
| 1494 | ∀pc. ∀pc_ok: pc < program_size. |
---|
| 1495 | nth_safe ? pc program pc_ok = \snd (next (load_code_memory program) (bitvector_of_nat … pc)). |
---|
| 1496 | #program elim program |
---|
| 1497 | [ #_ #pc #abs normalize in abs; @⊥ /2/ |
---|
| 1498 | | #hd #tl #IH #size_ok * |
---|
| 1499 | [ #pc_ok whd in ⊢ (??%?); whd in match (load_code_memory ?); |
---|
| 1500 | whd in match next; normalize nodelta |
---|
| 1501 | | #pc' #pc_ok' whd in ⊢ (??%?); whd in match (load_code_memory ?); |
---|
| 1502 | whd in match next; normalize nodelta |
---|
| 1503 | ] |
---|
| 1504 | cases daemon (*CSC: complete! *) |
---|
| 1505 | qed. |
---|
| 1506 | (*NO! Prima dimostrare tipo Russell di assembly in Assembly.ma |
---|
| 1507 | Poi dimostrare che per ogni i, se fetcho l'i-esimo bit di program |
---|
| 1508 | e' come fetchare l'i-esimo bit dalla memoria. |
---|
| 1509 | Concludere assembly_ok come semplice corollario. |
---|
| 1510 | *) |
---|
[1957] | 1511 | lemma assembly_ok: |
---|
[1948] | 1512 | ∀program. |
---|
[2075] | 1513 | ∀length_proof: |\snd program| < 2^16. |
---|
[1948] | 1514 | ∀sigma: Word → Word. |
---|
| 1515 | ∀policy: Word → bool. |
---|
[1966] | 1516 | ∀sigma_policy_witness: sigma_policy_specification program sigma policy. |
---|
[1948] | 1517 | ∀assembled. |
---|
[2021] | 1518 | ∀costs': BitVectorTrie costlabel 16. |
---|
[1957] | 1519 | let 〈preamble, instr_list〉 ≝ program in |
---|
| 1520 | let 〈labels, costs〉 ≝ create_label_cost_map instr_list in |
---|
[1966] | 1521 | let datalabels ≝ construct_datalabels preamble in |
---|
| 1522 | let lookup_datalabels ≝ λx. lookup_def … datalabels x (zero …) in |
---|
[2075] | 1523 | 〈assembled,costs'〉 = assembly program length_proof sigma policy → |
---|
[2021] | 1524 | (* costs = costs' ∧ CSC: costs != costs', costs' = sigma costs! *) |
---|
[1966] | 1525 | let code_memory ≝ load_code_memory assembled in |
---|
[2024] | 1526 | let lookup_labels ≝ λx. address_of_word_labels_code_mem instr_list x in |
---|
[2057] | 1527 | ∀ppc.∀ppc_ok. |
---|
| 1528 | let 〈pi, newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc ppc_ok in |
---|
[1966] | 1529 | let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi in |
---|
| 1530 | let pc ≝ sigma ppc in |
---|
| 1531 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
---|
| 1532 | encoding_check code_memory pc pc_plus_len assembled ∧ |
---|
| 1533 | sigma newppc = add … pc (bitvector_of_nat … len). |
---|
[2075] | 1534 | #program #length_proof #sigma #policy #sigma_policy_witness #assembled #costs' |
---|
| 1535 | cases (assembly program length_proof sigma policy) * #assembled' #costs'' |
---|
[2021] | 1536 | @pair_elim #preamble #instr_list #EQprogram whd in ⊢ (% → %); |
---|
| 1537 | @pair_elim #labels #costs #create_label_cost_refl whd in ⊢ (% → %); |
---|
| 1538 | #assembly_ok #Pair_eq destruct(Pair_eq) whd |
---|
[2057] | 1539 | #ppc #ppc_ok @pair_elim #pi #newppc #eq_fetch_pseudo_instruction |
---|
[2021] | 1540 | @pair_elim #len #assembled #eq_assembly_1_pseudoinstruction whd |
---|
| 1541 | lapply (assembly_ok ppc ?) [(*CSC: ????? WRONG HERE?*) cases daemon] -assembly_ok |
---|
| 1542 | >eq_fetch_pseudo_instruction |
---|
| 1543 | change with ((let 〈len0,assembledi〉 ≝ assembly_1_pseudoinstruction ????? pi in ∀j.∀H:j<|assembledi|.?) → ?) |
---|
| 1544 | >(? : assembly_1_pseudoinstruction ????? pi = 〈len,assembled〉) |
---|
| 1545 | [2: (*CSC: Provable, isn't it?*) cases daemon |
---|
| 1546 | | whd in ⊢ (% → ?); #assembly_ok |
---|
| 1547 | % |
---|
| 1548 | [2: (*CSC: Use Jaap's invariant here *) cases daemon |
---|
| 1549 | | lapply (load_code_memory_ok assembled' ?) [(*CSC: Jaap?*) cases daemon] |
---|
| 1550 | #load_code_memory_ok |
---|
| 1551 | -eq_assembly_1_pseudoinstruction -program -policy -costs'' -labels -preamble -instr_list -costs -pi -newppc |
---|
| 1552 | cut (len=|assembled|) [(*CSC: provable before cleaning*) cases daemon] #EQlen |
---|
| 1553 | (* Nice statement here *) |
---|
| 1554 | cut (∀j. ∀H: j < |assembled|. |
---|
| 1555 | nth_safe Byte j assembled H = |
---|
| 1556 | \snd (next (load_code_memory assembled') |
---|
| 1557 | (bitvector_of_nat 16 |
---|
| 1558 | (nat_of_bitvector … |
---|
| 1559 | (add … (sigma ppc) (bitvector_of_nat … j)))))) |
---|
| 1560 | [(*CSC: is it provable?*) cases daemon |
---|
| 1561 | | -assembly_ok -load_code_memory_ok generalize in match (sigma ppc); >EQlen -len |
---|
| 1562 | elim assembled |
---|
| 1563 | [ #pc #_ whd <add_zero % |
---|
| 1564 | | #hd #tl #IH #pc #H % |
---|
| 1565 | [ lapply (H 0 ?) [ normalize @le_S_S @le_O_n ] whd in ⊢ (??%? → ?); -H #H |
---|
| 1566 | >H -H whd in ⊢ (??%?); <add_zero // |
---|
| 1567 | | >(?: add … pc (bitvector_of_nat … (S (|tl|))) = add … (add … pc (bitvector_of_nat … 1)) (bitvector_of_nat … (|tl|))) |
---|
| 1568 | [2: (*CSC: TRUE, ARITHMETIC*) cases daemon] |
---|
| 1569 | @IH -IH #j #LTj lapply (H (S j) ?) [ @le_S_S @LTj ] |
---|
| 1570 | <(nth_safe_prepend … [hd] … LTj) #IH >IH |
---|
| 1571 | (*CSC: TRUE, ARITHMETICS*) |
---|
| 1572 | cases daemon |
---|
| 1573 | ] |
---|
[1957] | 1574 | ] |
---|
| 1575 | qed. |
---|
[906] | 1576 | |
---|
[1941] | 1577 | (* XXX: should we add that costs = costs'? *) |
---|
[1039] | 1578 | lemma fetch_assembly_pseudo2: |
---|
[1948] | 1579 | ∀program. |
---|
[2075] | 1580 | ∀length_proof: |\snd program| < 2^16. |
---|
[1948] | 1581 | ∀sigma. |
---|
| 1582 | ∀policy. |
---|
[1966] | 1583 | ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy. |
---|
[2057] | 1584 | ∀ppc.∀ppc_ok. |
---|
[1941] | 1585 | let 〈labels, costs〉 ≝ create_label_cost_map (\snd program) in |
---|
[2075] | 1586 | let 〈assembled, costs'〉 ≝ pi1 … (assembly program length_proof sigma policy) in |
---|
[1014] | 1587 | let code_memory ≝ load_code_memory assembled in |
---|
| 1588 | let data_labels ≝ construct_datalabels (\fst program) in |
---|
[2024] | 1589 | let lookup_labels ≝ λx. address_of_word_labels_code_mem (\snd program) x in |
---|
[1649] | 1590 | let lookup_datalabels ≝ λx. lookup_def ? ? data_labels x (zero ?) in |
---|
[2057] | 1591 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc ppc_ok in |
---|
[1948] | 1592 | let instructions ≝ expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi in |
---|
| 1593 | fetch_many code_memory (sigma newppc) (sigma ppc) instructions. |
---|
[2075] | 1594 | * #preamble #instr_list #length_proof #sigma #policy #sigma_policy_specification_witness #ppc #ppc_ok |
---|
[1941] | 1595 | @pair_elim #labels #costs #create_label_map_refl |
---|
| 1596 | @pair_elim #assembled #costs' #assembled_refl |
---|
| 1597 | letin code_memory ≝ (load_code_memory ?) |
---|
| 1598 | letin data_labels ≝ (construct_datalabels ?) |
---|
| 1599 | letin lookup_labels ≝ (λx. ?) |
---|
| 1600 | letin lookup_datalabels ≝ (λx. ?) |
---|
| 1601 | @pair_elim #pi #newppc #fetch_pseudo_refl |
---|
[2075] | 1602 | lapply (assembly_ok 〈preamble, instr_list〉 sigma ? policy sigma_policy_specification_witness assembled costs') |
---|
[1957] | 1603 | normalize nodelta |
---|
[1941] | 1604 | @pair_elim #labels' #costs' #create_label_map_refl' #H |
---|
[2021] | 1605 | lapply (H (sym_eq … assembled_refl)) -H |
---|
[1948] | 1606 | lapply (refl … (assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi)) |
---|
| 1607 | cases (assembly_1_pseudoinstruction ??????) in ⊢ (???% → ?); |
---|
[1941] | 1608 | #len #assembledi #EQ4 #H |
---|
| 1609 | lapply (H ppc) >fetch_pseudo_refl #H |
---|
[2057] | 1610 | lapply (fetch_assembly_pseudo 〈preamble,instr_list〉 sigma policy ppc ppc_ok (load_code_memory assembled)) |
---|
| 1611 | >EQ4 #H1 cases (H ppc_ok) |
---|
[1966] | 1612 | #H2 #H3 >H3 normalize nodelta in H1; normalize nodelta |
---|
[1941] | 1613 | >fetch_pseudo_refl in H1; #assm @assm assumption |
---|
[998] | 1614 | qed. |
---|
[985] | 1615 | |
---|
[906] | 1616 | (* OLD? |
---|
[877] | 1617 | definition assembly_specification: |
---|
| 1618 | ∀assembly_program: pseudo_assembly_program. |
---|
| 1619 | ∀code_mem: BitVectorTrie Byte 16. Prop ≝ |
---|
| 1620 | λpseudo_assembly_program. |
---|
| 1621 | λcode_mem. |
---|
| 1622 | ∀pc: Word. |
---|
| 1623 | let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in |
---|
| 1624 | let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in |
