[719] | 1 | include "ASM/ASM.ma". |
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| 2 | include "ASM/BitVectorTrie.ma". |
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| 3 | include "ASM/Arithmetic.ma". |
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| 4 | include "ASM/Fetch.ma". |
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| 5 | include "ASM/Status.ma". |
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[1459] | 6 | include alias "basics/logic.ma". |
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[719] | 7 | |
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[820] | 8 | definition assembly_preinstruction ≝ |
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| 9 | λA: Type[0]. |
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| 10 | λaddr_of: A → Byte. (* relative *) |
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| 11 | λpre: preinstruction A. |
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| 12 | match pre with |
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| 13 | [ ADD addr1 addr2 ⇒ |
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[757] | 14 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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[719] | 15 | [ REGISTER r ⇒ λ_.[ ([[false;false;true;false;true]]) @@ r ] |
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| 16 | | DIRECT b1 ⇒ λ_.[ ([[false;false;true;false;false;true;false;true]]); b1 ] |
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| 17 | | INDIRECT i1 ⇒ λ_. [ ([[false;false;true;false;false;true;true;i1]]) ] |
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| 18 | | DATA b1 ⇒ λ_. [ ([[false;false;true;false;false;true;false;false]]) ; b1 ] |
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| 19 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 20 | | ADDC addr1 addr2 ⇒ |
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[757] | 21 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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[719] | 22 | [ REGISTER r ⇒ λ_.[ ([[false;false;true;true;true]]) @@ r ] |
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| 23 | | DIRECT b1 ⇒ λ_.[ ([[false;false;true;true;false;true;false;true]]); b1 ] |
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| 24 | | INDIRECT i1 ⇒ λ_. [ ([[false;false;true;true;false;true;true;i1]]) ] |
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| 25 | | DATA b1 ⇒ λ_. [ ([[false;false;true;true;false;true;false;false]]) ; b1 ] |
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| 26 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 27 | | ANL addrs ⇒ |
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| 28 | match addrs with |
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| 29 | [ inl addrs ⇒ match addrs with |
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| 30 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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[757] | 31 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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[719] | 32 | [ REGISTER r ⇒ λ_.[ ([[false;true;false;true;true]]) @@ r ] |
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| 33 | | DIRECT b1 ⇒ λ_.[ ([[false;true;false;true;false;true;false;true]]); b1 ] |
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| 34 | | INDIRECT i1 ⇒ λ_. [ ([[false;true;false;true;false;true;true;i1]]) ] |
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| 35 | | DATA b1 ⇒ λ_. [ ([[false;true;false;true;false;true;false;false]]) ; b1 ] |
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| 36 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 37 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 38 | let b1 ≝ |
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| 39 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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| 40 | [ DIRECT b1 ⇒ λ_.b1 |
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| 41 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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| 42 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;data]] x) → ? with |
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| 43 | [ ACC_A ⇒ λ_.[ ([[false;true;false;true;false;false;true;false]]) ; b1 ] |
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| 44 | | DATA b2 ⇒ λ_. [ ([[false;true;false;true;false;false;true;true]]) ; b1 ; b2 ] |
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| 45 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 46 | ] |
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| 47 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 48 | match addr2 return λx. bool_to_Prop (is_in ? [[bit_addr;n_bit_addr]] x) → ? with |
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| 49 | [ BIT_ADDR b1 ⇒ λ_.[ ([[true;false;false;false;false;false;true;false]]) ; b1 ] |
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| 50 | | N_BIT_ADDR b1 ⇒ λ_. [ ([[true;false;true;true;false;false;false;false]]) ; b1 ] |
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| 51 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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| 52 | | CLR addr ⇒ |
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| 53 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;carry;bit_addr]] x) → ? with |
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| 54 | [ ACC_A ⇒ λ_. |
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| 55 | [ ([[true; true; true; false; false; true; false; false]]) ] |
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| 56 | | CARRY ⇒ λ_. |
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| 57 | [ ([[true; true; false; false; false; false; true; true]]) ] |
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| 58 | | BIT_ADDR b1 ⇒ λ_. |
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| 59 | [ ([[true; true; false; false; false; false; true; false]]) ; b1 ] |
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| 60 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 61 | | CPL addr ⇒ |
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| 62 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;carry;bit_addr]] x) → ? with |
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| 63 | [ ACC_A ⇒ λ_. |
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| 64 | [ ([[true; true; true; true; false; true; false; false]]) ] |
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| 65 | | CARRY ⇒ λ_. |
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| 66 | [ ([[true; false; true; true; false; false; true; true]]) ] |
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| 67 | | BIT_ADDR b1 ⇒ λ_. |
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| 68 | [ ([[true; false; true; true; false; false; true; false]]) ; b1 ] |
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| 69 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 70 | | DA addr ⇒ |
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| 71 | [ ([[true; true; false; true; false; true; false; false]]) ] |
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| 72 | | DEC addr ⇒ |
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[757] | 73 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;registr;direct;indirect]] x) → ? with |
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[719] | 74 | [ ACC_A ⇒ λ_. |
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| 75 | [ ([[false; false; false; true; false; true; false; false]]) ] |
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| 76 | | REGISTER r ⇒ λ_. |
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| 77 | [ ([[false; false; false; true; true]]) @@ r ] |
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| 78 | | DIRECT b1 ⇒ λ_. |
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| 79 | [ ([[false; false; false; true; false; true; false; true]]); b1 ] |
