[823] | 1 | include "ASM/Assembly.ma". |
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| 2 | include "ASM/Interpret.ma". |
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| 3 | |
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[861] | 4 | (* RUSSEL **) |
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| 5 | |
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| 6 | include "basics/jmeq.ma". |
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| 7 | |
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| 8 | notation > "hvbox(a break ≃ b)" |
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| 9 | non associative with precedence 45 |
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| 10 | for @{ 'jmeq ? $a ? $b }. |
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| 11 | |
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| 12 | notation < "hvbox(term 46 a break maction (≃) (≃\sub(t,u)) term 46 b)" |
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| 13 | non associative with precedence 45 |
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| 14 | for @{ 'jmeq $t $a $u $b }. |
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| 15 | |
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| 16 | interpretation "john major's equality" 'jmeq t x u y = (jmeq t x u y). |
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| 17 | |
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| 18 | lemma eq_to_jmeq: |
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| 19 | ∀A: Type[0]. |
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| 20 | ∀x, y: A. |
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| 21 | x = y → x ≃ y. |
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| 22 | // |
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| 23 | qed. |
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| 24 | |
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| 25 | definition inject : ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ λA,P,a,p. dp … a p. |
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| 26 | definition eject : ∀A.∀P: A → Prop.(Σx:A.P x) → A ≝ λA,P,c.match c with [ dp w p ⇒ w]. |
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| 27 | |
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| 28 | coercion inject nocomposites: ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ inject on a:? to Σx:?.?. |
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| 29 | coercion eject nocomposites: ∀A.∀P:A → Prop.∀c:Σx:A.P x.A ≝ eject on _c:Σx:?.? to ?. |
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| 30 | |
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| 31 | axiom VOID: Type[0]. |
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| 32 | axiom assert_false: VOID. |
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| 33 | definition bigbang: ∀A:Type[0].False → VOID → A. |
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| 34 | #A #abs cases abs |
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| 35 | qed. |
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| 36 | |
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| 37 | coercion bigbang nocomposites: ∀A:Type[0].False → ∀v:VOID.A ≝ bigbang on _v:VOID to ?. |
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| 38 | |
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| 39 | lemma sig2: ∀A.∀P:A → Prop. ∀p:Σx:A.P x. P (eject … p). |
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| 40 | #A #P #p cases p #w #q @q |
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| 41 | qed. |
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| 42 | |
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| 43 | lemma jmeq_to_eq: ∀A:Type[0]. ∀x,y:A. x≃y → x=y. |
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| 44 | #A #x #y #JMEQ @(jmeq_elim ? x … JMEQ) % |
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| 45 | qed. |
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| 46 | |
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| 47 | coercion jmeq_to_eq: ∀A:Type[0]. ∀x,y:A. ∀p:x≃y.x=y ≝ jmeq_to_eq on _p:?≃? to ?=?. |
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| 48 | |
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| 49 | (* END RUSSELL **) |
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| 50 | |
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[852] | 51 | let rec foldl_strong_internal |
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[853] | 52 | (A: Type[0]) (P: list A → Type[0]) (l: list A) |
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[851] | 53 | (H: ∀prefix. ∀hd. ∀tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd])) |
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| 54 | (prefix: list A) (suffix: list A) (acc: P prefix) on suffix: |
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| 55 | l = prefix @ suffix → P(prefix @ suffix) ≝ |
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| 56 | match suffix return λl'. l = prefix @ l' → P (prefix @ l') with |
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| 57 | [ nil ⇒ λprf. ? |
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| 58 | | cons hd tl ⇒ λprf. ? |
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| 59 | ]. |
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| 60 | [ > (append_nil ?) |
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| 61 | @ acc |
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[852] | 62 | | applyS (foldl_strong_internal A P l H (prefix @ [hd]) tl ? ?) |
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[851] | 63 | [ @ (H prefix hd tl prf acc) |
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| 64 | | applyS prf |
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| 65 | ] |
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| 66 | ] |
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| 67 | qed. |
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[852] | 68 | |
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| 69 | definition foldl_strong ≝ |
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| 70 | λA: Type[0]. |
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[853] | 71 | λP: list A → Type[0]. |
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[852] | 72 | λl: list A. |
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| 73 | λH: ∀prefix. ∀hd. ∀tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]). |
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| 74 | λacc: P [ ]. |
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[853] | 75 | foldl_strong_internal A P l H [ ] l acc (refl …). |
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[852] | 76 | |
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[855] | 77 | definition bit_elim: ∀P: bool → bool. bool ≝ |
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| 78 | λP. |
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| 79 | P true ∧ P false. |
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| 80 | |
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[854] | 81 | let rec bitvector_elim_internal |
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| 82 | (n: nat) (P: BitVector n → bool) (m: nat) on m: m ≤ n → BitVector (n - m) → bool ≝ |
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| 83 | match m return λm. m ≤ n → BitVector (n - m) → bool with |
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| 84 | [ O ⇒ λprf1. λprefix. P ? |
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| 85 | | S n' ⇒ λprf2. λprefix. bit_elim (λbit. bitvector_elim_internal n P n' ? ?) |
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| 86 | ]. |
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| 87 | [ applyS prefix |
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| 88 | | letin res ≝ (bit ::: prefix) |
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| 89 | < (minus_S_S ? ?) |
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| 90 | > (minus_Sn_m ? ?) |
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| 91 | [ @ res |
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| 92 | | @ prf2 |
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| 93 | ] |
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| 94 | | /2/ |
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| 95 | ]. |
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| 96 | qed. |
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[851] | 97 | |
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[854] | 98 | definition bitvector_elim ≝ |
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| 99 | λn: nat. |
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| 100 | λP: BitVector n → bool. |
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| 101 | bitvector_elim_internal n P n ? ?. |
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| 102 | [ @ (le_n ?) |
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| 103 | | < (minus_n_n ?) |
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| 104 | @ [[ ]] |
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| 105 | ] |
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| 106 | qed. |
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| 107 | |
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[867] | 108 | axiom vector_associative_append: |
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[859] | 109 | ∀A: Type[0]. |
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| 110 | ∀n, m, o: nat. |
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| 111 | ∀v: Vector A n. |
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| 112 | ∀q: Vector A m. |
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| 113 | ∀r: Vector A o. |
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| 114 | ((v @@ q) @@ r) |
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| 115 | ≃ |
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| 116 | (v @@ (q @@ r)). |
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| 117 | |
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[867] | 118 | lemma vector_cons_append: |
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[859] | 119 | ∀A: Type[0]. |
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| 120 | ∀n: nat. |
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[867] | 121 | ∀e: A. |
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[859] | 122 | ∀v: Vector A n. |
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[867] | 123 | e ::: v = [[ e ]] @@ v. |
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| 124 | # A # N # E # V |
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| 125 | elim V |
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| 126 | [ normalize % |
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| 127 | | # NN # AA # VV # IH |
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| 128 | normalize |
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| 129 | % |
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| 130 | ] |