---|
| 1625 | let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in |
---|
| 1626 | let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in |
---|
| 1627 | let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program |
---|
| 1628 | (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in |
---|
| 1629 | match pre_assembled with |
---|
| 1630 | [ None ⇒ True |
---|
| 1631 | | Some pc_code ⇒ |
---|
| 1632 | let 〈new_pc,code〉 ≝ pc_code in |
---|
| 1633 | encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ]. |
---|
| 1634 | |
---|
| 1635 | axiom assembly_meets_specification: |
---|
| 1636 | ∀pseudo_assembly_program. |
---|
| 1637 | match assembly pseudo_assembly_program with |
---|
| 1638 | [ None ⇒ True |
---|
| 1639 | | Some code_mem_cost ⇒ |
---|
| 1640 | let 〈code_mem, cost〉 ≝ code_mem_cost in |
---|
| 1641 | assembly_specification pseudo_assembly_program (load_code_memory code_mem) |
---|
| 1642 | ]. |
---|
| 1643 | (* |
---|
| 1644 | # PROGRAM |
---|
| 1645 | [ cases PROGRAM |
---|
| 1646 | # PREAMBLE |
---|
| 1647 | # INSTR_LIST |
---|
| 1648 | elim INSTR_LIST |
---|
| 1649 | [ whd |
---|
| 1650 | whd in ⊢ (∀_. %) |
---|
| 1651 | # PC |
---|
| 1652 | whd |
---|
| 1653 | | # INSTR |
---|
| 1654 | # INSTR_LIST_TL |
---|
| 1655 | # H |
---|
| 1656 | whd |
---|
| 1657 | whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
---|
| 1658 | ] |
---|
| 1659 | | cases not_implemented |
---|
| 1660 | ] *) |
---|
[906] | 1661 | *) |
---|
[877] | 1662 | |
---|
[2075] | 1663 | (* XXX: changed this type. Bool specifies whether byte is first or second |
---|
| 1664 | component of an address, and the Word is the pseudoaddress that it |
---|
| 1665 | corresponds to. Second component is the same principle for the accumulator |
---|
| 1666 | A. |
---|
| 1667 | *) |
---|
| 1668 | definition internal_pseudo_address_map ≝ list ((BitVector 8) × (bool × Word)) × (option (bool × Word)). |
---|
[942] | 1669 | |
---|
| 1670 | axiom low_internal_ram_of_pseudo_low_internal_ram: |
---|
| 1671 | ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7. |
---|
| 1672 | |
---|
| 1673 | axiom high_internal_ram_of_pseudo_high_internal_ram: |
---|
| 1674 | ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7. |
---|
| 1675 | |
---|
[2075] | 1676 | include "common/AssocList.ma". |
---|
| 1677 | |
---|
[942] | 1678 | axiom low_internal_ram_of_pseudo_internal_ram_hit: |
---|
[2075] | 1679 | ∀M:internal_pseudo_address_map.∀cm.∀s:PseudoStatus cm.∀sigma:Word → Word × bool.∀high: bool. ∀points_to: Word. ∀addr:BitVector 7. |
---|
| 1680 | assoc_list_lookup ?? (false:::addr) (eq_bv 8) (\fst M) = Some … (〈high, points_to〉) → |
---|
[942] | 1681 | let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in |
---|
[1666] | 1682 | let pbl ≝ lookup ? 7 addr (low_internal_ram … s) (zero 8) in |
---|
[942] | 1683 | let bl ≝ lookup ? 7 addr ram (zero 8) in |
---|
[2075] | 1684 | let 〈lower, higher〉 ≝ vsplit ? 8 8 points_to in |
---|
| 1685 | let 〈plower, phigher〉 ≝ vsplit ? 8 8 (\fst (sigma points_to)) in |
---|
| 1686 | if high then |
---|
| 1687 | (pbl = higher) ∧ (bl = phigher) |
---|
| 1688 | else |
---|
| 1689 | (pbl = lower) ∧ (bl = plower). |
---|
[942] | 1690 | |
---|
[949] | 1691 | (* changed from add to sub *) |
---|
[942] | 1692 | axiom low_internal_ram_of_pseudo_internal_ram_miss: |
---|
[1666] | 1693 | ∀T.∀M:internal_pseudo_address_map.∀cm.∀s:PreStatus T cm.∀addr:BitVector 7. |
---|
[2075] | 1694 | let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in |
---|
| 1695 | assoc_list_exists ?? (false:::addr) (eq_bv 8) (\fst M) = false → |
---|
| 1696 | lookup ? 7 addr ram (zero ?) = lookup ? 7 addr (low_internal_ram … s) (zero ?). |
---|
[942] | 1697 | |
---|
| 1698 | definition addressing_mode_ok ≝ |
---|
[1666] | 1699 | λT.λM:internal_pseudo_address_map.λcm.λs:PreStatus T cm. |
---|
[942] | 1700 | λaddr:addressing_mode. |
---|
| 1701 | match addr with |
---|
| 1702 | [ DIRECT d ⇒ |
---|
[2075] | 1703 | ¬(assoc_list_exists ?? d (eq_bv 8) (\fst M)) ∧ |
---|
| 1704 | ¬(assoc_list_exists ?? (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) (eq_bv 8) (\fst M)) |
---|
[942] | 1705 | | INDIRECT i ⇒ |
---|
[1666] | 1706 | let d ≝ get_register … s [[false;false;i]] in |
---|
[2075] | 1707 | ¬(assoc_list_exists ?? d (eq_bv 8) (\fst M)) ∧ |
---|
| 1708 | ¬(assoc_list_exists ?? (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) (eq_bv 8) (\fst M)) |
---|
[942] | 1709 | | EXT_INDIRECT _ ⇒ true |
---|
| 1710 | | REGISTER _ ⇒ true |
---|
[2075] | 1711 | | ACC_A ⇒ match \snd M with [ None ⇒ true | _ ⇒ false ] |
---|
[942] | 1712 | | ACC_B ⇒ true |
---|
| 1713 | | DPTR ⇒ true |
---|
| 1714 | | DATA _ ⇒ true |
---|
| 1715 | | DATA16 _ ⇒ true |
---|
| 1716 | | ACC_DPTR ⇒ true |
---|
| 1717 | | ACC_PC ⇒ true |
---|
| 1718 | | EXT_INDIRECT_DPTR ⇒ true |
---|
| 1719 | | INDIRECT_DPTR ⇒ true |
---|
| 1720 | | CARRY ⇒ true |
---|
| 1721 | | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *) |
---|
| 1722 | | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *) |
---|
| 1723 | | RELATIVE _ ⇒ true |
---|
| 1724 | | ADDR11 _ ⇒ true |
---|
| 1725 | | ADDR16 _ ⇒ true ]. |
---|
[944] | 1726 | |
---|
| 1727 | definition next_internal_pseudo_address_map0 ≝ |
---|
[950] | 1728 | λT. |
---|
[2075] | 1729 | λcm:T. |
---|
| 1730 | λaddr_of: Identifier → PreStatus T cm → Word. |
---|
[944] | 1731 | λfetched. |
---|
| 1732 | λM: internal_pseudo_address_map. |
---|
[1666] | 1733 | λs: PreStatus T cm. |
---|
[944] | 1734 | match fetched with |
---|
[942] | 1735 | [ Comment _ ⇒ Some ? M |
---|
| 1736 | | Cost _ ⇒ Some … M |
---|
| 1737 | | Jmp _ ⇒ Some … M |
---|
[2075] | 1738 | | Call a ⇒ |
---|
| 1739 | let a' ≝ addr_of a s in |
---|
| 1740 | let 〈callM, accM〉 ≝ M in |
---|
| 1741 | Some … 〈〈(add 8 (get_8051_sfr ?? s SFR_SP) (bitvector_of_nat 8 1)), 〈false, a'〉〉:: |
---|
| 1742 | 〈(add 8 (get_8051_sfr ?? s SFR_SP) (bitvector_of_nat 8 2)), 〈true, a'〉〉::callM, accM〉 |
---|
[942] | 1743 | | Mov _ _ ⇒ Some … M |
---|
| 1744 | | Instruction instr ⇒ |
---|
| 1745 | match instr with |
---|
| 1746 | [ ADD addr1 addr2 ⇒ |
---|
[1666] | 1747 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
[942] | 1748 | Some ? M |
---|
| 1749 | else |
---|
| 1750 | None ? |
---|
| 1751 | | ADDC addr1 addr2 ⇒ |
---|
[1666] | 1752 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
[942] | 1753 | Some ? M |
---|
| 1754 | else |
---|
| 1755 | None ? |
---|
| 1756 | | SUBB addr1 addr2 ⇒ |
---|
[1666] | 1757 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
[942] | 1758 | Some ? M |
---|
| 1759 | else |
---|
| 1760 | None ? |
---|
| 1761 | | _ ⇒ (* TO BE COMPLETED *) Some ? M ]]. |
---|
[944] | 1762 | |
---|
| 1763 | |
---|
| 1764 | definition next_internal_pseudo_address_map ≝ |
---|
| 1765 | λM:internal_pseudo_address_map. |
---|
[1666] | 1766 | λcm. |
---|
[2057] | 1767 | λs:PseudoStatus cm. λppc_ok. |
---|
[950] | 1768 | next_internal_pseudo_address_map0 ? |
---|
[2057] | 1769 | (\fst (fetch_pseudo_instruction (\snd cm) (program_counter … s) ppc_ok)) M cm s. |
---|
[1666] | 1770 | |
---|
| 1771 | definition code_memory_of_pseudo_assembly_program: |
---|
[1948] | 1772 | ∀pap:pseudo_assembly_program. |
---|
| 1773 | (Word → Word) → (Word → bool) → BitVectorTrie Byte 16 ≝ |
---|
| 1774 | λpap. |
---|
| 1775 | λsigma. |
---|
| 1776 | λpolicy. |
---|
[2021] | 1777 | let p ≝ pi1 … (assembly pap sigma policy) in |
---|
[1948] | 1778 | load_code_memory (\fst p). |
---|
[1666] | 1779 | |
---|
[2075] | 1780 | definition sfr_8051_of_pseudo_sfr_8051 ≝ |
---|
| 1781 | λM: internal_pseudo_address_map. |
---|
| 1782 | λsfrs: Vector Byte 19. |
---|
| 1783 | λsigma: Word → Word. |
---|
| 1784 | match \snd M with |
---|
| 1785 | [ None ⇒ sfrs |
---|
| 1786 | | Some s ⇒ |
---|
| 1787 | let 〈high, address〉 ≝ s in |
---|
| 1788 | let index ≝ sfr_8051_index SFR_ACC_A in |
---|
| 1789 | let 〈upper, lower〉 ≝ vsplit ? 8 8 (sigma address) in |
---|
| 1790 | if high then |
---|
| 1791 | set_index Byte 19 sfrs index upper ? |
---|
| 1792 | else |
---|
| 1793 | set_index Byte 19 sfrs index lower ? |
---|
| 1794 | ]. |
---|
| 1795 | // |
---|
| 1796 | qed. |
---|
| 1797 | |
---|
[942] | 1798 | definition status_of_pseudo_status: |
---|
[1948] | 1799 | internal_pseudo_address_map → ∀pap. ∀ps: PseudoStatus pap. |