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| 80 | | INDIRECT i1 ⇒ λ_. |
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| 81 | [ ([[false; false; false; true; false; true; true; i1]]) ] |
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| 82 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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[820] | 83 | | DJNZ addr1 addr2 ⇒ |
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| 84 | let b2 ≝ addr_of addr2 in |
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[757] | 85 | match addr1 return λx. bool_to_Prop (is_in ? [[registr;direct]] x) → ? with |
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[719] | 86 | [ REGISTER r ⇒ λ_. |
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| 87 | [ ([[true; true; false; true; true]]) @@ r ; b2 ] |
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| 88 | | DIRECT b1 ⇒ λ_. |
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| 89 | [ ([[true; true; false; true; false; true; false; true]]); b1; b2 ] |
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| 90 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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| 91 | | JC addr ⇒ |
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[820] | 92 | let b1 ≝ addr_of addr in |
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| 93 | [ ([[false; true; false; false; false; false; false; false]]); b1 ] |
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[719] | 94 | | JNC addr ⇒ |
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[820] | 95 | let b1 ≝ addr_of addr in |
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| 96 | [ ([[false; true; false; true; false; false; false; false]]); b1 ] |
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[719] | 97 | | JZ addr ⇒ |
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[820] | 98 | let b1 ≝ addr_of addr in |
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| 99 | [ ([[false; true; true; false; false; false; false; false]]); b1 ] |
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[719] | 100 | | JNZ addr ⇒ |
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[820] | 101 | let b1 ≝ addr_of addr in |
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| 102 | [ ([[false; true; true; true; false; false; false; false]]); b1 ] |
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[719] | 103 | | JB addr1 addr2 ⇒ |
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[820] | 104 | let b2 ≝ addr_of addr2 in |
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[719] | 105 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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| 106 | [ BIT_ADDR b1 ⇒ λ_. |
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| 107 | [ ([[false; false; true; false; false; false; false; false]]); b1; b2 ] |
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| 108 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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| 109 | | JNB addr1 addr2 ⇒ |
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[820] | 110 | let b2 ≝ addr_of addr2 in |
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[719] | 111 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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| 112 | [ BIT_ADDR b1 ⇒ λ_. |
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| 113 | [ ([[false; false; true; true; false; false; false; false]]); b1; b2 ] |
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| 114 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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| 115 | | JBC addr1 addr2 ⇒ |
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[820] | 116 | let b2 ≝ addr_of addr2 in |
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[719] | 117 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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| 118 | [ BIT_ADDR b1 ⇒ λ_. |
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| 119 | [ ([[false; false; false; true; false; false; false; false]]); b1; b2 ] |
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| 120 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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| 121 | | CJNE addrs addr3 ⇒ |
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[820] | 122 | let b3 ≝ addr_of addr3 in |
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[719] | 123 | match addrs with |
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| 124 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 125 | match addr2 return λx. bool_to_Prop (is_in ? [[direct;data]] x) → ? with |
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| 126 | [ DIRECT b1 ⇒ λ_. |
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| 127 | [ ([[true; false; true; true; false; true; false; true]]); b1; b3 ] |
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| 128 | | DATA b1 ⇒ λ_. |
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| 129 | [ ([[true; false; true; true; false; true; false; false]]); b1; b3 ] |
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| 130 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 131 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 132 | let b2 ≝ |
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| 133 | match addr2 return λx. bool_to_Prop (is_in ? [[data]] x) → ? with |
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| 134 | [ DATA b2 ⇒ λ_. b2 |
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| 135 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) in |
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[757] | 136 | match addr1 return λx. bool_to_Prop (is_in ? [[registr;indirect]] x) → list Byte with |
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[820] | 137 | [ REGISTER r ⇒ λ_. |
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[719] | 138 | [ ([[true; false; true; true; true]]) @@ r; b2; b3 ] |
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| 139 | | INDIRECT i1 ⇒ λ_. |
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| 140 | [ ([[true; false; true; true; false; true; true; i1]]); b2; b3 ] |
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[820] | 141 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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| 142 | ] |
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| 143 | | DIV addr1 addr2 ⇒ |
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| 144 | [ ([[true;false;false;false;false;true;false;false]]) ] |
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[719] | 145 | | INC addr ⇒ |
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[757] | 146 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;registr;direct;indirect;dptr]] x) → ? with |
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[719] | 147 | [ ACC_A ⇒ λ_. |
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| 148 | [ ([[false;false;false;false;false;true;false;false]]) ] |
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| 149 | | REGISTER r ⇒ λ_. |
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| 150 | [ ([[false;false;false;false;true]]) @@ r ] |
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| 151 | | DIRECT b1 ⇒ λ_. |
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| 152 | [ ([[false; false; false; false; false; true; false; true]]); b1 ] |
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| 153 | | INDIRECT i1 ⇒ λ_. |
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| 154 | [ ([[false; false; false; false; false; true; true; i1]]) ] |
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| 155 | | DPTR ⇒ λ_. |
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| 156 | [ ([[true;false;true;false;false;false;true;true]]) ] |
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| 157 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 158 | | MOV addrs ⇒ |