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| 131 | qed. |
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[859] | 132 | |
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| 133 | lemma super_rewrite2: |
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| 134 | ∀A:Type[0].∀n,m.∀v1: Vector A n.∀v2: Vector A m. |
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| 135 | ∀P: ∀m. Vector A m → Prop. |
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| 136 | n=m → v1 ≃ v2 → P n v1 → P m v2. |
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| 137 | #A #n #m #v1 #v2 #P #EQ <EQ in v2; #V #JMEQ >JMEQ // |
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| 138 | qed. |
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| 139 | |
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| 140 | lemma mem_middle_vector: |
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| 141 | ∀A: Type[0]. |
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| 142 | ∀m, o: nat. |
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| 143 | ∀eq: A → A → bool. |
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| 144 | ∀reflex: ∀a. eq a a = true. |
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| 145 | ∀p: Vector A m. |
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| 146 | ∀a: A. |
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| 147 | ∀r: Vector A o. |
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| 148 | mem A eq ? (p@@(a:::r)) a = true. |
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| 149 | # A # M # O # EQ # REFLEX # P # A |
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| 150 | elim P |
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| 151 | [ normalize |
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| 152 | > (REFLEX A) |
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| 153 | normalize |
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| 154 | # H |
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| 155 | % |
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| 156 | | # NN # AA # PP # IH |
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| 157 | normalize |
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| 158 | cases (EQ A AA) // |
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| 159 | @ IH |
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| 160 | ] |
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| 161 | qed. |
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| 162 | |
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| 163 | lemma mem_monotonic_wrt_append: |
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| 164 | ∀A: Type[0]. |
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| 165 | ∀m, o: nat. |
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| 166 | ∀eq: A → A → bool. |
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| 167 | ∀reflex: ∀a. eq a a = true. |
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| 168 | ∀p: Vector A m. |
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| 169 | ∀a: A. |
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| 170 | ∀r: Vector A o. |
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| 171 | mem A eq ? r a = true → mem A eq ? (p @@ r) a = true. |
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| 172 | # A # M # O # EQ # REFLEX # P # A |
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| 173 | elim P |
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| 174 | [ #R #H @H |
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| 175 | | #NN #AA # PP # IH #R #H |
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| 176 | normalize |
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| 177 | cases (EQ A AA) |
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| 178 | [ normalize % |
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| 179 | | @ IH @ H |
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| 180 | ] |
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| 181 | ] |
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| 182 | qed. |
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| 183 | |
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[864] | 184 | lemma subvector_multiple_append: |
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[859] | 185 | ∀A: Type[0]. |
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| 186 | ∀o, n: nat. |
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| 187 | ∀eq: A → A → bool. |
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| 188 | ∀refl: ∀a. eq a a = true. |
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| 189 | ∀h: Vector A o. |
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| 190 | ∀v: Vector A n. |
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| 191 | ∀m: nat. |
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| 192 | ∀q: Vector A m. |
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| 193 | bool_to_Prop (subvector_with A ? ? eq v (h @@ q @@ v)). |
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| 194 | # A # O # N # EQ # REFLEX # H # V |
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| 195 | elim V |
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| 196 | [ normalize |
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| 197 | # M # V % |
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| 198 | | # NN # AA # VV # IH # MM # QQ |
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| 199 | change with (bool_to_Prop (andb ??)) |
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| 200 | cut ((mem A EQ (O + (MM + S NN)) (H@@QQ@@AA:::VV) AA) = true) |
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| 201 | [ |
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| 202 | | # HH > HH |
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| 203 | > (vector_cons_append ? ? AA VV) |
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| 204 | change with (bool_to_Prop (subvector_with ??????)) |
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| 205 | @(super_rewrite2 A ((MM + 1)+ NN) (MM+S NN) ?? |
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| 206 | (λSS.λVS.bool_to_Prop (subvector_with ?? (O+SS) ?? (H@@VS))) |
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| 207 | ? |
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[867] | 208 | (vector_associative_append A ? ? ? QQ [[AA]] VV)) |
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[859] | 209 | [ >associative_plus // |
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| 210 | | @IH ] |
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| 211 | ] |
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| 212 | @(mem_monotonic_wrt_append) |
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| 213 | [ @ REFLEX |
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| 214 | | @(mem_monotonic_wrt_append) |
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| 215 | [ @ REFLEX |
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| 216 | | normalize |
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| 217 | > REFLEX |
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| 218 | normalize |
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| 219 | % |
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| 220 | ] |
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| 221 | ] |
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| 222 | qed. |
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[864] | 223 | |
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[867] | 224 | lemma vector_cons_empty: |
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[854] | 225 | ∀A: Type[0]. |
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[867] | 226 | ∀n: nat. |
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| 227 | ∀v: Vector A n. |
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| 228 | [[ ]] @@ v = v. |
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| 229 | # A # N # V |
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| 230 | elim V |
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| 231 | [ normalize % |
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| 232 | | # NN # HH # VV #H % |
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| 233 | ] |
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| 234 | qed. |
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| 235 | |
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| 236 | corollary subvector_hd_tl: |
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| 237 | ∀A: Type[0]. |
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[864] | 238 | ∀o: nat. |
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| 239 | ∀eq: A → A → bool. |
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| 240 | ∀refl: ∀a. eq a a = true. |
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[855] | 241 | ∀h: A. |
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[864] | 242 | ∀v: Vector A o. |
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| 243 | bool_to_Prop (subvector_with A ? ? eq v (h ::: v)). |
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[867] | 244 | # A # O # EQ # REFLEX # H # V |
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| 245 | > (vector_cons_append A ? H V) |
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| 246 | < (vector_cons_empty A ? ([[H]] @@ V)) |
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| 247 | @ (subvector_multiple_append A ? ? EQ REFLEX [[]] V ? [[ H ]]) |
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| 248 | qed. |
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[854] | 249 | |
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[867] | 250 | lemma eq_a_reflexive: |
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[864] | 251 | ∀a. eq_a a a = true. |
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[867] | 252 | # A |
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| 253 | cases A |
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| 254 | % |
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| 255 | qed. |
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[866] | 256 | |
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[867] | 257 | lemma is_in_monotonic_wrt_append: |
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| 258 | ∀m, n: nat. |
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| 259 | ∀p: Vector addressing_mode_tag m. |
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| 260 | ∀q: Vector addressing_mode_tag n. |
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| 261 | ∀to_search: addressing_mode. |
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| 262 | bool_to_Prop (is_in ? p to_search) → bool_to_Prop (is_in ? (q @@ p) to_search). |
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| 263 | # M # N # P # Q # TO_SEARCH |
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| 264 | # H |
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| 265 | elim Q |