---|
| 1800 | ∀sigma: Word → Word. ∀policy: Word → bool. |
---|
| 1801 | Status (code_memory_of_pseudo_assembly_program pap sigma policy) ≝ |
---|
| 1802 | λM. |
---|
| 1803 | λpap. |
---|
| 1804 | λps. |
---|
| 1805 | λsigma. |
---|
| 1806 | λpolicy. |
---|
| 1807 | let cm ≝ code_memory_of_pseudo_assembly_program … sigma policy in |
---|
| 1808 | let pc ≝ sigma (program_counter … ps) in |
---|
[1014] | 1809 | let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in |
---|
| 1810 | let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … ps) in |
---|
| 1811 | mk_PreStatus (BitVectorTrie Byte 16) |
---|
| 1812 | cm |
---|
| 1813 | lir |
---|
| 1814 | hir |
---|
| 1815 | (external_ram … ps) |
---|
| 1816 | pc |
---|
| 1817 | (special_function_registers_8051 … ps) |
---|
| 1818 | (special_function_registers_8052 … ps) |
---|
| 1819 | (p1_latch … ps) |
---|
| 1820 | (p3_latch … ps) |
---|
| 1821 | (clock … ps). |
---|
[877] | 1822 | |
---|
[909] | 1823 | (* |
---|
[877] | 1824 | definition write_at_stack_pointer': |
---|
| 1825 | ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝ |
---|
| 1826 | λM: Type[0]. |
---|
| 1827 | λs: PreStatus M. |
---|
| 1828 | λv: Byte. |
---|
[2032] | 1829 | let 〈 nu, nl 〉 ≝ vsplit … 4 4 (get_8051_sfr ? s SFR_SP) in |
---|
[877] | 1830 | let bit_zero ≝ get_index_v… nu O ? in |
---|
| 1831 | let bit_1 ≝ get_index_v… nu 1 ? in |
---|
| 1832 | let bit_2 ≝ get_index_v… nu 2 ? in |
---|
| 1833 | let bit_3 ≝ get_index_v… nu 3 ? in |
---|
| 1834 | if bit_zero then |
---|
| 1835 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
| 1836 | v (low_internal_ram ? s) in |
---|
| 1837 | set_low_internal_ram ? s memory |
---|
| 1838 | else |
---|
| 1839 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
| 1840 | v (high_internal_ram ? s) in |
---|
| 1841 | set_high_internal_ram ? s memory. |
---|
| 1842 | [ cases l0 % |
---|
| 1843 | |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ] |
---|
| 1844 | qed. |
---|
| 1845 | |
---|
| 1846 | definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus. |
---|
| 1847 | Σps':PseudoStatus.(code_memory … ps = code_memory … ps') |
---|
| 1848 | ≝ |
---|
| 1849 | λticks_of. |
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| 1850 | λs. |
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| 1851 | let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in |
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| 1852 | let ticks ≝ ticks_of (program_counter ? s) in |
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| 1853 | let s ≝ set_clock ? s (clock ? s + ticks) in |
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| 1854 | let s ≝ set_program_counter ? s pc in |
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| 1855 | match instr with |
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| 1856 | [ Instruction instr ⇒ |
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| 1857 | execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s |
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| 1858 | | Comment cmt ⇒ s |
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| 1859 | | Cost cst ⇒ s |
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| 1860 | | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp) |
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| 1861 | | Call call ⇒ |
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| 1862 | let a ≝ address_of_word_labels s call in |
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| 1863 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
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| 1864 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
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[2032] | 1865 | let 〈pc_bu, pc_bl〉 ≝ vsplit ? 8 8 (program_counter ? s) in |
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[877] | 1866 | let s ≝ write_at_stack_pointer' ? s pc_bl in |
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| 1867 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
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| 1868 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
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| 1869 | let s ≝ write_at_stack_pointer' ? s pc_bu in |
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| 1870 | set_program_counter ? s a |
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| 1871 | | Mov dptr ident ⇒ |
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| 1872 | set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr |
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| 1873 | ]. |
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| 1874 | [ |
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| 1875 | |2,3,4: % |
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| 1876 | | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) % |
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| 1877 | | |
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| 1878 | | % |
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| 1879 | ] |
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| 1880 | cases not_implemented |
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| 1881 | qed. |
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[909] | 1882 | *) |
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[911] | 1883 | |
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[877] | 1884 | (* |
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| 1885 | lemma execute_code_memory_unchanged: |
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| 1886 | ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps). |
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| 1887 | #ticks #ps whd in ⊢ (??? (??%)) |
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| 1888 | cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps)) |
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| 1889 | (program_counter pseudo_assembly_program ps)) #instr #pc |
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| 1890 | whd in ⊢ (??? (??%)) cases instr |
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| 1891 | [ #pre cases pre |
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| 1892 | [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
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[2032] | 1893 | cases (vsplit ????) #z1 #z2 % |
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[877] | 1894 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
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[2032] | 1895 | cases (vsplit ????) #z1 #z2 % |
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[877] | 1896 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
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[2032] | 1897 | cases (vsplit ????) #z1 #z2 % |
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[877] | 1898 | | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x |
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| 1899 | [ #x1 whd in ⊢ (??? (??%)) |
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| 1900 | | *: cases not_implemented |
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| 1901 | ] |
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| 1902 | | #comment % |
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| 1903 | | #cost % |
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| 1904 | | #label % |
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| 1905 | | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%)) |
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[2032] | 1906 | cases (vsplit ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2 |
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| 1907 | whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (vsplit ????) #w1 #w2 |
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[877] | 1908 | whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%)) |
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| 1909 | (* CSC: ??? *) |
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| 1910 | | #dptr #label (* CSC: ??? *) |
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| 1911 | ] |