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| 159 | match addrs with |
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| 160 | [ inl addrs ⇒ |
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| 161 | match addrs with |
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| 162 | [ inl addrs ⇒ |
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| 163 | match addrs with |
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| 164 | [ inl addrs ⇒ |
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| 165 | match addrs with |
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| 166 | [ inl addrs ⇒ |
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| 167 | match addrs with |
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| 168 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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[757] | 169 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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[719] | 170 | [ REGISTER r ⇒ λ_.[ ([[true;true;true;false;true]]) @@ r ] |
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| 171 | | DIRECT b1 ⇒ λ_.[ ([[true;true;true;false;false;true;false;true]]); b1 ] |
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| 172 | | INDIRECT i1 ⇒ λ_. [ ([[true;true;true;false;false;true;true;i1]]) ] |
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| 173 | | DATA b1 ⇒ λ_. [ ([[false;true;true;true;false;true;false;false]]) ; b1 ] |
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| 174 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 175 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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[757] | 176 | match addr1 return λx. bool_to_Prop (is_in ? [[registr;indirect]] x) → ? with |
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[719] | 177 | [ REGISTER r ⇒ λ_. |
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| 178 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;direct;data]] x) → ? with |
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| 179 | [ ACC_A ⇒ λ_.[ ([[true;true;true;true;true]]) @@ r ] |
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| 180 | | DIRECT b1 ⇒ λ_.[ ([[true;false;true;false;true]]) @@ r; b1 ] |
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| 181 | | DATA b1 ⇒ λ_. [ ([[false;true;true;true;true]]) @@ r; b1 ] |
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| 182 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 183 | | INDIRECT i1 ⇒ λ_. |
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| 184 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;direct;data]] x) → ? with |
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| 185 | [ ACC_A ⇒ λ_.[ ([[true;true;true;true;false;true;true;i1]]) ] |
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| 186 | | DIRECT b1 ⇒ λ_.[ ([[true;false;true;false;false;true;true;i1]]); b1 ] |
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| 187 | | DATA b1 ⇒ λ_. [ ([[false;true;true;true;false;true;true;i1]]) ; b1 ] |
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| 188 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 189 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1)] |
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| 190 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 191 | let b1 ≝ |
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| 192 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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| 193 | [ DIRECT b1 ⇒ λ_. b1 |
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| 194 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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[757] | 195 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;registr;direct;indirect;data]] x) → ? with |
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[719] | 196 | [ ACC_A ⇒ λ_.[ ([[true;true;true;true;false;true;false;true]]); b1] |
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| 197 | | REGISTER r ⇒ λ_.[ ([[true;false;false;false;true]]) @@ r; b1 ] |
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| 198 | | DIRECT b2 ⇒ λ_.[ ([[true;false;false;false;false;true;false;true]]); b1; b2 ] |
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| 199 | | INDIRECT i1 ⇒ λ_. [ ([[true;false;false;false;false;true;true;i1]]); b1 ] |
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| 200 | | DATA b2 ⇒ λ_. [ ([[false;true;true;true;false;true;false;true]]); b1; b2 ] |
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| 201 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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| 202 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 203 | match addr2 return λx. bool_to_Prop (is_in ? [[data16]] x) → ? with |
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| 204 | [ DATA16 w ⇒ λ_. |
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[938] | 205 | let b1_b2 ≝ split ? 8 8 w in |
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| 206 | let b1 ≝ \fst b1_b2 in |
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[1040] | 207 | let b2 ≝ \snd b1_b2 in |
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[719] | 208 | [ ([[true;false;false;true;false;false;false;false]]); b1; b2] |
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| 209 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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| 210 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 211 | match addr2 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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| 212 | [ BIT_ADDR b1 ⇒ λ_. |
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| 213 | [ ([[true;false;true;false;false;false;true;false]]); b1 ] |
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| 214 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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| 215 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 216 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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| 217 | [ BIT_ADDR b1 ⇒ λ_. |
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| 218 | [ ([[true;false;false;true;false;false;true;false]]); b1 ] |
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| 219 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1)] |
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| 220 | | MOVX addrs ⇒ |
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| 221 | match addrs with |
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| 222 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 223 | match addr2 return λx. bool_to_Prop (is_in ? [[ext_indirect;ext_indirect_dptr]] x) → ? with |
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| 224 | [ EXT_INDIRECT i1 ⇒ λ_. |
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| 225 | [ ([[true;true;true;false;false;false;true;i1]]) ] |
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| 226 | | EXT_INDIRECT_DPTR ⇒ λ_. |
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| 227 | [ ([[true;true;true;false;false;false;false;false]]) ] |
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| 228 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 229 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 230 | match addr1 return λx. bool_to_Prop (is_in ? [[ext_indirect;ext_indirect_dptr]] x) → ? with |
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| 231 | [ EXT_INDIRECT i1 ⇒ λ_. |
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| 232 | [ ([[true;true;true;true;false;false;true;i1]]) ] |
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| 233 | | EXT_INDIRECT_DPTR ⇒ λ_. |
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| 234 | [ ([[true;true;true;true;false;false;false;false]]) ] |