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| 266 | [ normalize |
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| 267 | @ H |
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| 268 | | # NN # PP # QQ # IH |
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| 269 | normalize |
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| 270 | cases (is_a PP TO_SEARCH) |
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| 271 | [ normalize |
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| 272 | % |
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| 273 | | normalize |
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| 274 | normalize in IH |
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| 275 | @ IH |
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| 276 | ] |
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| 277 | ] |
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| 278 | qed. |
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| 279 | |
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| 280 | corollary is_in_hd_tl: |
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| 281 | ∀to_search: addressing_mode. |
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| 282 | ∀hd: addressing_mode_tag. |
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| 283 | ∀n: nat. |
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| 284 | ∀v: Vector addressing_mode_tag n. |
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| 285 | bool_to_Prop (is_in ? v to_search) → bool_to_Prop (is_in ? (hd:::v) to_search). |
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| 286 | # TO_SEARCH # HD # N # V |
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| 287 | elim V |
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| 288 | [ # H |
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| 289 | normalize in H; |
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| 290 | cases H |
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| 291 | | # NN # HHD # VV # IH # HH |
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| 292 | > vector_cons_append |
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| 293 | > (vector_cons_append ? ? HHD VV) |
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| 294 | @ (is_in_monotonic_wrt_append ? 1 ([[HHD]]@@VV) [[HD]] TO_SEARCH) |
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| 295 | @ HH |
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| 296 | ] |
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| 297 | qed. |
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| 298 | |
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[854] | 299 | let rec list_addressing_mode_tags_elim |
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| 300 | (n: nat) (l: Vector addressing_mode_tag (S n)) on l: (l → bool) → bool ≝ |
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| 301 | match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x'). |
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| 302 | (l → bool) → bool ] with |
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| 303 | [ VEmpty ⇒ true |
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| 304 | | VCons len hd tl ⇒ λP. |
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| 305 | let process_hd ≝ |
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| 306 | match hd return λhd. ∀P: hd:::tl → bool. bool with |
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| 307 | [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x)) |
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| 308 | | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x)) |
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| 309 | | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x)) |
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| 310 | | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x)) |
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| 311 | | acc_a ⇒ λP.P ACC_A |
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| 312 | | acc_b ⇒ λP.P ACC_B |
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| 313 | | dptr ⇒ λP.P DPTR |
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| 314 | | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x)) |
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| 315 | | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x)) |
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| 316 | | acc_dptr ⇒ λP.P ACC_DPTR |
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| 317 | | acc_pc ⇒ λP.P ACC_PC |
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| 318 | | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR |
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| 319 | | indirect_dptr ⇒ λP.P INDIRECT_DPTR |
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| 320 | | carry ⇒ λP.P CARRY |
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| 321 | | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x)) |
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| 322 | | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x)) |
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| 323 | | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x)) |
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| 324 | | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x)) |
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| 325 | | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x)) |
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| 326 | ] |
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| 327 | in |
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| 328 | andb (process_hd P) |
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[867] | 329 | (match len return λx. x = len → bool with |
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| 330 | [ O ⇒ λprf. true |
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| 331 | | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len)) |
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[854] | 332 | ]. |
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[867] | 333 | try % |
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| 334 | [ 2: cases (sym_eq ??? prf); @tl |
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| 335 | | cases prf in tl H; #tl |
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| 336 | normalize in ⊢ (∀_: %. ?) |
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| 337 | # H |
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| 338 | whd |
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| 339 | normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
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| 340 | cases (is_a hd (subaddressing_modeel y tl H)) whd // ] |
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| 341 | qed. |
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[854] | 342 | |
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[867] | 343 | definition product_elim ≝ |
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| 344 | λm, n: nat. |
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| 345 | λv: Vector addressing_mode_tag (S m). |
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| 346 | λq: Vector addressing_mode_tag (S n). |
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| 347 | λP: (Vector addressing_mode_tag (S m) × (Vector addressing_mode_tag (S n))) → bool. |
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| 348 | P 〈v, q〉. |
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| 349 | |
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| 350 | axiom union_elim: |
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| 351 | ∀m, n: nat. ((Vector addressing_mode_tag m ⊎ Vector addressing_mode_tag n) → bool) → bool. |
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| 352 | |
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[871] | 353 | (* |
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[854] | 354 | definition preinstruction_elim: ∀P: preinstruction [[ relative ]] → bool. bool ≝ |
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[867] | 355 | λP. |
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| 356 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADD ? ACC_A addr)) ∧ |
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| 357 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADDC ? ACC_A addr)) ∧ |
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| 358 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (SUBB ? ACC_A addr)) ∧ |
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| 359 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ; dptr ]] (λaddr. P (INC ? addr)) ∧ |
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| 360 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (DEC ? addr)) ∧ |
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| 361 | list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (MUL ? ACC_A addr)) ∧ |
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| 362 | list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (DIV ? ACC_A addr)) ∧ |
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| 363 | list_addressing_mode_tags_elim ? [[ registr ; direct ]] (λaddr. P (DJNZ ? ? addr)) ∧ |
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| 364 | list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CLR ? addr)) ∧ |
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| 365 | list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CPL ? addr)) ∧ |
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| 366 | P (DA ? ACC_A) ∧ |
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[870] | 367 | bitvector_elim 8 (λr. P (JC ? (RELATIVE r))) ∧ |
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| 368 | bitvector_elim 8 (λr. P (JNC ? (RELATIVE r))) ∧ |
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| 369 | bitvector_elim 8 (λr. P (JZ ? (RELATIVE r))) ∧ |
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| 370 | bitvector_elim 8 (λr. P (JNZ ? (RELATIVE r))) ∧ |
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| 371 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JB ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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| 372 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JNB ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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| 373 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JBC ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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| 374 | list_addressing_mode_tags_elim ? [[ registr; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧ |
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| 375 | P (RL ? ACC_A) ∧ |
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| 376 | P (RLC ? ACC_A) ∧ |
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| 377 | P (RR ? ACC_A) ∧ |
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| 378 | P (RRC ? ACC_A) ∧ |