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| 1912 | cases not_implemented |
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| 1913 | qed. |
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| 1914 | *) |
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| 1915 | |
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[1014] | 1916 | (* DEAD CODE? |
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[877] | 1917 | lemma status_of_pseudo_status_failure_depends_only_on_code_memory: |
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[942] | 1918 | ∀M:internal_pseudo_address_map. |
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[877] | 1919 | ∀ps,ps': PseudoStatus. |
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[987] | 1920 | ∀pol. |
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| 1921 | ∀prf:code_memory … ps = code_memory … ps'. |
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| 1922 | let pol' ≝ ? in |
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| 1923 | match status_of_pseudo_status M ps pol with |
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| 1924 | [ None ⇒ status_of_pseudo_status M ps' pol' = None … |
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| 1925 | | Some _ ⇒ ∃w. status_of_pseudo_status M ps' pol' = Some … w |
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[877] | 1926 | ]. |
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[987] | 1927 | [2: <prf @pol] |
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| 1928 | #M #ps #ps' #pol #H normalize nodelta; whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ]) |
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| 1929 | generalize in match (refl … (assembly (code_memory … ps) pol)) |
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| 1930 | cases (assembly ??) in ⊢ (???% → %) |
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[877] | 1931 | [ #K whd whd in ⊢ (??%?) <H >K % |
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| 1932 | | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ] |
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[909] | 1933 | qed. |
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[1014] | 1934 | *) |
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[877] | 1935 | |
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[1948] | 1936 | definition ticks_of0: |
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| 1937 | ∀p:pseudo_assembly_program. |
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| 1938 | (Word → Word) → (Word → bool) → Word → pseudo_instruction → nat × nat ≝ |
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| 1939 | λprogram: pseudo_assembly_program. |
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| 1940 | λsigma. |
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| 1941 | λpolicy. |
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[929] | 1942 | λppc: Word. |
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[1953] | 1943 | λfetched. |
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[929] | 1944 | match fetched with |
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| 1945 | [ Instruction instr ⇒ |
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| 1946 | match instr with |
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[1953] | 1947 | [ JC lbl ⇒ ? (* |
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[1649] | 1948 | match pol lookup_labels ppc with |
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[929] | 1949 | [ short_jump ⇒ 〈2, 2〉 |
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| 1950 | | medium_jump ⇒ ? |
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| 1951 | | long_jump ⇒ 〈4, 4〉 |
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[1953] | 1952 | ] *) |
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| 1953 | | JNC lbl ⇒ ? (* |
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[1649] | 1954 | match pol lookup_labels ppc with |
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[929] | 1955 | [ short_jump ⇒ 〈2, 2〉 |
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| 1956 | | medium_jump ⇒ ? |
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| 1957 | | long_jump ⇒ 〈4, 4〉 |
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[1953] | 1958 | ] *) |
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| 1959 | | JB bit lbl ⇒ ? (* |
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[1649] | 1960 | match pol lookup_labels ppc with |
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[929] | 1961 | [ short_jump ⇒ 〈2, 2〉 |
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| 1962 | | medium_jump ⇒ ? |
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| 1963 | | long_jump ⇒ 〈4, 4〉 |
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[1953] | 1964 | ] *) |
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| 1965 | | JNB bit lbl ⇒ ? (* |
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[1649] | 1966 | match pol lookup_labels ppc with |
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[929] | 1967 | [ short_jump ⇒ 〈2, 2〉 |
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| 1968 | | medium_jump ⇒ ? |
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| 1969 | | long_jump ⇒ 〈4, 4〉 |
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[1953] | 1970 | ] *) |
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| 1971 | | JBC bit lbl ⇒ ? (* |
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[1649] | 1972 | match pol lookup_labels ppc with |
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[929] | 1973 | [ short_jump ⇒ 〈2, 2〉 |
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| 1974 | | medium_jump ⇒ ? |
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| 1975 | | long_jump ⇒ 〈4, 4〉 |
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[1953] | 1976 | ] *) |
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| 1977 | | JZ lbl ⇒ ? (* |
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[1649] | 1978 | match pol lookup_labels ppc with |
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[929] | 1979 | [ short_jump ⇒ 〈2, 2〉 |
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| 1980 | | medium_jump ⇒ ? |
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| 1981 | | long_jump ⇒ 〈4, 4〉 |
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[1953] | 1982 | ] *) |
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| 1983 | | JNZ lbl ⇒ ? (* |
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[1649] | 1984 | match pol lookup_labels ppc with |
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[929] | 1985 | [ short_jump ⇒ 〈2, 2〉 |
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| 1986 | | medium_jump ⇒ ? |
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| 1987 | | long_jump ⇒ 〈4, 4〉 |
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[1953] | 1988 | ] *) |
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| 1989 | | CJNE arg lbl ⇒ ? (* |
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[1649] | 1990 | match pol lookup_labels ppc with |
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[929] | 1991 | [ short_jump ⇒ 〈2, 2〉 |
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| 1992 | | medium_jump ⇒ ? |
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| 1993 | | long_jump ⇒ 〈4, 4〉 |
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[1953] | 1994 | ] *) |
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| 1995 | | DJNZ arg lbl ⇒ ? (* |
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[1649] | 1996 | match pol lookup_labels ppc with |
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[929] | 1997 | [ short_jump ⇒ 〈2, 2〉 |
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| 1998 | | medium_jump ⇒ ? |
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| 1999 | | long_jump ⇒ 〈4, 4〉 |
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[1953] | 2000 | ] *) |
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[936] | 2001 | | ADD arg1 arg2 ⇒ |
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| 2002 | let ticks ≝ ticks_of_instruction (ADD ? arg1 arg2) in |
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| 2003 | 〈ticks, ticks〉 |