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| 235 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1)] |
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| 236 | | MUL addr1 addr2 ⇒ |
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| 237 | [ ([[true;false;true;false;false;true;false;false]]) ] |
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| 238 | | NOP ⇒ |
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| 239 | [ ([[false;false;false;false;false;false;false;false]]) ] |
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| 240 | | ORL addrs ⇒ |
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| 241 | match addrs with |
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| 242 | [ inl addrs ⇒ |
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| 243 | match addrs with |
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| 244 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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[757] | 245 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;data;direct;indirect]] x) → ? with |
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[719] | 246 | [ REGISTER r ⇒ λ_.[ ([[false;true;false;false;true]]) @@ r ] |
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| 247 | | DIRECT b1 ⇒ λ_.[ ([[false;true;false;false;false;true;false;true]]); b1 ] |
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| 248 | | INDIRECT i1 ⇒ λ_. [ ([[false;true;false;false;false;true;true;i1]]) ] |
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| 249 | | DATA b1 ⇒ λ_. [ ([[false;true;false;false;false;true;false;false]]) ; b1 ] |
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| 250 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 251 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 252 | let b1 ≝ |
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| 253 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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| 254 | [ DIRECT b1 ⇒ λ_. b1 |
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| 255 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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| 256 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;data]] x) → ? with |
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| 257 | [ ACC_A ⇒ λ_. |
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| 258 | [ ([[false;true;false;false;false;false;true;false]]); b1 ] |
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| 259 | | DATA b2 ⇒ λ_. |
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| 260 | [ ([[false;true;false;false;false;false;true;true]]); b1; b2 ] |
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| 261 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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| 262 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 263 | match addr2 return λx. bool_to_Prop (is_in ? [[bit_addr;n_bit_addr]] x) → ? with |
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| 264 | [ BIT_ADDR b1 ⇒ λ_. |
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| 265 | [ ([[false;true;true;true;false;false;true;false]]); b1 ] |
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| 266 | | N_BIT_ADDR b1 ⇒ λ_. |
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| 267 | [ ([[true;false;true;false;false;false;false;false]]); b1 ] |
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| 268 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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| 269 | | POP addr ⇒ |
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| 270 | match addr return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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| 271 | [ DIRECT b1 ⇒ λ_. |
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| 272 | [ ([[true;true;false;true;false;false;false;false]]) ; b1 ] |
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| 273 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 274 | | PUSH addr ⇒ |
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| 275 | match addr return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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| 276 | [ DIRECT b1 ⇒ λ_. |
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| 277 | [ ([[true;true;false;false;false;false;false;false]]) ; b1 ] |
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| 278 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 279 | | RET ⇒ |
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| 280 | [ ([[false;false;true;false;false;false;true;false]]) ] |
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| 281 | | RETI ⇒ |
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| 282 | [ ([[false;false;true;true;false;false;true;false]]) ] |
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| 283 | | RL addr ⇒ |
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| 284 | [ ([[false;false;true;false;false;false;true;true]]) ] |
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| 285 | | RLC addr ⇒ |
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| 286 | [ ([[false;false;true;true;false;false;true;true]]) ] |
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| 287 | | RR addr ⇒ |
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| 288 | [ ([[false;false;false;false;false;false;true;true]]) ] |
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| 289 | | RRC addr ⇒ |
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| 290 | [ ([[false;false;false;true;false;false;true;true]]) ] |
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| 291 | | SETB addr ⇒ |
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| 292 | match addr return λx. bool_to_Prop (is_in ? [[carry;bit_addr]] x) → ? with |
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| 293 | [ CARRY ⇒ λ_. |
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| 294 | [ ([[true;true;false;true;false;false;true;true]]) ] |
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| 295 | | BIT_ADDR b1 ⇒ λ_. |
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| 296 | [ ([[true;true;false;true;false;false;true;false]]); b1 ] |
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| 297 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 298 | | SUBB addr1 addr2 ⇒ |
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[757] | 299 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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[719] | 300 | [ REGISTER r ⇒ λ_. |
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| 301 | [ ([[true;false;false;true;true]]) @@ r ] |
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| 302 | | DIRECT b1 ⇒ λ_. |
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| 303 | [ ([[true;false;false;true;false;true;false;true]]); b1] |
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| 304 | | INDIRECT i1 ⇒ λ_. |
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| 305 | [ ([[true;false;false;true;false;true;true;i1]]) ] |
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| 306 | | DATA b1 ⇒ λ_. |
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| 307 | [ ([[true;false;false;true;false;true;false;false]]); b1] |
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| 308 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 309 | | SWAP addr ⇒ |
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| 310 | [ ([[true;true;false;false;false;true;false;false]]) ] |
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| 311 | | XCH addr1 addr2 ⇒ |
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[757] | 312 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect]] x) → ? with |
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[719] | 313 | [ REGISTER r ⇒ λ_. |
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| 314 | [ ([[true;true;false;false;true]]) @@ r ] |
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| 315 | | DIRECT b1 ⇒ λ_. |