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| 379 | P (SWAP ? ACC_A) ∧ |
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| 380 | P (RET ?) ∧ |
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| 381 | P (RETI ?) ∧ |
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| 382 | P (NOP ?) ∧ |
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| 383 | list_addressing_mode_tags_elim ? [[ carry; bit_addr ]] (λaddr. P (SETB ? addr)) ∧ |
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| 384 | bitvector_elim 8 (λaddr. P (PUSH ? (DIRECT addr))) ∧ |
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| 385 | bitvector_elim 8 (λaddr. P (POP ? (DIRECT addr))). |
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[867] | 386 | |
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| 387 | |
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| 388 | |
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| 389 | |
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| 390 | |
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| 391 | list_addressing_mode_tags_elim ? [[ data ]] (λaddr. P (CJNE ? (inl ? ? (〈ACC_A, addr)). |
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| 392 | |
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| 393 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (ANL ? addr)) ∧ |
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| 394 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (ORL ? addr)) ∧ |
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| 395 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (XRL ? addr)) ∧ |
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| 396 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (SWAP ? addr)) ∧ |
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| 397 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (MOV ? addr)) ∧ |
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| 398 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (MOVX ? addr)) ∧ |
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| 399 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (SETB ? addr)) ∧ |
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| 400 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (PUSH ? addr)) ∧ |
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| 401 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (POP ? addr)) ∧ |
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| 402 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (XCH ? addr)) ∧ |
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| 403 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (XCHD ? addr)) ∧ |
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| 404 | P (RET ?) ∧ |
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| 405 | P (RETI ?) ∧ |
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| 406 | P (NOP ?). |
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[854] | 407 | |
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| 408 | |
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[870] | 409 | axiom instruction_elim: ∀P: instruction → bool. bool. |
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[854] | 410 | |
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| 411 | |
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| 412 | lemma instruction_elim_correct: |
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| 413 | ∀i: instruction. |
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| 414 | ∀P: instruction → bool. |
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| 415 | instruction_elim P = true → ∀j. P j = true. |
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| 416 | |
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| 417 | lemma test: |
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| 418 | ∀i: instruction. |
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| 419 | ∃pc. |
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| 420 | let assembled ≝ assembly1 i in |
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| 421 | let code_memory ≝ load_code_memory assembled in |
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| 422 | let fetched ≝ fetch code_memory pc in |
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| 423 | let 〈instr_pc, ticks〉 ≝ fetched in |
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| 424 | \fst instr_pc = i. |
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| 425 | # INSTR |
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| 426 | @ (ex_intro ?) |
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| 427 | [ @ (zero 16) |
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| 428 | | @ (instruction_elim INSTR) |
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| 429 | ]. |
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[871] | 430 | *) |
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| 431 | |
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[840] | 432 | (* This establishes the correspondence between pseudo program counters and |
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| 433 | program counters. It is at the heart of the proof. *) |
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| 434 | (*CSC: code taken from build_maps *) |
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[845] | 435 | definition sigma0: pseudo_assembly_program → option (nat × (nat × (BitVectorTrie Word 16))) ≝ |
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[840] | 436 | λinstr_list. |
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[845] | 437 | foldl ?? |
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[840] | 438 | (λt. λi. |
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| 439 | match t with |
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| 440 | [ None ⇒ None ? |
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| 441 | | Some ppc_pc_map ⇒ |
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| 442 | let 〈ppc,pc_map〉 ≝ ppc_pc_map in |
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| 443 | let 〈program_counter, sigma_map〉 ≝ pc_map in |
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| 444 | let 〈label, i〉 ≝ i in |
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| 445 | match construct_costs instr_list program_counter (λx. zero ?) (λx. zero ?) (Stub …) i with |
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| 446 | [ None ⇒ None ? |
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| 447 | | Some pc_ignore ⇒ |
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| 448 | let 〈pc,ignore〉 ≝ pc_ignore in |
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| 449 | Some … 〈S ppc,〈pc, insert ? ? (bitvector_of_nat ? ppc) (bitvector_of_nat ? pc) sigma_map〉〉 ] |
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[845] | 450 | ]) (Some ? 〈0, 〈0, (Stub ? ?)〉〉) (\snd instr_list). |
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| 451 | |
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[866] | 452 | definition tech_pc_sigma0: pseudo_assembly_program → option (nat × (BitVectorTrie Word 16)) ≝ |
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[845] | 453 | λinstr_list. |
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| 454 | match sigma0 instr_list with |
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| 455 | [ None ⇒ None … |
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| 456 | | Some result ⇒ |
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| 457 | let 〈ppc,pc_sigma_map〉 ≝ result in |
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[866] | 458 | Some … pc_sigma_map ]. |
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[845] | 459 | |
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[848] | 460 | definition sigma_safe: pseudo_assembly_program → option (Word → Word) ≝ |
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[845] | 461 | λinstr_list. |
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| 462 | match sigma0 instr_list with |
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[840] | 463 | [ None ⇒ None ? |
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| 464 | | Some result ⇒ |
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| 465 | let 〈ppc,pc_sigma_map〉 ≝ result in |
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| 466 | let 〈pc, sigma_map〉 ≝ pc_sigma_map in |
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| 467 | if gtb pc (2^16) then |
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| 468 | None ? |
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| 469 | else |
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[841] | 470 | Some ? (λx.lookup ?? x sigma_map (zero …)) ]. |
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[840] | 471 | |
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[848] | 472 | axiom policy_ok: ∀p. sigma_safe p ≠ None …. |
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[840] | 473 | |
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[848] | 474 | definition sigma: pseudo_assembly_program → Word → Word ≝ |
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| 475 | λp. |
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| 476 | match sigma_safe p return λr:option (Word → Word). r ≠ None … → Word → Word with |
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| 477 | [ None ⇒ λabs. ⊥ |
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| 478 | | Some r ⇒ λ_.r] (policy_ok p). |
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| 479 | cases abs // |
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| 480 | qed. |
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[845] | 481 | |
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[856] | 482 | lemma length_append: |
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| 483 | ∀A.∀l1,l2:list A. |
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| 484 | |l1 @ l2| = |l1| + |l2|. |
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| 485 | #A #l1 elim l1 |
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| 486 | [ // |
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| 487 | | #hd #tl #IH #l2 normalize <IH //] |
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| 488 | qed. |
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| 489 | |
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[862] | 490 | let rec does_not_occur (id:Identifier) (l:list labelled_instruction) on l: bool ≝ |
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| 491 | match l with |
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| 492 | [ nil ⇒ true |
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[865] | 493 | | cons hd tl ⇒ notb (instruction_matches_identifier id hd) ∧ does_not_occur id tl]. |
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[860] | 494 | |
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[862] | 495 | lemma does_not_occur_None: |
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| 496 | ∀id,i,list_instr. |
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| 497 | does_not_occur id (list_instr@[〈None …,i〉]) = |