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[929] | 2004 | | ADDC arg1 arg2 ⇒ |
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[936] | 2005 | let ticks ≝ ticks_of_instruction (ADDC ? arg1 arg2) in |
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| 2006 | 〈ticks, ticks〉 |
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[929] | 2007 | | SUBB arg1 arg2 ⇒ |
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[936] | 2008 | let ticks ≝ ticks_of_instruction (SUBB ? arg1 arg2) in |
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| 2009 | 〈ticks, ticks〉 |
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[929] | 2010 | | INC arg ⇒ |
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[936] | 2011 | let ticks ≝ ticks_of_instruction (INC ? arg) in |
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| 2012 | 〈ticks, ticks〉 |
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[929] | 2013 | | DEC arg ⇒ |
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[936] | 2014 | let ticks ≝ ticks_of_instruction (DEC ? arg) in |
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| 2015 | 〈ticks, ticks〉 |
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[929] | 2016 | | MUL arg1 arg2 ⇒ |
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[936] | 2017 | let ticks ≝ ticks_of_instruction (MUL ? arg1 arg2) in |
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| 2018 | 〈ticks, ticks〉 |
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[929] | 2019 | | DIV arg1 arg2 ⇒ |
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[936] | 2020 | let ticks ≝ ticks_of_instruction (DIV ? arg1 arg2) in |
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| 2021 | 〈ticks, ticks〉 |
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[929] | 2022 | | DA arg ⇒ |
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[936] | 2023 | let ticks ≝ ticks_of_instruction (DA ? arg) in |
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| 2024 | 〈ticks, ticks〉 |
---|
[929] | 2025 | | ANL arg ⇒ |
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[936] | 2026 | let ticks ≝ ticks_of_instruction (ANL ? arg) in |
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| 2027 | 〈ticks, ticks〉 |
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[929] | 2028 | | ORL arg ⇒ |
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[936] | 2029 | let ticks ≝ ticks_of_instruction (ORL ? arg) in |
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| 2030 | 〈ticks, ticks〉 |
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[929] | 2031 | | XRL arg ⇒ |
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[936] | 2032 | let ticks ≝ ticks_of_instruction (XRL ? arg) in |
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| 2033 | 〈ticks, ticks〉 |
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[929] | 2034 | | CLR arg ⇒ |
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[936] | 2035 | let ticks ≝ ticks_of_instruction (CLR ? arg) in |
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| 2036 | 〈ticks, ticks〉 |
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[929] | 2037 | | CPL arg ⇒ |
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[936] | 2038 | let ticks ≝ ticks_of_instruction (CPL ? arg) in |
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| 2039 | 〈ticks, ticks〉 |
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[929] | 2040 | | RL arg ⇒ |
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[936] | 2041 | let ticks ≝ ticks_of_instruction (RL ? arg) in |
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| 2042 | 〈ticks, ticks〉 |
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[929] | 2043 | | RLC arg ⇒ |
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[936] | 2044 | let ticks ≝ ticks_of_instruction (RLC ? arg) in |
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| 2045 | 〈ticks, ticks〉 |
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[929] | 2046 | | RR arg ⇒ |
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[936] | 2047 | let ticks ≝ ticks_of_instruction (RR ? arg) in |
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| 2048 | 〈ticks, ticks〉 |
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[929] | 2049 | | RRC arg ⇒ |
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[936] | 2050 | let ticks ≝ ticks_of_instruction (RRC ? arg) in |
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| 2051 | 〈ticks, ticks〉 |
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[929] | 2052 | | SWAP arg ⇒ |
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[936] | 2053 | let ticks ≝ ticks_of_instruction (SWAP ? arg) in |
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| 2054 | 〈ticks, ticks〉 |
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[929] | 2055 | | MOV arg ⇒ |
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[936] | 2056 | let ticks ≝ ticks_of_instruction (MOV ? arg) in |
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| 2057 | 〈ticks, ticks〉 |
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[929] | 2058 | | MOVX arg ⇒ |
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[936] | 2059 | let ticks ≝ ticks_of_instruction (MOVX ? arg) in |
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| 2060 | 〈ticks, ticks〉 |
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[929] | 2061 | | SETB arg ⇒ |
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[936] | 2062 | let ticks ≝ ticks_of_instruction (SETB ? arg) in |
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| 2063 | 〈ticks, ticks〉 |
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[929] | 2064 | | PUSH arg ⇒ |
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[936] | 2065 | let ticks ≝ ticks_of_instruction (PUSH ? arg) in |
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| 2066 | 〈ticks, ticks〉 |
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[929] | 2067 | | POP arg ⇒ |
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[936] | 2068 | let ticks ≝ ticks_of_instruction (POP ? arg) in |
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| 2069 | 〈ticks, ticks〉 |
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[929] | 2070 | | XCH arg1 arg2 ⇒ |
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[936] | 2071 | let ticks ≝ ticks_of_instruction (XCH ? arg1 arg2) in |
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| 2072 | 〈ticks, ticks〉 |
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[929] | 2073 | | XCHD arg1 arg2 ⇒ |
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[936] | 2074 | let ticks ≝ ticks_of_instruction (XCHD ? arg1 arg2) in |
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| 2075 | 〈ticks, ticks〉 |
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[929] | 2076 | | RET ⇒ |
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[936] | 2077 | let ticks ≝ ticks_of_instruction (RET ?) in |
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| 2078 | 〈ticks, ticks〉 |
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[929] | 2079 | | RETI ⇒ |
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[936] | 2080 | let ticks ≝ ticks_of_instruction (RETI ?) in |
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| 2081 | 〈ticks, ticks〉 |
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[929] | 2082 | | NOP ⇒ |
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[936] | 2083 | let ticks ≝ ticks_of_instruction (NOP ?) in |
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| 2084 | 〈ticks, ticks〉 |
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[929] | 2085 | ] |
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| 2086 | | Comment comment ⇒ 〈0, 0〉 |
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| 2087 | | Cost cost ⇒ 〈0, 0〉 |
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[938] | 2088 | | Jmp jmp ⇒ 〈2, 2〉 |
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| 2089 | | Call call ⇒ 〈2, 2〉 |
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| 2090 | | Mov dptr tgt ⇒ 〈2, 2〉 |
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[929] | 2091 | ]. |
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[1953] | 2092 | cases daemon |
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| 2093 | qed. |
