[3381] | 1 | include "Common.ma". |
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| 2 | |
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| 3 | inductive classification: Type[0] ≝ |
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| 4 | sequential: classification |
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| 5 | | conditional: nat → classification |
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| 6 | | call: ident → classification |
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| 7 | | ret: classification. |
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| 8 | |
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| 9 | record object_code_def: Type[1] ≝ |
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| 10 | { status: Type[0] |
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| 11 | ; trans: status → status |
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| 12 | ; max: nat |
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| 13 | ; pc: status → nat |
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| 14 | ; succ: nat → nat |
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| 15 | ; labelled: nat → nat → bool |
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| 16 | ; classify: nat → classification |
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| 17 | ; classification_ok1: |
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| 18 | ∀s. classify (pc s) = sequential → pc (trans s) = succ (pc s) |
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| 19 | ; code_closed: |
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| 20 | ∀pc. pc < max → classify pc = sequential → succ pc < max |
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| 21 | (* serve per jmp e call??? *) |
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| 22 | }. |
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| 23 | |
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| 24 | record abelian_monoid: Type[1] ≝ |
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| 25 | { carrier :> Type[0] |
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| 26 | ; op: carrier → carrier → carrier |
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| 27 | ; e: carrier |
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| 28 | ; neutral: ∀x. op … x e = x |
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| 29 | ; associative: ∀x,y,z. op … (op … x y) z = op … x (op … y z) |
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| 30 | ; commutative: ∀x,y. op … x y = op … y x |
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| 31 | }. |
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| 32 | |
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| 33 | instr_cost: nat → M |
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| 34 | |
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| 35 | k: n:nat → pc:nat → M ≝ |
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| 36 | match n with |
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| 37 | [ O ⇒ whatever |
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| 38 | | S n' ⇒ |
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| 39 | match classify pc with |
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| 40 | [ seq ⇒ |
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| 41 | if labelelled pc (succ pc) then |
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| 42 | instr_cost pc |
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| 43 | else |
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| 44 | instr_cost pc + k n' (succ pc) |
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| 45 | | call ⇒ |
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| 46 | if postlabelled pc then |
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| 47 | instr_cost pc |
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| 48 | else |
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| 49 | instr_cost pc + k n' (succ pc) |
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| 50 | | _ ⇒ instr_cost pc |
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| 51 | ]] |
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| 52 | |
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| 53 | theorem strong: |
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| 54 | ∀τ: so -pm→ sn. τ finisce con label o return → |
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| 55 | Σ_(pc → pc' ∈ τ) instr_cost pc = |
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| 56 | k max (pc s0) + Σ_(pc -L→ pc' ∈ τ senza l'ultima) K max pc'. |
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| 57 | |
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| 58 | definition x ≤ y ≝ ∃z. op x z = y. |
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| 59 | |
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| 60 | theorem weak: |
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| 61 | ∀τ: so -pm→ sn. τ estendibile fino a trovare una label o una ret. |
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| 62 | Σ_(pc → pc' ∈ τ) instr_cost pc ≤ |
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| 63 | k max (pc s0) + Σ_(pc -L→ pc' ∈ τ senza l'ultima se finisce con label o return labelled) K max pc'. |
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| 64 | |
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| 65 | theorem superweak: |
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| 66 | come strong o weak, ma |
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| 67 | 1. tutte le call sono postlabelled |
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| 68 | 2. il monoide non e' abeliano |
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| 69 | |
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| 70 | =================================================== |
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| 71 | |
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| 72 | record cost_monoid ≝ λ nat → |
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| 73 | |
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| 74 | let rec cost_from (pc: nat): |
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| 75 | |
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| 76 | inductive instr: Type[0] := |
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| 77 | | Iconst: nat → instr |
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| 78 | | Ivar: option ident → instr |
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| 79 | | Isetvar: option ident → instr |
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| 80 | | Iadd: instr |
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| 81 | | Isub: instr |
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| 82 | | Ijmp: nat → instr |
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| 83 | | Ibne: nat → instr |
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| 84 | | Ibge: nat → instr |
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| 85 | | Ihalt: instr |
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| 86 | | Iio: instr |
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| 87 | | Icall: fname → instr |
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| 88 | | Iret: fname → instr. |