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| 498 | does_not_occur id list_instr. |
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| 499 | #id #i #list_instr elim list_instr |
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| 500 | [ % | #hd #tl #IH whd in ⊢ (??%%) >IH %] |
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| 501 | qed. |
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| 502 | |
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| 503 | let rec occurs_exactly_once (id:Identifier) (l:list labelled_instruction) on l : bool ≝ |
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| 504 | match l with |
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| 505 | [ nil ⇒ false |
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| 506 | | cons hd tl ⇒ |
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[865] | 507 | if instruction_matches_identifier id hd then |
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[862] | 508 | does_not_occur id tl |
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| 509 | else |
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| 510 | occurs_exactly_once id tl ]. |
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| 511 | |
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| 512 | lemma occurs_exactly_once_None: |
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| 513 | ∀id,i,list_instr. |
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| 514 | occurs_exactly_once id (list_instr@[〈None …,i〉]) = |
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| 515 | occurs_exactly_once id list_instr. |
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| 516 | #id #i #list_instr elim list_instr |
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| 517 | [ % | #hd #tl #IH whd in ⊢ (??%%) >IH >does_not_occur_None %] |
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| 518 | qed. |
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| 519 | |
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| 520 | coercion bool_to_Prop: ∀b:bool. Prop ≝ bool_to_Prop on _b:bool to Type[0]. |
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| 521 | |
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[861] | 522 | lemma index_of_internal_None: ∀i,id,instr_list,n. |
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| 523 | occurs_exactly_once id (instr_list@[〈None …,i〉]) → |
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[865] | 524 | index_of_internal ? (instruction_matches_identifier id) instr_list n = |
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| 525 | index_of_internal ? (instruction_matches_identifier id) (instr_list@[〈None …,i〉]) n. |
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[861] | 526 | #i #id #instr_list elim instr_list |
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[862] | 527 | [ #n #abs whd in abs; cases abs |
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| 528 | | #hd #tl #IH #n whd in ⊢ (% → ??%%); whd in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?] → ?) |
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[865] | 529 | cases (instruction_matches_identifier id hd) whd in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?] → ??%%) |
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[862] | 530 | [ #H % |
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| 531 | | #H @IH whd in H; cases (occurs_exactly_once ??) in H ⊢ % |
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| 532 | [ #_ % | #abs cases abs ]]] |
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[861] | 533 | qed. |
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| 534 | |
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| 535 | lemma address_of_word_labels_code_mem_None: ∀i,id,instr_list. |
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| 536 | occurs_exactly_once id (instr_list@[〈None …,i〉]) → |
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| 537 | address_of_word_labels_code_mem instr_list id = |
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| 538 | address_of_word_labels_code_mem (instr_list@[〈None …,i〉]) id. |
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| 539 | #i #id #instr_list #H whd in ⊢ (??%%) whd in ⊢ (??(??%?)(??%?)) |
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[862] | 540 | >(index_of_internal_None … H) % |
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| 541 | qed. |
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[861] | 542 | |
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[866] | 543 | axiom tech_pc_sigma0_append: |
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| 544 | ∀preamble,instr_list,prefix,label,i,pc',code,pc,costs,costs'. |
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| 545 | Some … 〈pc,costs〉 = tech_pc_sigma0 〈preamble,prefix〉 → |
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| 546 | construct_costs 〈preamble,instr_list〉 … pc (λx.zero 16) (λx. zero 16) costs i = Some … 〈pc',code〉 → |
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| 547 | tech_pc_sigma0 〈preamble,prefix@[〈label,i〉]〉 = Some … 〈pc',costs'〉. |
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| 548 | |
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| 549 | axiom tech_pc_sigma0_append_None: |
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| 550 | ∀preamble,instr_list,prefix,i,pc,costs. |
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| 551 | Some … 〈pc,costs〉 = tech_pc_sigma0 〈preamble,prefix〉 → |
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| 552 | construct_costs 〈preamble,instr_list〉 … pc (λx.zero 16) (λx. zero 16) costs i = None … |
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| 553 | → False. |
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| 554 | |
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[871] | 555 | lemma BitVectorTrie_O: |
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| 556 | ∀A:Type[0].∀v:BitVectorTrie A 0.(∃w. v ≃ Leaf A w) ∨ v ≃ Stub A 0. |
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| 557 | #A #v generalize in match (refl … O) cases v in ⊢ (??%? → (?(??(λ_.?%%??)))(?%%??)) |
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| 558 | [ #w #_ %1 %[@w] % |
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| 559 | | #n #l #r #abs @⊥ // |
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| 560 | | #n #EQ %2 >EQ %] |
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| 561 | qed. |
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| 562 | |
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| 563 | lemma BitVectorTrie_Sn: |
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| 564 | ∀A:Type[0].∀n.∀v:BitVectorTrie A (S n).(∃l,r. v ≃ Node A n l r) ∨ v ≃ Stub A (S n). |
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| 565 | #A #n #v generalize in match (refl … (S n)) cases v in ⊢ (??%? → (?(??(λ_.??(λ_.?%%??))))%) |
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| 566 | [ #m #abs @⊥ // |
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| 567 | | #m #l #r #EQ %1 <(injective_S … EQ) %[@l] %[@r] // |
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| 568 | | #m #EQ %2 // ] |
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| 569 | qed. |
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| 570 | |
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| 571 | lemma lookup_prepare_trie_for_insertion_hit: |
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| 572 | ∀A:Type[0].∀a,v:A.∀n.∀b:BitVector n. |
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| 573 | lookup … b (prepare_trie_for_insertion … b v) a = v. |
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| 574 | #A #a #v #n #b elim b // #m #hd #tl #IH cases hd normalize // |
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| 575 | qed. |
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| 576 | |
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| 577 | lemma lookup_insert_hit: |
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| 578 | ∀A:Type[0].∀a,v:A.∀n.∀b:BitVector n.∀t:BitVectorTrie A n. |
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| 579 | lookup … b (insert … b v t) a = v. |
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| 580 | #A #a #v #n #b elim b -b -n // |
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| 581 | #n #hd #tl #IH #t cases(BitVectorTrie_Sn … t) |
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| 582 | [ * #l * #r #JMEQ >JMEQ cases hd normalize // |
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| 583 | | #JMEQ >JMEQ cases hd normalize @lookup_prepare_trie_for_insertion_hit ] |
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| 584 | qed. |
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| 585 | |
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| 586 | lemma BitVector_O: ∀v:BitVector 0. v ≃ VEmpty bool. |
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| 587 | #v generalize in match (refl … 0) cases v in ⊢ (??%? → ?%%??) // |
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| 588 | #n #hd #tl #abs @⊥ // |
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| 589 | qed. |
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| 590 | |
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| 591 | lemma BitVector_Sn: ∀n.∀v:BitVector (S n). |
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| 592 | ∃hd.∃tl.v ≃ VCons bool n hd tl. |
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| 593 | #n #v generalize in match (refl … (S n)) cases v in ⊢ (??%? → ??(λ_.??(λ_.?%%??))) |
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| 594 | [ #abs @⊥ // |
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| 595 | | #m #hd #tl #EQ <(injective_S … EQ) %[@hd] %[@tl] // ] |
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| 596 | qed. |
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| 597 | |
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| 598 | lemma lookup_prepare_trie_for_insertion_miss: |
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| 599 | ∀A:Type[0].∀a,v:A.∀n.∀c,b:BitVector n. |
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| 600 | (notb (eq_bv ? b c)) → lookup … b (prepare_trie_for_insertion … c v) a = a. |
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| 601 | #A #a #v #n #c elim c |
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| 602 | [ #b >(BitVector_O … b) normalize #abs @⊥ // |
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| 603 | | #m #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ |
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| 604 | cases hd cases hd' normalize |
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| 605 | [2,3: #_ cases tl' // |
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| 606 | |*: change with (bool_to_Prop (notb (eq_bv ???)) → ?) /2/ ]] |
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| 607 | qed. |
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| 608 | |
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| 609 | lemma lookup_insert_miss: |