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[929] | 2094 | |
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[1948] | 2095 | definition ticks_of: |
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| 2096 | ∀p:pseudo_assembly_program. |
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[2057] | 2097 | (Word → Word) → (Word → bool) → ∀ppc:Word. |
---|
| 2098 | nat_of_bitvector … ppc < |\snd p| → nat × nat ≝ |
---|
[1948] | 2099 | λprogram: pseudo_assembly_program. |
---|
| 2100 | λsigma. |
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| 2101 | λpolicy. |
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[2057] | 2102 | λppc: Word. λppc_ok. |
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| 2103 | let pseudo ≝ \snd program in |
---|
| 2104 | let 〈fetched, new_ppc〉 ≝ fetch_pseudo_instruction pseudo ppc ppc_ok in |
---|
[1948] | 2105 | ticks_of0 program sigma policy ppc fetched. |
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[930] | 2106 | |
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[935] | 2107 | lemma eq_rect_Type1_r: |
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| 2108 | ∀A: Type[1]. |
---|
[1948] | 2109 | ∀a: A. |
---|
[935] | 2110 | ∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p. |
---|
| 2111 | #A #a #P #H #x #p |
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[1649] | 2112 | generalize in match H; |
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| 2113 | generalize in match P; |
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[1948] | 2114 | cases p // |
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[935] | 2115 | qed. |
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| 2116 | |
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[2032] | 2117 | axiom vsplit_append: |
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[949] | 2118 | ∀A: Type[0]. |
---|
| 2119 | ∀m, n: nat. |
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| 2120 | ∀v, v': Vector A m. |
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| 2121 | ∀q, q': Vector A n. |
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[2032] | 2122 | let 〈v', q'〉 ≝ vsplit A m n (v@@q) in |
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[949] | 2123 | v = v' ∧ q = q'. |
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| 2124 | |
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[2032] | 2125 | lemma vsplit_vector_singleton: |
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[949] | 2126 | ∀A: Type[0]. |
---|
| 2127 | ∀n: nat. |
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| 2128 | ∀v: Vector A (S n). |
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| 2129 | ∀rest: Vector A n. |
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| 2130 | ∀s: Vector A 1. |
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| 2131 | v = s @@ rest → |
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[1953] | 2132 | ((get_index_v A ? v 0 ?) ::: rest) = v. |
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| 2133 | [1: |
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| 2134 | #A #n #v cases daemon (* XXX: !!! *) |
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| 2135 | |2: |
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| 2136 | @le_S_S @le_O_n |
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| 2137 | ] |
---|
| 2138 | qed. |
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[949] | 2139 | |
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[985] | 2140 | example sub_minus_one_seven_eight: |
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[949] | 2141 | ∀v: BitVector 7. |
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| 2142 | false ::: (\fst (sub_7_with_carry v (bitvector_of_nat ? 1) false)) = |
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| 2143 | \fst (sub_8_with_carry (false ::: v) (bitvector_of_nat ? 1) false). |
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[985] | 2144 | cases daemon. |
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| 2145 | qed. |
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[949] | 2146 | |
---|
[950] | 2147 | (* |
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[945] | 2148 | lemma blah: |
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[949] | 2149 | ∀m: internal_pseudo_address_map. |
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[945] | 2150 | ∀s: PseudoStatus. |
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| 2151 | ∀arg: Byte. |
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[949] | 2152 | ∀b: bool. |
---|
| 2153 | addressing_mode_ok m s (DIRECT arg) = true → |
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| 2154 | get_arg_8 ? (set_low_internal_ram ? s (low_internal_ram_of_pseudo_low_internal_ram m (low_internal_ram ? s))) b (DIRECT arg) = |
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| 2155 | get_arg_8 ? s b (DIRECT arg). |
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| 2156 | [2, 3: normalize % ] |
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| 2157 | #m #s #arg #b #hyp |
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| 2158 | whd in ⊢ (??%%) |
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[2032] | 2159 | @vsplit_elim'' |
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[949] | 2160 | #nu' #nl' #arg_nu_nl_eq |
---|
| 2161 | normalize nodelta |
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| 2162 | generalize in match (refl ? (get_index_v bool 4 nu' ? ?)) |
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| 2163 | cases (get_index_v bool 4 nu' ? ?) in ⊢ (??%? → %) |
---|
| 2164 | #get_index_v_eq |
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| 2165 | normalize nodelta |
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| 2166 | [2: |
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| 2167 | normalize nodelta |
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[2032] | 2168 | @vsplit_elim'' |
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[949] | 2169 | #bit_one' #three_bits' #bit_one_three_bit_eq |
---|
| 2170 | generalize in match (low_internal_ram_of_pseudo_internal_ram_miss m s (three_bits'@@nl')) |
---|
| 2171 | normalize nodelta |
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| 2172 | generalize in match (refl ? (sub_7_with_carry ? ? ?)) |
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| 2173 | cases (sub_7_with_carry ? ? ?) in ⊢ (??%? → %) |
---|
| 2174 | #Saddr #carr' #Saddr_carr_eq |
---|
| 2175 | normalize nodelta |
---|
| 2176 | #carr_hyp' |
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| 2177 | @carr_hyp' |
---|
| 2178 | [1: |
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| 2179 | |2: whd in hyp:(??%?); generalize in match hyp; -hyp; |
---|
| 2180 | generalize in match (refl ? (¬(member (BitVector 8) ? arg m))) |
---|
| 2181 | cases (¬(member (BitVector 8) ? arg m)) in ⊢ (??%? → %) |
---|
| 2182 | #member_eq |
---|
| 2183 | normalize nodelta |
---|
| 2184 | [2: #destr destruct(destr) |
---|
| 2185 | |1: -carr_hyp'; |
---|
| 2186 | >arg_nu_nl_eq |
---|
[2032] | 2187 | <(vsplit_vector_singleton ? ? nu' ? ? ? bit_one_three_bit_eq) |
---|
[949] | 2188 | [1: >get_index_v_eq in ⊢ (??%? → ?) |
---|
| 2189 | |2: @le_S @le_S @le_S @le_n |
---|
| 2190 | ] |
---|
| 2191 | cases (member (BitVector 8) ? (\fst ?) ?) |
---|
| 2192 | [1: #destr normalize in destr; destruct(destr) |
---|
| 2193 | |2: |
---|
| 2194 | ] |
---|
| 2195 | ] |
---|
| 2196 | |3: >get_index_v_eq in ⊢ (??%?) |
---|
| 2197 | change in ⊢ (??(???%?)?) with ((? ::: three_bits') @@ nl') |
---|
[2032] | 2198 | >(vsplit_vector_singleton … bit_one_three_bit_eq) |
---|
[949] | 2199 | <arg_nu_nl_eq |
---|
| 2200 | whd in hyp:(??%?) |
---|
| 2201 | cases (member (BitVector 8) (eq_bv 8) arg m) in hyp |
---|
| 2202 | normalize nodelta [*: #ignore @sym_eq ] |
---|
| 2203 | ] |
---|
| 2204 | | |
---|
| 2205 | ]. |
---|
[950] | 2206 | *) |
---|
[949] | 2207 | (* |
---|
[945] | 2208 | map_address0 ... (DIRECT arg) = Some .. → |
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| 2209 | get_arg_8 (map_address0 ... (internal_ram ...) (DIRECT arg) = |
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| 2210 | get_arg_8 (internal_ram ...) (DIRECT arg) |