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| 316 | [ ([[true;true;false;false;false;true;false;true]]); b1] |
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| 317 | | INDIRECT i1 ⇒ λ_. |
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| 318 | [ ([[true;true;false;false;false;true;true;i1]]) ] |
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| 319 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 320 | | XCHD addr1 addr2 ⇒ |
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| 321 | match addr2 return λx. bool_to_Prop (is_in ? [[indirect]] x) → ? with |
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| 322 | [ INDIRECT i1 ⇒ λ_. |
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| 323 | [ ([[true;true;false;true;false;true;true;i1]]) ] |
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| 324 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 325 | | XRL addrs ⇒ |
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| 326 | match addrs with |
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| 327 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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[757] | 328 | match addr2 return λx. bool_to_Prop (is_in ? [[data;registr;direct;indirect]] x) → ? with |
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[719] | 329 | [ REGISTER r ⇒ λ_. |
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| 330 | [ ([[false;true;true;false;true]]) @@ r ] |
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| 331 | | DIRECT b1 ⇒ λ_. |
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| 332 | [ ([[false;true;true;false;false;true;false;true]]); b1] |
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| 333 | | INDIRECT i1 ⇒ λ_. |
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| 334 | [ ([[false;true;true;false;false;true;true;i1]]) ] |
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| 335 | | DATA b1 ⇒ λ_. |
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| 336 | [ ([[false;true;true;false;false;true;false;false]]); b1] |
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| 337 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 338 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 339 | let b1 ≝ |
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| 340 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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| 341 | [ DIRECT b1 ⇒ λ_. b1 |
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| 342 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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| 343 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;data]] x) → ? with |
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| 344 | [ ACC_A ⇒ λ_. |
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| 345 | [ ([[false;true;true;false;false;false;true;false]]); b1 ] |
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| 346 | | DATA b2 ⇒ λ_. |
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| 347 | [ ([[false;true;true;false;false;false;true;true]]); b1; b2 ] |
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[820] | 348 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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| 349 | ]. |
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[719] | 350 | |
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[820] | 351 | definition assembly1 ≝ |
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| 352 | λi: instruction. |
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| 353 | match i with |
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| 354 | [ ACALL addr ⇒ |
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| 355 | match addr return λx. bool_to_Prop (is_in ? [[addr11]] x) → ? with |
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| 356 | [ ADDR11 w ⇒ λ_. |
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[938] | 357 | let v1_v2 ≝ split ? 3 8 w in |
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| 358 | let v1 ≝ \fst v1_v2 in |
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| 359 | let v2 ≝ \snd v1_v2 in |
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[820] | 360 | [ (v1 @@ [[true; false; false; false; true]]) ; v2 ] |
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| 361 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 362 | | AJMP addr ⇒ |
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| 363 | match addr return λx. bool_to_Prop (is_in ? [[addr11]] x) → ? with |
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| 364 | [ ADDR11 w ⇒ λ_. |
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[938] | 365 | let v1_v2 ≝ split ? 3 8 w in |
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| 366 | let v1 ≝ \fst v1_v2 in |
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| 367 | let v2 ≝ \snd v1_v2 in |
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[820] | 368 | [ (v1 @@ [[false; false; false; false; true]]) ; v2 ] |
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| 369 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 370 | | JMP adptr ⇒ |
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| 371 | [ ([[false;true;true;true;false;false;true;true]]) ] |
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| 372 | | LCALL addr ⇒ |
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| 373 | match addr return λx. bool_to_Prop (is_in ? [[addr16]] x) → ? with |
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| 374 | [ ADDR16 w ⇒ λ_. |
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[938] | 375 | let b1_b2 ≝ split ? 8 8 w in |
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| 376 | let b1 ≝ \fst b1_b2 in |
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| 377 | let b2 ≝ \snd b1_b2 in |
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[820] | 378 | [ ([[false;false;false;true;false;false;true;false]]); b1; b2 ] |
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| 379 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 380 | | LJMP addr ⇒ |
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| 381 | match addr return λx. bool_to_Prop (is_in ? [[addr16]] x) → ? with |
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| 382 | [ ADDR16 w ⇒ λ_. |
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[938] | 383 | let b1_b2 ≝ split ? 8 8 w in |
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| 384 | let b1 ≝ \fst b1_b2 in |
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| 385 | let b2 ≝ \snd b1_b2 in |
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[820] | 386 | [ ([[false;false;false;false;false;false;true;false]]); b1; b2 ] |
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| 387 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 388 | | MOVC addr1 addr2 ⇒ |
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| 389 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_dptr;acc_pc]] x) → ? with |
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| 390 | [ ACC_DPTR ⇒ λ_. |
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| 391 | [ ([[true;false;false;true;false;false;true;true]]) ] |
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| 392 | | ACC_PC ⇒ λ_. |
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| 393 | [ ([[true;false;false;false;false;false;true;true]]) ] |
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| 394 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 395 | | SJMP addr ⇒ |
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| 396 | match addr return λx. bool_to_Prop (is_in ? [[relative]] x) → ? with |
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| 397 | [ RELATIVE b1 ⇒ λ_. |
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| 398 | [ ([[true;false;false;false;false;false;false;false]]); b1 ] |
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| 399 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 400 | | RealInstruction instr ⇒ |