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| 610 | ∀A:Type[0].∀a,v:A.∀n.∀c,b:BitVector n.∀t:BitVectorTrie A n. |
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| 611 | (notb (eq_bv ? b c)) → lookup … b (insert … c v t) a = lookup … b t a. |
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| 612 | #A #a #v #n #c elim c -c -n |
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| 613 | [ #b #t #DIFF @⊥ whd in DIFF; >(BitVector_O … b) in DIFF // |
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| 614 | | #n #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ |
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| 615 | #t cases(BitVectorTrie_Sn … t) |
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| 616 | [ * #l * #r #JMEQ >JMEQ cases hd cases hd' #H normalize in H; |
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| 617 | [1,4: change in H with (bool_to_Prop (notb (eq_bv ???))) ] normalize // @IH // |
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| 618 | | #JMEQ >JMEQ cases hd cases hd' #H normalize in H; |
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| 619 | [1,4: change in H with (bool_to_Prop (notb (eq_bv ???))) ] normalize |
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| 620 | [3,4: cases tl' // | *: @lookup_prepare_trie_for_insertion_miss //]]] |
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| 621 | qed. |
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| 622 | |
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[845] | 623 | definition build_maps' ≝ |
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| 624 | λpseudo_program. |
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| 625 | let 〈preamble,instr_list〉 ≝ pseudo_program in |
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| 626 | let result ≝ |
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[853] | 627 | foldl_strong |
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[848] | 628 | (option Identifier × pseudo_instruction) |
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[853] | 629 | (λpre. Σres:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))). |
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| 630 | let pre' ≝ 〈preamble,pre〉 in |
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| 631 | let 〈labels,pc_costs〉 ≝ res in |
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[866] | 632 | tech_pc_sigma0 pre' = Some … pc_costs ∧ |
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[861] | 633 | ∀id. occurs_exactly_once id pre → |
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| 634 | lookup ?? id labels (zero …) = sigma pre' (address_of_word_labels_code_mem pre id)) |
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[853] | 635 | instr_list |
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| 636 | (λprefix,i,tl,prf,t. |
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[848] | 637 | let 〈labels, pc_costs〉 ≝ t in |
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| 638 | let 〈program_counter, costs〉 ≝ pc_costs in |
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| 639 | let 〈label, i'〉 ≝ i in |
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| 640 | let labels ≝ |
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| 641 | match label with |
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| 642 | [ None ⇒ labels |
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| 643 | | Some label ⇒ |
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| 644 | let program_counter_bv ≝ bitvector_of_nat ? program_counter in |
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| 645 | insert ? ? label program_counter_bv labels |
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| 646 | ] |
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| 647 | in |
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[866] | 648 | match construct_costs 〈preamble,instr_list〉 program_counter (λx. zero ?) (λx. zero ?) costs i' with |
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[848] | 649 | [ None ⇒ |
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| 650 | let dummy ≝ 〈labels,pc_costs〉 in |
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| 651 | dummy |
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| 652 | | Some construct ⇒ 〈labels, construct〉 |
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| 653 | ] |
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[853] | 654 | ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 |
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[848] | 655 | in |
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| 656 | let 〈labels, pc_costs〉 ≝ result in |
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| 657 | let 〈pc, costs〉 ≝ pc_costs in |
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| 658 | 〈labels, costs〉. |
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[866] | 659 | [3: whd % // #id normalize in ⊢ (% → ?) #abs @⊥ // |
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| 660 | | whd cases construct in p3 #PC #CODE #JMEQ % |
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| 661 | [ @(tech_pc_sigma0_append ??????????? (jmeq_to_eq ??? JMEQ)) | #id #Hid ] |
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| 662 | | (* dummy case *) @⊥ |
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| 663 | @(tech_pc_sigma0_append_None ?? prefix ???? (jmeq_to_eq ??? p3)) ] |
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| 664 | [*: generalize in match (sig2 … t) whd in ⊢ (% → ?) |
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| 665 | >p whd in ⊢ (% → ?) >p1 * #IH0 #IH1 >IH0 // ] |
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| 666 | whd in ⊢ (??(????%?)?) -labels1; |
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| 667 | cases label in Hid |
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| 668 | [ #Hid whd in ⊢ (??(????%?)?) >IH1 -IH1 |
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| 669 | [ >(address_of_word_labels_code_mem_None … Hid) |
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| 670 | (* MANCA LEMMA: INDIRIZZO TROVATO NEL PROGRAMMA! *) |
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| 671 | | whd in Hid >occurs_exactly_once_None in Hid // ] |
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| 672 | | -label #label #Hid whd in ⊢ (??(????%?)?) |
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| 673 | |
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| 674 | ] |
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[853] | 675 | qed. |
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[848] | 676 | |
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| 677 | (* |
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[853] | 678 | (* |
---|
[848] | 679 | notation < "hvbox('let' 〈ident x,ident y〉 ≝ t 'in' s)" |
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| 680 | with precedence 10 |
---|
| 681 | for @{ match $t with [ pair ${ident x} ${ident y} ⇒ $s ] }. |
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| 682 | *) |
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| 683 | |
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| 684 | lemma build_maps_ok: |
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| 685 | ∀p:pseudo_assembly_program. |
---|
| 686 | let 〈labels,costs〉 ≝ build_maps' p in |
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| 687 | ∀pc. |
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| 688 | (nat_of_bitvector … pc) < length … (\snd p) → |
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| 689 | lookup ?? pc labels (zero …) = sigma p (\snd (fetch_pseudo_instruction (\snd p) pc)). |
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| 690 | #p cases p #preamble #instr_list |
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| 691 | elim instr_list |
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| 692 | [ whd #pc #abs normalize in abs; cases (not_le_Sn_O ?) [#H cases (H abs) ] |
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| 693 | | #hd #tl #IH |
---|
| 694 | whd in ⊢ (match % with [ _ ⇒ ?]) |
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| 695 | ] |
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| 696 | qed. |
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| 697 | *) |
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| 698 | |
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| 699 | (* |
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| 700 | lemma list_elim_rev: |
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| 701 | ∀A:Type[0].∀P:list A → Prop. |
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| 702 | P [ ] → (∀n,l. length l = n → P l → |
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| 703 | P [ ] → (∀l,a. P l → P (l@[a])) → |
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| 704 | ∀l. P l. |
---|
| 705 | #A #P |
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| 706 | qed.*) |
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| 707 | |
---|
| 708 | lemma rev_preserves_length: |
---|
| 709 | ∀A.∀l. length … (rev A l) = length … l. |
---|
| 710 | #A #l elim l |
---|
| 711 | [ % |
---|
| 712 | | #hd #tl #IH normalize >length_append normalize /2/ ] |
---|
| 713 | qed. |
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| 714 | |
---|
| 715 | lemma rev_append: |
---|
| 716 | ∀A.∀l1,l2. |
---|
| 717 | rev A (l1@l2) = rev A l2 @ rev A l1. |
---|
| 718 | #A #l1 elim l1 normalize // |
---|
| 719 | qed. |
---|
| 720 | |
---|
| 721 | lemma rev_rev: ∀A.∀l. rev … (rev A l) = l. |
---|
| 722 | #A #l elim l |
---|
| 723 | [ // |
---|
| 724 | | #hd #tl #IH normalize >rev_append normalize // ] |
---|
| 725 | qed. |
---|
| 726 | |
---|
| 727 | lemma split_len_Sn: |
---|
| 728 | ∀A:Type[0].∀l:list A.∀len. |
---|
| 729 | length … l = S len → |
---|
| 730 | Σl'.Σa. l = l'@[a] ∧ length … l' = len. |
---|
| 731 | #A #l elim l |
---|
| 732 | [ normalize #len #abs destruct |
---|
| 733 | | #hd #tl #IH #len |
---|
| 734 | generalize in match (rev_rev … tl) |
---|
| 735 | cases (rev A tl) in ⊢ (??%? → ?) |
---|
| 736 | [ #H <H normalize #EQ % [@[ ]] % [@hd] normalize /2/ |
---|
| 737 | | #a #l' #H <H normalize #EQ |
---|
| 738 | %[@(hd::rev … l')] %[@a] % // |
---|
| 739 | >length_append in EQ #EQ normalize in EQ; normalize; |
---|
| 740 | generalize in match (injective_S … EQ) #EQ2 /2/ ]] |
---|
| 741 | qed. |
---|
| 742 | |
---|
| 743 | lemma list_elim_rev: |
---|
| 744 | ∀A:Type[0].∀P:list A → Type[0]. |
---|
| 745 | P [ ] → (∀l,a. P l → P (l@[a])) → |
---|
| 746 | ∀l. P l. |
---|
| 747 | #A #P #H1 #H2 #l |
---|
| 748 | generalize in match (refl … (length … l)) |
---|
| 749 | generalize in ⊢ (???% → ?) #n generalize in match l |
---|
| 750 | elim n |
---|
| 751 | [ #L cases L [ // | #x #w #abs (normalize in abs) @⊥ // ] |
---|
| 752 | | #m #IH #L #EQ |
---|
| 753 | cases (split_len_Sn … EQ) #l' * #a * /3/ ] |
---|
| 754 | qed. |
---|
| 755 | |
---|
| 756 | axiom is_prefix: ∀A:Type[0]. list A → list A → Prop. |