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| 89 | |
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| 90 | definition programT ≝ fname → list instr. |
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| 91 | |
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| 92 | definition fetch: list instr → nat → option instr ≝ λl,n. nth_opt ? n l. |
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| 93 | |
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| 94 | definition stackT: Type[0] ≝ list nat. |
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| 95 | |
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| 96 | definition vmstate ≝ λS:storeT. (list instr) × nat × (stackT × S). |
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| 97 | |
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| 98 | definition pc: ∀S. vmstate S → nat ≝ λS,s. \snd (\fst s). |
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| 99 | |
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| 100 | inductive vmstep (p: programT) (S: storeT) : vmstate S → vmstate S → Prop := |
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| 101 | | vmstep_const: ∀c,pc,stk,s,n. fetch c pc = Some … (Iconst n) → |
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| 102 | vmstep … |
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| 103 | 〈c, pc, stk, s〉 |
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| 104 | 〈c, 1 + pc, n :: stk, s〉 |
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| 105 | | vmstep_var: ∀c,pc,stk,s,x. fetch c pc = Some … (Ivar x) → |
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| 106 | vmstep … |
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| 107 | 〈c, pc, stk, s〉 |
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| 108 | 〈c, 1 + pc, get … s x :: stk, s〉 |
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| 109 | | vmstep_setvar: ∀c,pc,stk,s,x,n. fetch c pc = Some … (Isetvar x) → |
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| 110 | vmstep … |
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| 111 | 〈c, pc, n :: stk, s〉 |
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| 112 | 〈c, 1 + pc, stk, set … s x n〉 |
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| 113 | | vmstep_add: ∀c,pc,stk,s,n1,n2. fetch c pc = Some … Iadd → |
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| 114 | vmstep … |
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| 115 | 〈c, pc, n2 :: n1 :: stk, s〉 |
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| 116 | 〈c, 1 + pc, (n1 + n2) :: stk, s〉 |
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| 117 | | vmstep_sub: ∀c,pc,stk,s,n1,n2. fetch c pc = Some … Isub → |
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| 118 | vmstep … |
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| 119 | 〈c, pc, n2 :: n1 :: stk, s〉 |
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| 120 | 〈c, 1 + pc, (n1 - n2) :: stk, s〉 |
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| 121 | | vmstep_bne: ∀c,pc,stk,s,ofs,n1,n2. fetch c pc = Some … (Ibne ofs) → |
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| 122 | let pc' ≝ if eqb n1 n2 then 1 + pc else 1 + pc + ofs in |
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| 123 | vmstep … |
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| 124 | 〈c, pc, n2 :: n1 :: stk, s〉 |
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| 125 | 〈c, pc', stk, s〉 |
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| 126 | | vmstep_bge: ∀c,pc,stk,s,ofs,n1,n2. fetch c pc = Some … (Ibge ofs) → |
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| 127 | let pc' ≝ if ltb n1 n2 then 1 + pc else 1 + pc + ofs in |
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| 128 | vmstep … |
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| 129 | 〈c, pc, n2 :: n1 :: stk, s〉 |
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| 130 | 〈c, pc', stk, s〉 |
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| 131 | | vmstep_branch: ∀c,pc,stk,s,ofs. fetch c pc = Some … (Ijmp ofs) → |
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| 132 | let pc' ≝ 1 + pc + ofs in |
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| 133 | vmstep … |
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| 134 | 〈c, pc, stk, s〉 |
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| 135 | 〈c, 1 + pc + ofs, stk, s〉 |
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| 136 | | vmstep_io: ∀c,pc,stk,s. fetch c pc = Some … Iio → |
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| 137 | vmstep … |
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| 138 | 〈c, pc, stk, s〉 |
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| 139 | 〈c, 1 + pc, stk, s〉. |
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| 140 | |
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| 141 | definition emitterT ≝ nat → nat → option label. |
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| 142 | |
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| 143 | definition vmlstep: ∀p: programT. ∀S: storeT. ∀emit: emitterT. |
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| 144 | vmstate S → vmstate S → list label → Prop ≝ |
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| 145 | λp,S,emitter,s1,s2,ll. ∀l. |
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| 146 | vmstep p S s1 s2 ∧ ll = [l] ∧ emitter (pc … s1) (pc … s2) = Some … l. |
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| 147 | |
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| 148 | (* |
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| 149 | Definition star_vm (lbl: Type) (c: code (instr_vm lbl)) := star (trans_vm lbl c). |
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| 150 | |
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| 151 | Definition term_vm_lbl (lbl: Type) (c: code (instr_vm lbl)) (s_init s_fin: store) (trace: list lbl) := |
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| 152 | exists pc, |
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| 153 | code_at c pc = Some (Ihalt lbl) /\ |
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| 154 | star_vm lbl c (0, nil, s_init) (pc, nil, s_fin) trace. |
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| 155 | |
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| 156 | Definition term_vm (c: code_vm) (s_init s_fin: store):= |
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| 157 | exists pc, |
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| 158 | code_at c pc = Some (Ihalt False) /\ |
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| 159 | star_vm False c (0, nil, s_init) (pc, nil, s_fin) nil. |
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| 160 | |
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| 161 | Definition term_vml (c: code_vml) (s_init s_fin: store) (trace: list label) := |
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| 162 | exists pc, |
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| 163 | code_at c pc = Some (Ihalt label) /\ |
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| 164 | star_vm label c (0, nil, s_init) (pc, nil, s_fin) trace. |
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| 165 | *) |
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