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| 401 | assembly_preinstruction [[ relative ]] |
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| 402 | (λx. |
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| 403 | match x return λs. bool_to_Prop (is_in ? [[ relative ]] s) → ? with |
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| 404 | [ RELATIVE r ⇒ λ_. r |
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| 405 | | _ ⇒ λabsd. ⊥ |
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| 406 | ] (subaddressing_modein … x)) instr |
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| 407 | ]. |
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| 408 | cases absd |
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| 409 | qed. |
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| 410 | |
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[832] | 411 | inductive jump_length: Type[0] ≝ |
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| 412 | | short_jump: jump_length |
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| 413 | | medium_jump: jump_length |
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| 414 | | long_jump: jump_length. |
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| 415 | |
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[1309] | 416 | (* jump_expansion_policy: instruction number ↦ 〈pc, jump_length〉 *) |
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[1393] | 417 | definition jump_expansion_policy ≝ BitVectorTrie (ℕ × jump_length) 16. |
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[907] | 418 | |
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[842] | 419 | definition expand_relative_jump_internal: |
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| 420 | (Identifier → Word) → jump_length → Identifier → Word → ([[relative]] → preinstruction [[relative]]) → |
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| 421 | option (list instruction) |
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| 422 | ≝ |
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| 423 | λlookup_labels,jmp_len.λjmp:Identifier.λpc,i. |
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[832] | 424 | match jmp_len with |
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[842] | 425 | [ short_jump ⇒ |
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| 426 | let lookup_address ≝ lookup_labels jmp in |
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| 427 | let 〈result, flags〉 ≝ sub_16_with_carry pc lookup_address false in |
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| 428 | let 〈upper, lower〉 ≝ split ? 8 8 result in |
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| 429 | if eq_bv ? upper (zero 8) then |
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| 430 | let address ≝ RELATIVE lower in |
---|
| 431 | Some ? [ RealInstruction (i address) ] |
---|
| 432 | else |
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| 433 | None ? |
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| 434 | | medium_jump ⇒ None … |
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[832] | 435 | | long_jump ⇒ |
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[842] | 436 | Some ? |
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| 437 | [ RealInstruction (i (RELATIVE (bitvector_of_nat ? 2))); |
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[832] | 438 | SJMP (RELATIVE (bitvector_of_nat ? 3)); (* LJMP size? *) |
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[842] | 439 | LJMP (ADDR16 (lookup_labels jmp)) |
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[832] | 440 | ] |
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| 441 | ]. |
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| 442 | @ I |
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| 443 | qed. |
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[825] | 444 | |
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[842] | 445 | definition expand_relative_jump: (Identifier → Word) → jump_length → Word → preinstruction Identifier → option (list instruction) ≝ |
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| 446 | λlookup_labels. |
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[832] | 447 | λjmp_len: jump_length. |
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[842] | 448 | λpc. |
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[832] | 449 | λi: preinstruction Identifier. |
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| 450 | let rel_jmp ≝ RELATIVE (bitvector_of_nat ? 2) in |
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| 451 | match i with |
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[842] | 452 | [ JC jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JC ?) |
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| 453 | | JNC jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JNC ?) |
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| 454 | | JB baddr jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JB ? baddr) |
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| 455 | | JZ jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JZ ?) |
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| 456 | | JNZ jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JNZ ?) |
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| 457 | | JBC baddr jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JBC ? baddr) |
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| 458 | | JNB baddr jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (JNB ? baddr) |
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| 459 | | CJNE addr jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (CJNE ? addr) |
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| 460 | | DJNZ addr jmp ⇒ expand_relative_jump_internal lookup_labels jmp_len jmp pc (DJNZ ? addr) |
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[832] | 461 | | ADD arg1 arg2 ⇒ Some ? [ ADD ? arg1 arg2 ] |
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| 462 | | ADDC arg1 arg2 ⇒ Some ? [ ADDC ? arg1 arg2 ] |
---|
| 463 | | SUBB arg1 arg2 ⇒ Some ? [ SUBB ? arg1 arg2 ] |
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| 464 | | INC arg ⇒ Some ? [ INC ? arg ] |
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| 465 | | DEC arg ⇒ Some ? [ DEC ? arg ] |
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| 466 | | MUL arg1 arg2 ⇒ Some ? [ MUL ? arg1 arg2 ] |
---|
| 467 | | DIV arg1 arg2 ⇒ Some ? [ DIV ? arg1 arg2 ] |
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| 468 | | DA arg ⇒ Some ? [ DA ? arg ] |
---|
| 469 | | ANL arg ⇒ Some ? [ ANL ? arg ] |
---|
| 470 | | ORL arg ⇒ Some ? [ ORL ? arg ] |
---|
| 471 | | XRL arg ⇒ Some ? [ XRL ? arg ] |
---|
| 472 | | CLR arg ⇒ Some ? [ CLR ? arg ] |
---|
| 473 | | CPL arg ⇒ Some ? [ CPL ? arg ] |
---|
| 474 | | RL arg ⇒ Some ? [ RL ? arg ] |
---|
| 475 | | RR arg ⇒ Some ? [ RR ? arg ] |
---|
| 476 | | RLC arg ⇒ Some ? [ RLC ? arg ] |
---|
| 477 | | RRC arg ⇒ Some ? [ RRC ? arg ] |
---|
| 478 | | SWAP arg ⇒ Some ? [ SWAP ? arg ] |
---|
| 479 | | MOV arg ⇒ Some ? [ MOV ? arg ] |
---|
| 480 | | MOVX arg ⇒ Some ? [ MOVX ? arg ] |
---|
| 481 | | SETB arg ⇒ Some ? [ SETB ? arg ] |
---|
| 482 | | PUSH arg ⇒ Some ? [ PUSH ? arg ] |
---|
| 483 | | POP arg ⇒ Some ? [ POP ? arg ] |
---|
| 484 | | XCH arg1 arg2 ⇒ Some ? [ XCH ? arg1 arg2 ] |
---|
| 485 | | XCHD arg1 arg2 ⇒ Some ? [ XCHD ? arg1 arg2 ] |
---|
| 486 | | RET ⇒ Some ? [ RET ? ] |
---|
| 487 | | RETI ⇒ Some ? [ RETI ? ] |
---|
| 488 | | NOP ⇒ Some ? [ RealInstruction (NOP ?) ] |
---|
| 489 | ]. |
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| 490 | |
---|
[1309] | 491 | definition expand_pseudo_instruction_safe: ? → ? → Word → jump_length → pseudo_instruction → option (list instruction) ≝ |
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[825] | 492 | λlookup_labels. |