---|
| 757 | axiom prefix_of_append: |
---|
| 758 | ∀A:Type[0].∀l,l1,l2:list A. |
---|
| 759 | is_prefix … l l1 → is_prefix … l (l1@l2). |
---|
[853] | 760 | axiom prefix_reflexive: ∀A,l. is_prefix A l l. |
---|
| 761 | axiom nil_prefix: ∀A,l. is_prefix A [ ] l. |
---|
[848] | 762 | |
---|
[853] | 763 | record Propify (A:Type[0]) : Type[0] (*Prop*) ≝ { in_propify: A }. |
---|
[849] | 764 | |
---|
| 765 | definition Propify_elim: ∀A. ∀P:Prop. (A → P) → (Propify A → P) ≝ |
---|
| 766 | λA,P,H,x. match x with [ mk_Propify p ⇒ H p ]. |
---|
| 767 | |
---|
[850] | 768 | definition app ≝ |
---|
| 769 | λA:Type[0].λl1:Propify (list A).λl2:list A. |
---|
| 770 | match l1 with |
---|
| 771 | [ mk_Propify l1 ⇒ mk_Propify … (l1@l2) ]. |
---|
| 772 | |
---|
| 773 | lemma app_nil: ∀A,l1. app A l1 [ ] = l1. |
---|
| 774 | #A * /3/ |
---|
| 775 | qed. |
---|
| 776 | |
---|
| 777 | lemma app_assoc: ∀A,l1,l2,l3. app A (app A l1 l2) l3 = app A l1 (l2@l3). |
---|
| 778 | #A * #l1 normalize // |
---|
| 779 | qed. |
---|
| 780 | |
---|
| 781 | let rec foldli (A: Type[0]) (B: Propify (list A) → Type[0]) |
---|
| 782 | (f: ∀prefix. B prefix → ∀x.B (app … prefix [x])) |
---|
| 783 | (prefix: Propify (list A)) (b: B prefix) (l: list A) on l : |
---|
| 784 | B (app … prefix l) ≝ |
---|
[849] | 785 | match l with |
---|
[850] | 786 | [ nil ⇒ ? (* b *) |
---|
| 787 | | cons hd tl ⇒ ? (*foldli A B f (prefix@[hd]) (f prefix b hd) tl*) |
---|
[849] | 788 | ]. |
---|
[850] | 789 | [ applyS b |
---|
| 790 | | <(app_assoc ?? [hd]) @(foldli A B f (app … prefix [hd]) (f prefix b hd) tl) ] |
---|
| 791 | qed. |
---|
[849] | 792 | |
---|
[850] | 793 | (* |
---|
| 794 | let rec foldli (A: Type[0]) (B: list A → Type[0]) (f: ∀prefix. B prefix → ∀x. B (prefix@[x])) |
---|
| 795 | (prefix: list A) (b: B prefix) (l: list A) on l : B (prefix@l) ≝ |
---|
| 796 | match l with |
---|
| 797 | [ nil ⇒ ? (* b *) |
---|
| 798 | | cons hd tl ⇒ |
---|
| 799 | ? (*foldli A B f (prefix@[hd]) (f prefix b hd) tl*) |
---|
| 800 | ]. |
---|
| 801 | [ applyS b |
---|
| 802 | | applyS (foldli A B f (prefix@[hd]) (f prefix b hd) tl) ] |
---|
| 803 | qed. |
---|
| 804 | *) |
---|
| 805 | |
---|
| 806 | definition foldll: |
---|
| 807 | ∀A:Type[0].∀B: Propify (list A) → Type[0]. |
---|
| 808 | (∀prefix. B prefix → ∀x. B (app … prefix [x])) → |
---|
| 809 | B (mk_Propify … []) → ∀l: list A. B (mk_Propify … l) |
---|
| 810 | ≝ λA,B,f. foldli A B f (mk_Propify … [ ]). |
---|
| 811 | |
---|
[853] | 812 | axiom is_pprefix: ∀A:Type[0]. Propify (list A) → list A → Prop. |
---|
| 813 | axiom pprefix_of_append: |
---|
| 814 | ∀A:Type[0].∀l,l1,l2. |
---|
| 815 | is_pprefix A l l1 → is_pprefix A l (l1@l2). |
---|
| 816 | axiom pprefix_reflexive: ∀A,l. is_pprefix A (mk_Propify … l) l. |
---|
| 817 | axiom nil_pprefix: ∀A,l. is_pprefix A (mk_Propify … [ ]) l. |
---|
| 818 | |
---|
| 819 | |
---|
| 820 | axiom foldll': |
---|
| 821 | ∀A:Type[0].∀l: list A. |
---|
| 822 | ∀B: ∀prefix:Propify (list A). is_pprefix ? prefix l → Type[0]. |
---|
| 823 | (∀prefix,proof. B prefix proof → ∀x,proof'. B (app … prefix [x]) proof') → |
---|
| 824 | B (mk_Propify … [ ]) (nil_pprefix …) → B (mk_Propify … l) (pprefix_reflexive … l). |
---|
| 825 | #A #l #B |
---|
| 826 | generalize in match (foldll A (λprefix. is_pprefix ? prefix l)) #HH |
---|
| 827 | |
---|
| 828 | |
---|
| 829 | #H #acc |
---|
| 830 | @foldll |
---|
| 831 | [ |
---|
| 832 | | |
---|
| 833 | ] |
---|
| 834 | |
---|
| 835 | ≝ λA,B,f. foldli A B f (mk_Propify … [ ]). |
---|
| 836 | |
---|
| 837 | |
---|
| 838 | (* |
---|
| 839 | record subset (A:Type[0]) (P: A → Prop): Type[0] ≝ |
---|
| 840 | { subset_wit:> A; |
---|
| 841 | subset_proof: P subset_wit |
---|
| 842 | }. |
---|
| 843 | *) |
---|
| 844 | |
---|
[848] | 845 | definition build_maps' ≝ |
---|
| 846 | λpseudo_program. |
---|
| 847 | let 〈preamble,instr_list〉 ≝ pseudo_program in |
---|
| 848 | let result ≝ |
---|
[853] | 849 | foldll |
---|
| 850 | (option Identifier × pseudo_instruction) |
---|
| 851 | (λprefix. |
---|
| 852 | Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))). |
---|
| 853 | match prefix return λ_.Prop with [mk_Propify prefix ⇒ tech_pc_sigma0 〈preamble,prefix〉 ≠ None ?]) |
---|
| 854 | (λprefix,t,i. |
---|
[845] | 855 | let 〈labels, pc_costs〉 ≝ t in |
---|
| 856 | let 〈program_counter, costs〉 ≝ pc_costs in |
---|
| 857 | let 〈label, i'〉 ≝ i in |
---|
| 858 | let labels ≝ |
---|
| 859 | match label with |
---|
| 860 | [ None ⇒ labels |
---|
| 861 | | Some label ⇒ |
---|
| 862 | let program_counter_bv ≝ bitvector_of_nat ? program_counter in |
---|
| 863 | insert ? ? label program_counter_bv labels |
---|
| 864 | ] |
---|
| 865 | in |
---|
| 866 | match construct_costs pseudo_program program_counter (λx. zero ?) (λx. zero ?) costs i' with |
---|
[848] | 867 | [ None ⇒ |
---|
| 868 | let dummy ≝ 〈labels,pc_costs〉 in |
---|
| 869 | dummy |
---|
[845] | 870 | | Some construct ⇒ 〈labels, construct〉 |
---|
| 871 | ] |
---|
[853] | 872 | ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 instr_list |
---|
[845] | 873 | in |
---|
| 874 | let 〈labels, pc_costs〉 ≝ result in |
---|
| 875 | let 〈pc, costs〉 ≝ pc_costs in |
---|
| 876 | 〈labels, costs〉. |
---|
[853] | 877 | [ |
---|
| 878 | | @⊥ |
---|
| 879 | | normalize % // |
---|
| 880 | ] |
---|
| 881 | qed. |
---|
[850] | 882 | |
---|
| 883 | definition build_maps' ≝ |
---|
| 884 | λpseudo_program. |
---|
| 885 | let 〈preamble,instr_list〉 ≝ pseudo_program in |
---|
| 886 | let result ≝ |
---|
| 887 | foldl |
---|
| 888 | (Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))). |
---|
| 889 | ∃instr_list_prefix. is_prefix ? instr_list_prefix instr_list ∧ |
---|
| 890 | tech_pc_sigma0 〈preamble,instr_list_prefix〉 = Some ? (\fst (\snd t))) |
---|
| 891 | (Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix. |
---|
| 892 | let instr_list_prefix' ≝ instr_list_prefix @ [i] in |
---|
| 893 | is_prefix ? instr_list_prefix' instr_list → |
---|
| 894 | tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ?) |
---|
| 895 | (λt: Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))). |
---|
| 896 | ∃instr_list_prefix. is_prefix ? instr_list_prefix instr_list ∧ |
---|
| 897 | tech_pc_sigma0 〈preamble,instr_list_prefix〉 = Some ? (\fst (\snd t)). |
---|
| 898 | λi: Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix. |
---|
| 899 | let instr_list_prefix' ≝ instr_list_prefix @ [i] in |
---|
| 900 | is_prefix ? instr_list_prefix' instr_list → |
---|
| 901 | tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ? . |
---|
| 902 | let 〈labels, pc_costs〉 ≝ t in |
---|
| 903 | let 〈program_counter, costs〉 ≝ pc_costs in |
---|
| 904 | let 〈label, i'〉 ≝ i in |
---|
| 905 | let labels ≝ |
---|
| 906 | match label with |
---|
| 907 | [ None ⇒ labels |
---|
| 908 | | Some label ⇒ |
---|
| 909 | let program_counter_bv ≝ bitvector_of_nat ? program_counter in |
---|
| 910 | insert ? ? label program_counter_bv labels |
---|
| 911 | ] |
---|
| 912 | in |
---|
| 913 | match construct_costs pseudo_program program_counter (λx. zero ?) (λx. zero ?) costs i' with |
---|
| 914 | [ None ⇒ |
---|
| 915 | let dummy ≝ 〈labels,pc_costs〉 in |
---|
| 916 | dummy |
---|
| 917 | | Some construct ⇒ 〈labels, construct〉 |
---|
| 918 | ] |
---|
| 919 | ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 ?(*instr_list*) |
---|
| 920 | in |
---|
| 921 | let 〈labels, pc_costs〉 ≝ result in |
---|
| 922 | let 〈pc, costs〉 ≝ pc_costs in |
---|
| 923 | 〈labels, costs〉. |
---|
[848] | 924 | [4: @(list_elim_rev ? |
---|
| 925 | (λinstr_list. list ( |
---|
| 926 | (Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix. |
---|
| 927 | let instr_list_prefix' ≝ instr_list_prefix @ [i] in |
---|
| 928 | is_prefix ? instr_list_prefix' instr_list → |
---|
| 929 | tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ?))) |
---|
| 930 | ?? instr_list) (* CSC: BAD ORDER FOR CODE EXTRACTION *) |
---|
| 931 | [ @[ ] |
---|
| 932 | | #l' #a #limage %2 |
---|
| 933 | [ %[@a] #PREFIX #PREFIX_OK |
---|
| 934 | | (* CSC: EVEN WORST CODE FOR EXTRACTION: WE SHOULD STRENGTHEN |
---|
| 935 | THE INDUCTION HYPOTHESIS INSTEAD *) |
---|
| 936 | elim limage |
---|
| 937 | [ %1 |
---|
| 938 | | #HD #TL #IH @(?::IH) cases HD #ELEM #K1 %[@ELEM] #K2 #K3 |
---|
| 939 | @K1 @(prefix_of_append ???? K3) |
---|
| 940 | ] |
---|
| 941 | ] |
---|
| 942 | |
---|
| 943 | |
---|
| 944 | |
---|
| 945 | |
---|
| 946 | cases t in c2 ⊢ % #t' * #LIST_PREFIX * #H1t' #H2t' #HJMt' |
---|
[845] | 947 | % [@ (LIST_PREFIX @ [i])] % |
---|
| 948 | [ cases (sig2 … i LIST_PREFIX) #K1 #K2 @K1 |
---|
| 949 | | (* DOABLE IN PRINCIPLE *) |
---|
| 950 | ] |
---|
| 951 | | (* assert false case *) |
---|
| 952 | |3: % [@ ([ ])] % [2: % | (* DOABLE *)] |
---|
[848] | 953 | | |
---|
[845] | 954 | |
---|
[825] | 955 | let rec encoding_check (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word) |
---|
| 956 | (encoding: list Byte) on encoding: Prop ≝ |
---|
| 957 | match encoding with |
---|
| 958 | [ nil ⇒ final_pc = pc |
---|
| 959 | | cons hd tl ⇒ |
---|
| 960 | let 〈new_pc, byte〉 ≝ next code_memory pc in |
---|
| 961 | hd = byte ∧ encoding_check code_memory new_pc final_pc tl |
---|
| 962 | ]. |
---|
| 963 | |
---|
[826] | 964 | definition assembly_specification: |
---|
[838] | 965 | ∀assembly_program: pseudo_assembly_program. |
---|
[825] | 966 | ∀code_mem: BitVectorTrie Byte 16. Prop ≝ |
---|
| 967 | λpseudo_assembly_program. |
---|
| 968 | λcode_mem. |
---|
| 969 | ∀pc: Word. |
---|
| 970 | let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in |
---|
| 971 | let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in |
---|
[846] | 972 | let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in |
---|
| 973 | let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in |
---|
[838] | 974 | let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program |
---|
[846] | 975 | (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in |
---|
[838] | 976 | match pre_assembled with |
---|
| 977 | [ None ⇒ True |
---|
| 978 | | Some pc_code ⇒ |
---|
| 979 | let 〈new_pc,code〉 ≝ pc_code in |
---|
[846] | 980 | encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ]. |
---|
[826] | 981 | |
---|
[827] | 982 | axiom assembly_meets_specification: |
---|
[826] | 983 | ∀pseudo_assembly_program. |
---|
| 984 | match assembly pseudo_assembly_program with |
---|
| 985 | [ None ⇒ True |
---|
| 986 | | Some code_mem_cost ⇒ |
---|
| 987 | let 〈code_mem, cost〉 ≝ code_mem_cost in |
---|
[838] | 988 | assembly_specification pseudo_assembly_program (load_code_memory code_mem) |
---|
[826] | 989 | ]. |
---|
[838] | 990 | (* |
---|
[826] | 991 | # PROGRAM |
---|
| 992 | [ cases PROGRAM |
---|
| 993 | # PREAMBLE |
---|
| 994 | # INSTR_LIST |
---|
| 995 | elim INSTR_LIST |
---|
| 996 | [ whd |
---|
| 997 | whd in ⊢ (∀_. %) |
---|
| 998 | # PC |
---|
| 999 | whd |
---|
| 1000 | | # INSTR |
---|
| 1001 | # INSTR_LIST_TL |
---|
| 1002 | # H |
---|
| 1003 | whd |
---|
[832] | 1004 | whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
---|
[826] | 1005 | ] |
---|
| 1006 | | cases not_implemented |
---|
[827] | 1007 | ] *) |