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| 2211 | |
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| 2212 | ((if addressing_mode_ok M ps ACC_A∧addressing_mode_ok M ps (DIRECT ARG2) |
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| 2213 | then Some internal_pseudo_address_map M |
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| 2214 | else None internal_pseudo_address_map ) |
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| 2215 | =Some internal_pseudo_address_map M') |
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| 2216 | |
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[1014] | 2217 | axiom low_internal_ram_write_at_stack_pointer: |
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[1953] | 2218 | ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3.∀sigma: Word → Word.∀policy: Word → bool. |
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| 2219 | ∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8. |
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[1945] | 2220 | get_8051_sfr T2 cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP → |
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[1666] | 2221 | low_internal_ram ? cm2 s2 = low_internal_ram T2 cm3 s3 → |
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[1946] | 2222 | sp1 = add ? (get_8051_sfr … cm1 s1 SFR_SP) (bitvector_of_nat 8 1) → |
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| 2223 | sp2 = add ? sp1 (bitvector_of_nat 8 1) → |
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[1953] | 2224 | bu@@bl = sigma (pbu@@pbl) → |
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[1666] | 2225 | low_internal_ram T1 cm1 |
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| 2226 | (write_at_stack_pointer … |
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| 2227 | (set_8051_sfr … |
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| 2228 | (write_at_stack_pointer … |
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| 2229 | (set_8051_sfr … |
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| 2230 | (set_low_internal_ram … s1 |
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| 2231 | (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s2))) |
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[1014] | 2232 | SFR_SP sp1) |
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| 2233 | bl) |
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| 2234 | SFR_SP sp2) |
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| 2235 | bu) |
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| 2236 | = low_internal_ram_of_pseudo_low_internal_ram (sp1::M) |
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[1666] | 2237 | (low_internal_ram … |
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| 2238 | (write_at_stack_pointer … |
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| 2239 | (set_8051_sfr … |
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| 2240 | (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl) |
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[1014] | 2241 | SFR_SP sp2) |
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| 2242 | pbu)). |
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| 2243 | |
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[1953] | 2244 | lemma high_internal_ram_write_at_stack_pointer: |
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| 2245 | ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3.∀sigma:Word → Word.∀policy: Word → bool. |
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| 2246 | ∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8. |
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[1945] | 2247 | get_8051_sfr T2 cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP → |
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[1666] | 2248 | high_internal_ram ?? s2 = high_internal_ram T2 cm3 s3 → |
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[1946] | 2249 | sp1 = add ? (get_8051_sfr ? cm1 s1 SFR_SP) (bitvector_of_nat 8 1) → |
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| 2250 | sp2 = add ? sp1 (bitvector_of_nat 8 1) → |
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[1953] | 2251 | bu@@bl = sigma (pbu@@pbl) → |
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[1666] | 2252 | high_internal_ram T1 cm1 |
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| 2253 | (write_at_stack_pointer … |
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| 2254 | (set_8051_sfr … |
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| 2255 | (write_at_stack_pointer … |
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| 2256 | (set_8051_sfr … |
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| 2257 | (set_high_internal_ram … s1 |
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| 2258 | (high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … s2))) |
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[1014] | 2259 | SFR_SP sp1) |
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| 2260 | bl) |
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| 2261 | SFR_SP sp2) |
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| 2262 | bu) |
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| 2263 | = high_internal_ram_of_pseudo_high_internal_ram (sp1::M) |
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[1666] | 2264 | (high_internal_ram … |
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| 2265 | (write_at_stack_pointer … |
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| 2266 | (set_8051_sfr … |
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| 2267 | (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl) |
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[1014] | 2268 | SFR_SP sp2) |
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| 2269 | pbu)). |
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[1953] | 2270 | #T1 #T2 #M #cm1 #s1 #cm2 #s2 #cm3 #s3 #sigma #policy #pbu #pbl #bu #bl #sp1 #sp2 |
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| 2271 | #get_8051_sfr_refl #high_internal_ram_refl #sp1_refl #sp2_refl #sigma_refl |
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| 2272 | cases daemon (* XXX: !!! *) |
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| 2273 | qed. |
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[2075] | 2274 | *) |
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[1014] | 2275 | |
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[1037] | 2276 | lemma Some_Some_elim: |
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| 2277 | ∀T:Type[0].∀x,y:T.∀P:Type[2]. (x=y → P) → Some T x = Some T y → P. |
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| 2278 | #T #x #y #P #H #K @H @option_destruct_Some // |
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| 2279 | qed. |
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| 2280 | |
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[1957] | 2281 | lemma pair_destruct_right: |
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| 2282 | ∀A: Type[0]. |
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| 2283 | ∀B: Type[0]. |
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| 2284 | ∀a, c: A. |
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| 2285 | ∀b, d: B. |
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| 2286 | 〈a, b〉 = 〈c, d〉 → b = d. |
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| 2287 | #A #B #a #b #c #d // |
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| 2288 | qed. |
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[1955] | 2289 | |
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[1945] | 2290 | (*CSC: ???*) |
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[2024] | 2291 | (* XXX: we need a precondition here stating that the PPC is within the |
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| 2292 | bounds of the instruction list in order for Jaap's specification to |
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| 2293 | apply. |
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| 2294 | *) |
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[1953] | 2295 | lemma snd_assembly_1_pseudoinstruction_ok: |