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| 493 | λlookup_datalabels. |
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[832] | 494 | λpc. |
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| 495 | λjmp_len. |
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| 496 | λi. |
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| 497 | match i with |
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| 498 | [ Cost cost ⇒ Some ? [ ] |
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| 499 | | Comment comment ⇒ Some ? [ ] |
---|
| 500 | | Call call ⇒ |
---|
| 501 | match jmp_len with |
---|
| 502 | [ short_jump ⇒ None ? |
---|
| 503 | | medium_jump ⇒ |
---|
| 504 | let 〈ignore, address〉 ≝ split ? 5 11 (lookup_labels call) in |
---|
| 505 | let 〈fst_5, rest〉 ≝ split ? 5 11 pc in |
---|
| 506 | if eq_bv ? ignore fst_5 then |
---|
| 507 | let address ≝ ADDR11 address in |
---|
| 508 | Some ? ([ ACALL address ]) |
---|
| 509 | else |
---|
| 510 | None ? |
---|
| 511 | | long_jump ⇒ |
---|
| 512 | let address ≝ ADDR16 (lookup_labels call) in |
---|
| 513 | Some ? [ LCALL address ] |
---|
| 514 | ] |
---|
| 515 | | Mov d trgt ⇒ |
---|
| 516 | let address ≝ DATA16 (lookup_datalabels trgt) in |
---|
| 517 | Some ? [ RealInstruction (MOV ? (inl ? ? (inl ? ? (inr ? ? 〈DPTR, address〉))))] |
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[842] | 518 | | Instruction instr ⇒ expand_relative_jump lookup_labels jmp_len pc instr |
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[832] | 519 | | Jmp jmp ⇒ |
---|
| 520 | match jmp_len with |
---|
| 521 | [ short_jump ⇒ |
---|
| 522 | let lookup_address ≝ lookup_labels jmp in |
---|
| 523 | let 〈result, flags〉 ≝ sub_16_with_carry pc lookup_address false in |
---|
| 524 | let 〈upper, lower〉 ≝ split ? 8 8 result in |
---|
| 525 | if eq_bv ? upper (zero 8) then |
---|
| 526 | let address ≝ RELATIVE lower in |
---|
| 527 | Some ? [ SJMP address ] |
---|
| 528 | else |
---|
| 529 | None ? |
---|
| 530 | | medium_jump ⇒ |
---|
| 531 | let address ≝ lookup_labels jmp in |
---|
| 532 | let 〈fst_5_addr, rest_addr〉 ≝ split ? 5 11 address in |
---|
| 533 | let 〈fst_5_pc, rest_pc〉 ≝ split ? 5 11 pc in |
---|
| 534 | if eq_bv ? fst_5_addr fst_5_pc then |
---|
| 535 | let address ≝ ADDR11 rest_addr in |
---|
| 536 | Some ? ([ AJMP address ]) |
---|
| 537 | else |
---|
| 538 | None ? |
---|
| 539 | | long_jump ⇒ |
---|
| 540 | let address ≝ ADDR16 (lookup_labels jmp) in |
---|
| 541 | Some ? [ LJMP address ] |
---|
| 542 | ] |
---|
| 543 | ]. |
---|
[825] | 544 | @ I |
---|
| 545 | qed. |
---|
| 546 | |
---|
[1309] | 547 | (* label_trie: identifier ↦ 〈instruction number, address〉 *) |
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[1393] | 548 | definition label_trie ≝ BitVectorTrie (nat × nat) 16. |
---|
[1309] | 549 | |
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[1393] | 550 | definition add_instruction_size: ℕ → jump_length → pseudo_instruction → ℕ ≝ |
---|
[1309] | 551 | λpc.λjmp_len.λinstr. |
---|
[1393] | 552 | let bv_pc ≝ bitvector_of_nat 16 pc in |
---|
| 553 | let ilist ≝ expand_pseudo_instruction_safe (λx.bv_pc) (λx.bv_pc) bv_pc jmp_len instr in |
---|
[1309] | 554 | let bv: list (BitVector 8) ≝ match ilist with |
---|
| 555 | [ None ⇒ (* hmm, this shouldn't happen *) [ ] |
---|
| 556 | | Some l ⇒ flatten … (map … assembly1 l) |
---|
| 557 | ] in |
---|
[1393] | 558 | pc + (|bv|). |
---|
| 559 | |
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[1363] | 560 | definition is_label ≝ |
---|
| 561 | λx:labelled_instruction.λl:Identifier. |
---|
| 562 | let 〈lbl,instr〉 ≝ x in |
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| 563 | match lbl with |
---|
| 564 | [ Some l' ⇒ l' = l |
---|
| 565 | | _ ⇒ False |
---|
| 566 | ]. |
---|
[1309] | 567 | |
---|
[1459] | 568 | lemma label_does_not_occur: |
---|
[1363] | 569 | ∀i,p,l. |
---|
| 570 | is_label (nth i ? p 〈None ?, Comment [ ]〉) l → does_not_occur l p = false. |
---|
| 571 | #i #p #l generalize in match i; elim p |
---|
| 572 | [ #i >nth_nil #H @⊥ @H |
---|
| 573 | | #h #t #IH #i cases i -i |
---|
| 574 | [ cases h #hi #hp cases hi normalize |
---|
| 575 | [ #H @⊥ @H |
---|
| 576 | | #l' #Heq change in match (eq_v ??? l' l) with (eq_bv ? l' l) |
---|
| 577 | >(eq_eq_bv … Heq) // |
---|
| 578 | ] |
---|
| 579 | | #i #H whd in match (does_not_occur ??) |
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| 580 | whd in match (instruction_matches_identifier ??) |
---|
| 581 | cases h #hi #hp cases hi normalize |
---|
| 582 | [ @(IH i) @H |
---|
| 583 | | #l' cases (eq_v ??? l' l) |
---|
| 584 | [ normalize // |
---|
| 585 | | normalize @(IH i) @H |
---|
| 586 | ] |
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| 587 | ] |
---|
| 588 | ] |
---|
| 589 | ] |
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| 590 | qed. |
---|
[1393] | 591 | |
---|
[1459] | 592 | (* lemma coerc_sigma: |
---|
| 593 | ∀A,P.∀p:A.P p → Σx:A.P x. |
---|
| 594 | #A #P #a #p % [ @a | /2/] |
---|
| 595 | qed. |
---|
| 596 | coercion coerc_sigma:∀A,P.∀p:A.P p → Σx:A.P x |
---|
| 597 | ≝ coerc_sigma on p: ? to (Sig ??). *) |
---|
| 598 | |
---|
[1393] | 599 | lemma coerc_pair_sigma: |
---|
| 600 | ∀A,B,P. ∀p:A × B. P (\snd p) → A × (Σx:B.P x). |
---|
| 601 | #A #B #P * #a #b #p % [@a | /2/] |
---|
| 602 | qed. |
---|
| 603 | coercion coerc_pair_sigma:∀A,B,P. ∀p:A × B. P (\snd p) → A × (Σx:B.P x) |
---|
| 604 | ≝ coerc_pair_sigma on p: (? × ?) to (? × (Sig ??)). |
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[1363] | 605 | |
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[1393] | 606 | let rec create_label_trie (program: list labelled_instruction) |
---|
| 607 | (policy: jump_expansion_policy): |
---|
| 608 | (Σlabels:label_trie. |
---|
| 609 | ∀i:ℕ.i < |program| → |
---|
| 610 | ∀l.occurs_exactly_once l program → |
---|
| 611 | is_label (nth i ? program 〈None ?, Comment [ ]〉) l → |
---|
| 612 | ∃a.lookup_opt … l labels = Some ? 〈i,a〉 |
---|
| 613 | ) ≝ |
---|
[1363] | 614 | let 〈final_pc, final_labels〉 ≝ |
---|
[1309] | 615 | foldl_strong (option Identifier × pseudo_instruction) |
---|
[1393] | 616 | (λprefix.ℕ × (Σlabels. |
---|
[1363] | 617 | ∀i:ℕ.i < |prefix| → |
---|
[1393] | 618 | ∀l.occurs_exactly_once l prefix → |
---|
[1363] | 619 | is_label (nth i ? prefix 〈None ?, Comment [ ]〉) l → |
---|
[1393] | 620 | ∃a.lookup_opt … l labels = Some ? 〈i,a〉) |
---|
[1363] | 621 | ) |
---|
[1309] | 622 | program |
---|
| 623 | (λprefix.λx.λtl.λprf.λacc. |
---|
[1363] | 624 | let 〈pc,labels〉 ≝ acc in |
---|
[1309] | 625 | let 〈label,instr〉 ≝ x in |
---|
[1393] | 626 | let new_labels ≝ |
---|
| 627 | match label with |
---|
| 628 | [ None ⇒ labels |
---|
| 629 | | Some l ⇒ insert … l 〈|prefix|, pc〉 labels |
---|
| 630 | ] in |
---|
| 631 | let jmp_len ≝ \snd (lookup … (bitvector_of_nat 16 (|prefix|)) policy 〈pc, long_jump〉) in |
---|
| 632 | 〈add_instruction_size pc jmp_len instr, new_labels〉 |
---|
| 633 | ) 〈0, Stub ? ?〉 in |
---|
[1309] | 634 | final_labels. |
---|
[1393] | 635 | [ #i >append_length <commutative_plus #Hi normalize in Hi; cases (le_to_or_lt_eq … Hi) -Hi; |
---|
| 636 | [ #Hi #l normalize nodelta; cases label normalize nodelta |
---|
[1363] | 637 | [ >occurs_exactly_once_None #Hocc >(nth_append_first ? ? prefix ? ? (le_S_S_to_le ? ? Hi)) #Hl |
---|
[1393] | 638 | lapply (sig2 … labels) #Hacc elim (Hacc i (le_S_S_to_le … Hi) l Hocc Hl) #addr #Haddr |
---|
[1363] | 639 | % [ @addr | @Haddr ] |
---|
[1459] | 640 | | #l' #Hocc #Hl lapply (occurs_exactly_once_Some_stronger … Hocc) -Hocc; #Hocc |
---|
[1363] | 641 | lapply (refl ? (eq_bv 16 l' l)) cases (eq_bv ? l' l) in ⊢ (???% → %) #Heq |
---|
| 642 | [ >Heq in Hocc; #Hocc normalize in Hocc; |
---|
| 643 | >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) in Hl; #Hl |
---|
| 644 | @⊥ @(absurd … Hocc) |
---|
[1393] | 645 | | >Heq in Hocc; normalize nodelta #Hocc lapply (sig2 … labels) #Hacc elim (Hacc i (le_S_S_to_le … Hi) l Hocc ?) |
---|