---|
| 1008 | |
---|
| 1009 | definition status_of_pseudo_status: PseudoStatus → option Status ≝ |
---|
| 1010 | λps. |
---|
[828] | 1011 | let pap ≝ code_memory … ps in |
---|
| 1012 | match assembly pap with |
---|
| 1013 | [ None ⇒ None … |
---|
| 1014 | | Some p ⇒ |
---|
| 1015 | let cm ≝ load_code_memory (\fst p) in |
---|
[846] | 1016 | let pc ≝ sigma' pap (program_counter ? ps) in |
---|
[828] | 1017 | Some … |
---|
| 1018 | (mk_PreStatus (BitVectorTrie Byte 16) |
---|
| 1019 | cm |
---|
| 1020 | (low_internal_ram … ps) |
---|
| 1021 | (high_internal_ram … ps) |
---|
| 1022 | (external_ram … ps) |
---|
| 1023 | pc |
---|
| 1024 | (special_function_registers_8051 … ps) |
---|
| 1025 | (special_function_registers_8052 … ps) |
---|
| 1026 | (p1_latch … ps) |
---|
| 1027 | (p3_latch … ps) |
---|
| 1028 | (clock … ps)) ]. |
---|
[847] | 1029 | |
---|
[839] | 1030 | definition write_at_stack_pointer': |
---|
[834] | 1031 | ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝ |
---|
| 1032 | λM: Type[0]. |
---|
| 1033 | λs: PreStatus M. |
---|
| 1034 | λv: Byte. |
---|
| 1035 | let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in |
---|
| 1036 | let bit_zero ≝ get_index_v… nu O ? in |
---|
| 1037 | let bit_1 ≝ get_index_v… nu 1 ? in |
---|
| 1038 | let bit_2 ≝ get_index_v… nu 2 ? in |
---|
| 1039 | let bit_3 ≝ get_index_v… nu 3 ? in |
---|
| 1040 | if bit_zero then |
---|
| 1041 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
| 1042 | v (low_internal_ram ? s) in |
---|
| 1043 | set_low_internal_ram ? s memory |
---|
| 1044 | else |
---|
| 1045 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
| 1046 | v (high_internal_ram ? s) in |
---|
| 1047 | set_high_internal_ram ? s memory. |
---|
| 1048 | [ cases l0 % |
---|
[847] | 1049 | |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ] |
---|
[834] | 1050 | qed. |
---|
| 1051 | |
---|
[833] | 1052 | definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus. |
---|
| 1053 | Σps':PseudoStatus.(code_memory … ps = code_memory … ps') |
---|
| 1054 | ≝ |
---|
| 1055 | λticks_of. |
---|
| 1056 | λs. |
---|
| 1057 | let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in |
---|
| 1058 | let ticks ≝ ticks_of (program_counter ? s) in |
---|
| 1059 | let s ≝ set_clock ? s (clock ? s + ticks) in |
---|
| 1060 | let s ≝ set_program_counter ? s pc in |
---|
| 1061 | match instr with |
---|
| 1062 | [ Instruction instr ⇒ |
---|
[834] | 1063 | execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s |
---|
| 1064 | | Comment cmt ⇒ s |
---|
| 1065 | | Cost cst ⇒ s |
---|
| 1066 | | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp) |
---|
[833] | 1067 | | Call call ⇒ |
---|
| 1068 | let a ≝ address_of_word_labels s call in |
---|
| 1069 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
| 1070 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
| 1071 | let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in |
---|
[839] | 1072 | let s ≝ write_at_stack_pointer' ? s pc_bl in |
---|
[833] | 1073 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
| 1074 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
[839] | 1075 | let s ≝ write_at_stack_pointer' ? s pc_bu in |
---|
[834] | 1076 | set_program_counter ? s a |
---|
[833] | 1077 | | Mov dptr ident ⇒ |
---|
[834] | 1078 | set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr |
---|
| 1079 | ]. |
---|
[833] | 1080 | [ |
---|
| 1081 | |2,3,4: % |
---|
[834] | 1082 | | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) % |
---|
| 1083 | | |
---|
| 1084 | | % |
---|
| 1085 | ] |
---|
[839] | 1086 | cases not_implemented |
---|
[834] | 1087 | qed. |
---|
[833] | 1088 | |
---|
[839] | 1089 | (* |
---|
[829] | 1090 | lemma execute_code_memory_unchanged: |
---|
| 1091 | ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps). |
---|
| 1092 | #ticks #ps whd in ⊢ (??? (??%)) |
---|
| 1093 | cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps)) |
---|
| 1094 | (program_counter pseudo_assembly_program ps)) #instr #pc |
---|
| 1095 | whd in ⊢ (??? (??%)) cases instr |
---|
| 1096 | [ #pre cases pre |
---|
| 1097 | [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
| 1098 | cases (split ????) #z1 #z2 % |
---|
[831] | 1099 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
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| 1100 | cases (split ????) #z1 #z2 % |
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| 1101 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
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| 1102 | cases (split ????) #z1 #z2 % |
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| 1103 | | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x |
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[833] | 1104 | [ #x1 whd in ⊢ (??? (??%)) |
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[829] | 1105 | | *: cases not_implemented |
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| 1106 | ] |
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| 1107 | | #comment % |
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| 1108 | | #cost % |
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| 1109 | | #label % |
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[831] | 1110 | | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%)) |
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| 1111 | cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2 |
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| 1112 | whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2 |
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| 1113 | whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%)) |
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| 1114 | (* CSC: ??? *) |
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[829] | 1115 | | #dptr #label (* CSC: ??? *) |
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| 1116 | ] |
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| 1117 | cases not_implemented |
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| 1118 | qed. |
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[839] | 1119 | *) |
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[829] | 1120 | |
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[839] | 1121 | lemma status_of_pseudo_status_failure_depends_only_on_code_memory: |
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| 1122 | ∀ps,ps': PseudoStatus. |
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| 1123 | code_memory … ps = code_memory … ps' → |
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| 1124 | match status_of_pseudo_status ps with |
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| 1125 | [ None ⇒ status_of_pseudo_status ps' = None … |
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| 1126 | | Some _ ⇒ ∃w. status_of_pseudo_status ps' = Some … w |
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| 1127 | ]. |
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[846] | 1128 | #ps #ps' #H whd in ⊢ (mat |
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| 1129 | ch % with [ _ ⇒ ? | _ ⇒ ? ]) |
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[839] | 1130 | generalize in match (refl … (assembly (code_memory … ps))) |
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| 1131 | cases (assembly ?) in ⊢ (???% → %) |
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| 1132 | [ #K whd whd in ⊢ (??%?) <H >K % |
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| 1133 | | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ] |
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[846] | 1134 | qed.*) |
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| 1135 | |
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| 1136 | let rec encoding_check' (code_memory: BitVectorTrie Byte 16) (pc: Word) (encoding: list Byte) on encoding: Prop ≝ |
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| 1137 | match encoding with |
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| 1138 | [ nil ⇒ True |
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| 1139 | | cons hd tl ⇒ |
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| 1140 | let 〈new_pc, byte〉 ≝ next code_memory pc in |
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| 1141 | hd = byte ∧ encoding_check' code_memory new_pc tl |
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| 1142 | ]. |
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| 1143 | |
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[851] | 1144 | (* prove later *) |
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| 1145 | axiom test: |
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[846] | 1146 | ∀pc: Word. |
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| 1147 | ∀code_memory: BitVectorTrie Byte 16. |
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| 1148 | ∀i: instruction. |
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| 1149 | let assembled ≝ assembly1 i in |
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| 1150 | encoding_check' code_memory pc assembled → |
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| 1151 | let 〈instr_pc, ignore〉 ≝ fetch code_memory pc in |
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| 1152 | let 〈instr, pc〉 ≝ instr_pc in |
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| 1153 | instr = i. |
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[839] | 1154 | |
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| 1155 | lemma main_thm: |
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[827] | 1156 | ∀ticks_of. |
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| 1157 | ∀ps: PseudoStatus. |
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[829] | 1158 | match status_of_pseudo_status ps with [ None ⇒ True | Some s ⇒ |
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[827] | 1159 | let ps' ≝ execute_1_pseudo_instruction ticks_of ps in |
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[828] | 1160 | match status_of_pseudo_status ps' with [ None ⇒ True | Some s'' ⇒ |
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[827] | 1161 | let s' ≝ execute_1 s in |
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[829] | 1162 | s = s'']]. |
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| 1163 | #ticks_of #ps |
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| 1164 | whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ]) |
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| 1165 | cases (assembly (code_memory pseudo_assembly_program ps)) [%] * #cm #costs whd |
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| 1166 | whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ]) |
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[839] | 1167 | generalize in match (sig2 … (execute_1_pseudo_instruction' ticks_of ps)) |
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| 1168 | |
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[847] | 1169 | cases (status_of_pseudo_status (execute_1_pseudo_instruction ticks_of ps)) [%] #s'' whd |
---|