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[1948] | 2296 | ∀program: pseudo_assembly_program. |
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| 2297 | ∀sigma: Word → Word. |
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| 2298 | ∀policy: Word → bool. |
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[1966] | 2299 | ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy. |
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[1948] | 2300 | ∀ppc: Word. |
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[2057] | 2301 | ∀ppc_in_bounds: nat_of_bitvector 16 ppc < |\snd program|. |
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[1948] | 2302 | ∀pi. |
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| 2303 | ∀lookup_labels. |
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| 2304 | ∀lookup_datalabels. |
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[2024] | 2305 | lookup_labels = (λx. (address_of_word_labels_code_mem (\snd program) x)) → |
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[1948] | 2306 | lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) → |
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[2057] | 2307 | \fst (fetch_pseudo_instruction (\snd program) ppc ppc_in_bounds) = pi → |
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[2024] | 2308 | let len ≝ \fst (assembly_1_pseudoinstruction lookup_labels sigma policy (*(sigma ppc)*) ppc lookup_datalabels pi) in |
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[1948] | 2309 | sigma (add … ppc (bitvector_of_nat ? 1)) = add … (sigma ppc) (bitvector_of_nat ? len). |
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[2024] | 2310 | #program #sigma #policy #sigma_policy_specification_witness #ppc #ppc_in_bounds #pi |
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[1966] | 2311 | #lookup_labels #lookup_datalabels |
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| 2312 | #lookup_labels_refl #lookup_datalabels_refl #fetch_pseudo_refl |
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[1955] | 2313 | normalize nodelta |
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| 2314 | generalize in match fetch_pseudo_refl; -fetch_pseudo_refl |
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| 2315 | #fetch_pseudo_refl |
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[2021] | 2316 | letin assembled ≝ (\fst (pi1 … (assembly program sigma policy))) |
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| 2317 | letin costs ≝ (\snd (pi1 … (assembly program sigma policy))) |
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[1966] | 2318 | lapply (assembly_ok program sigma policy sigma_policy_specification_witness assembled costs) |
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[1957] | 2319 | @pair_elim #preamble #instr_list #program_refl |
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[1955] | 2320 | @pair_elim #labels #costs' #create_label_cost_map_refl |
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[2021] | 2321 | <eq_pair_fst_snd #H lapply (H (refl …)) -H #H |
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[2057] | 2322 | lapply (H ppc ppc_in_bounds) -H |
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[1955] | 2323 | @pair_elim #pi' #newppc #fetch_pseudo_refl' |
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[1966] | 2324 | @pair_elim #len #assembled #assembly1_refl #H |
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| 2325 | cases H |
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[1955] | 2326 | #encoding_check_assm #sigma_newppc_refl |
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| 2327 | >fetch_pseudo_refl' in fetch_pseudo_refl; #pi_refl' |
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| 2328 | >pi_refl' in assembly1_refl; #assembly1_refl |
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[1966] | 2329 | >lookup_labels_refl >lookup_datalabels_refl |
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| 2330 | >program_refl normalize nodelta |
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| 2331 | >assembly1_refl |
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[1955] | 2332 | <sigma_newppc_refl |
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| 2333 | generalize in match fetch_pseudo_refl'; |
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[2057] | 2334 | whd in match (fetch_pseudo_instruction ???); |
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[1955] | 2335 | @pair_elim #lbl #instr #nth_refl normalize nodelta |
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[2057] | 2336 | #G cases (pair_destruct_right ?????? G) % |
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[1953] | 2337 | qed. |
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[1667] | 2338 | |
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| 2339 | lemma pose: ∀A:Type[0].∀B:A → Type[0].∀a:A. (∀a':A. a'=a → B a') → B a. |
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[1948] | 2340 | /2/ |
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[1667] | 2341 | qed. |
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| 2342 | |
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[1668] | 2343 | (* To be moved in ProofStatus *) |
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| 2344 | lemma program_counter_set_program_counter: |
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[1948] | 2345 | ∀T. |
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| 2346 | ∀cm. |
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| 2347 | ∀s. |
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| 2348 | ∀x. |
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| 2349 | program_counter T cm (set_program_counter T cm s x) = x. |
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| 2350 | // |
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[1668] | 2351 | qed. |
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[1984] | 2352 | |
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| 2353 | (* XXX: easy but tedious *) |
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| 2354 | axiom assembly1_lt_128: |
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| 2355 | ∀i: instruction. |
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| 2356 | |(assembly1 i)| < 128. |
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| 2357 | |
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| 2358 | axiom most_significant_bit_zero: |
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| 2359 | ∀size, m: nat. |
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| 2360 | ∀size_proof: 0 < size. |
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| 2361 | m < 2^size → get_index_v bool (S size) (bitvector_of_nat (S size) m) 1 ? = false. |
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| 2362 | normalize in size_proof; /2/ |
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| 2363 | qed. |
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| 2364 | |
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| 2365 | lemma bitvector_of_nat_sign_extension_equivalence: |
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| 2366 | ∀m: nat. |
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| 2367 | ∀size_proof: m < 128. |
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| 2368 | sign_extension … (bitvector_of_nat 8 m) = bitvector_of_nat 16 m. |
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| 2369 | #m #size_proof whd in ⊢ (??%?); |
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| 2370 | >most_significant_bit_zero |
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| 2371 | [1: |
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| 2372 | cases daemon |
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| 2373 | |2: |
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| 2374 | assumption |
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| 2375 | |3: |
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| 2376 | // |
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| 2377 | ] |
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[2078] | 2378 | qed. |
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