| 646 | [ #addr #Haddr % [ @addr | <Haddr @lookup_opt_insert_miss >eq_bv_sym >Heq // ] |
---|
| 647 | | >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) in Hl; // |
---|
[1363] | 648 | ] |
---|
| 649 | ] |
---|
[1459] | 650 | >(label_does_not_occur i … Hl) normalize nodelta @nmk // |
---|
[1363] | 651 | ] |
---|
| 652 | | #Hi #l #Hocc >(injective_S … Hi) >nth_append_second |
---|
[1393] | 653 | [ <minus_n_n #Hl normalize in Hl; normalize nodelta cases label in Hl; |
---|
[1363] | 654 | [ normalize nodelta #H @⊥ @H |
---|
[1393] | 655 | | #l' normalize nodelta #Heq % [ @pc | <Heq normalize nodelta @lookup_opt_insert_hit ] |
---|
[1363] | 656 | ] |
---|
| 657 | | @le_n |
---|
| 658 | ] |
---|
| 659 | ] |
---|
| 660 | | #i #Hi #l #Hl @⊥ @Hl |
---|
| 661 | ] |
---|
[1103] | 662 | qed. |
---|
| 663 | |
---|
[1393] | 664 | definition select_reljump_length: label_trie → ℕ → Identifier → jump_length ≝ |
---|
[1309] | 665 | λlabels.λpc.λlbl. |
---|
| 666 | let 〈n, addr〉 ≝ lookup … lbl labels 〈0, pc〉 in |
---|
[1393] | 667 | if leb pc addr (* forward jump *) |
---|
| 668 | then if leb (addr - pc) 126 |
---|
| 669 | then short_jump |
---|
| 670 | else long_jump |
---|
| 671 | else if leb (pc - addr) 129 |
---|
| 672 | then short_jump |
---|
| 673 | else long_jump. |
---|
[1103] | 674 | |
---|
[1393] | 675 | definition select_call_length: label_trie → ℕ → Identifier → jump_length ≝ |
---|
| 676 | λlabels.λpc_nat.λlbl. |
---|
| 677 | let pc ≝ bitvector_of_nat 16 pc_nat in |
---|
| 678 | let addr ≝ bitvector_of_nat 16 (\snd (lookup ? ? lbl labels 〈0, pc_nat〉)) in |
---|
[1309] | 679 | let 〈fst_5_addr, rest_addr〉 ≝ split ? 5 11 addr in |
---|
| 680 | let 〈fst_5_pc, rest_pc〉 ≝ split ? 5 11 pc in |
---|
| 681 | if eq_bv ? fst_5_addr fst_5_pc |
---|
| 682 | then medium_jump |
---|
| 683 | else long_jump. |
---|
| 684 | |
---|
[1393] | 685 | definition select_jump_length: label_trie → ℕ → Identifier → jump_length ≝ |
---|
[1309] | 686 | λlabels.λpc.λlbl. |
---|
| 687 | let 〈n, addr〉 ≝ lookup … lbl labels 〈0, pc〉 in |
---|
[1393] | 688 | if leb pc addr |
---|
| 689 | then if leb (addr - pc) 126 |
---|
| 690 | then short_jump |
---|
| 691 | else select_call_length labels pc lbl |
---|
| 692 | else if leb (pc - addr) 129 |
---|
| 693 | then short_jump |
---|
| 694 | else select_call_length labels pc lbl. |
---|
[1309] | 695 | |
---|
[1393] | 696 | definition jump_expansion_step_instruction: label_trie → ℕ → |
---|
[1309] | 697 | preinstruction Identifier → option jump_length ≝ |
---|
| 698 | λlabels.λpc.λi. |
---|
| 699 | match i with |
---|
| 700 | [ JC j ⇒ Some ? (select_reljump_length labels pc j) |
---|
| 701 | | JNC j ⇒ Some ? (select_reljump_length labels pc j) |
---|
| 702 | | JZ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
| 703 | | JNZ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
| 704 | | JB _ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
| 705 | | JBC _ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
| 706 | | JNB _ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
| 707 | | CJNE _ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
| 708 | | DJNZ _ j ⇒ Some ? (select_reljump_length labels pc j) |
---|
| 709 | | _ ⇒ None ? |
---|
| 710 | ]. |
---|
[1103] | 711 | |
---|
[1309] | 712 | definition max_length: jump_length → jump_length → jump_length ≝ |
---|
| 713 | λj1.λj2. |
---|
| 714 | match j1 with |
---|
| 715 | [ long_jump ⇒ long_jump |
---|
| 716 | | medium_jump ⇒ |
---|
| 717 | match j2 with |
---|
| 718 | [ long_jump ⇒ long_jump |
---|
| 719 | | _ ⇒ medium_jump |
---|
| 720 | ] |
---|
| 721 | | short_jump ⇒ j2 |
---|
| 722 | ]. |
---|
[1393] | 723 | |
---|
| 724 | definition jmpleq: jump_length → jump_length → Prop ≝ |
---|
| 725 | λj1.λj2. |
---|
| 726 | match j1 with |
---|
| 727 | [ short_jump ⇒ True |
---|
| 728 | | medium_jump ⇒ |
---|
| 729 | match j2 with |
---|
| 730 | [ short_jump ⇒ False |
---|
| 731 | | _ ⇒ True |
---|
| 732 | ] |
---|
| 733 | | long_jump ⇒ |
---|
| 734 | match j2 with |
---|
| 735 | [ long_jump ⇒ True |
---|
| 736 | | _ ⇒ False |
---|
| 737 | ] |
---|
| 738 | ]. |
---|
| 739 | |
---|
[1417] | 740 | lemma jmpleq_max_length: ∀x,y.jmpleq y (max_length y x). |
---|
| 741 | #x #y |
---|
| 742 | cases y |
---|
| 743 | [ // |
---|
| 744 | | cases x // |
---|
| 745 | | // |
---|
| 746 | ] |
---|
| 747 | qed. |
---|
| 748 | |
---|
[1393] | 749 | definition is_jump' ≝ |
---|
| 750 | λx:preinstruction Identifier. |
---|
| 751 | match x with |
---|
| 752 | [ JC _ ⇒ True |
---|
| 753 | | JNC _ ⇒ True |
---|
| 754 | | JZ _ ⇒ True |
---|
| 755 | | JNZ _ ⇒ True |
---|
| 756 | | JB _ _ ⇒ True |
---|
| 757 | | JNB _ _ ⇒ True |
---|
| 758 | | JBC _ _ ⇒ True |
---|
| 759 | | CJNE _ _ ⇒ True |
---|
| 760 | | DJNZ _ _ ⇒ True |
---|
| 761 | | _ ⇒ False |
---|
| 762 | ]. |
---|
| 763 | |
---|
| 764 | definition is_jump ≝ |
---|
| 765 | λx:labelled_instruction. |
---|
| 766 | let 〈label,instr〉 ≝ x in |
---|
| 767 | match instr with |
---|
| 768 | [ Instruction i ⇒ is_jump' i |
---|
| 769 | | Call _ ⇒ True |
---|
| 770 | | Jmp _ ⇒ True |
---|
| 771 | | _ ⇒ False |
---|
| 772 | ]. |
---|
[1404] | 773 | |
---|
| 774 | definition jump_in_policy ≝ |
---|
| 775 | λprefix:list labelled_instruction.λpolicy:jump_expansion_policy. |
---|
| 776 | ∀i:ℕ.i < |prefix| → is_jump (nth i ? prefix 〈None ?, Comment []〉) → |
---|
| 777 | ∃p,j.lookup_opt … (bitvector_of_nat 16 i) policy = Some ? 〈p,j〉. |
---|
[1459] | 778 | |
---|
| 779 | axiom bitvector_of_nat_abs: |
---|
| 780 | ∀x,y:ℕ.x ≠ y → ¬eq_bv 16 (bitvector_of_nat 16 x) (bitvector_of_nat 16 y). |
---|
| 781 | |
---|
| 782 | let rec jump_expansion_start (program: list labelled_instruction): |
---|
| 783 | (Σpolicy.jump_in_policy program policy) ≝ |
---|
| 784 | foldl_strong (option Identifier × pseudo_instruction) |
---|
| 785 | (λprefix.Σpolicy.jump_in_policy prefix policy) |
---|
| 786 | program |
---|
| 787 | (λprefix.λx.λtl.λprf.λpolicy. |
---|
| 788 | let 〈label,instr〉 ≝ x in |
---|
| 789 | match instr with |
---|
| 790 | [ Instruction i ⇒ match i with |
---|
| 791 | [ JC _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
| 792 | | JNC _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
| 793 | | JZ _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
| 794 | | JNZ _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
| 795 | | JB _ _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
| 796 | | JNB _ _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
| 797 | | JBC _ _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
| 798 | | CJNE _ _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
| 799 | | DJNZ _ _ ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
| 800 | | _ ⇒ (eject … policy) |
---|
| 801 | ] |
---|
| 802 | | Call c ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
| 803 | | Jmp j ⇒ insert … (bitvector_of_nat 16 (|prefix|)) 〈0,short_jump〉 policy |
---|
| 804 | | _ ⇒ (eject … policy) |
---|
| 805 | ] |
---|
| 806 | ) (Stub ? ?). |
---|
| 807 | [43: normalize #i #Hi @⊥ @(absurd (i < 0)) [ @Hi | @not_le_Sn_O ] ] |
---|
| 808 | whd #i >append_length <commutative_plus #Hi normalize in Hi; cases (le_to_or_lt_eq … Hi) -Hi; |
---|
| 809 | #Hi |
---|
| 810 | [2,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,70,72,74,76,78,80,82,84: |
---|
| 811 | >nth_append_second >(injective_S … Hi) |
---|
| 812 | [2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62: @le_n] |
---|
| 813 | <minus_n_n #Hj normalize in Hj; @⊥ @Hj |
---|
| 814 | |4,6,52,54,56,58,60,62,64,66,68: >nth_append_second >(injective_S … Hi) |
---|
| 815 | [2,4,6,8,10,12,14,16,18,20,22: @le_n] |
---|
| 816 | <minus_n_n #Hj % [1,3,5,7,9,11,13,15,17,19,21: @O ] % |
---|
| 817 | [1,3,5,7,9,11,13,15,17,19,21: @short_jump ] @lookup_opt_insert_hit |
---|
| 818 | ] |
---|
| 819 | >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) #Hj |
---|
| 820 | lapply (sig2 … policy) #Hpolicy elim (Hpolicy i (le_S_S_to_le … Hi) Hj) |
---|
| 821 | #p #H elim H #j #Hpj % |
---|
| 822 | [1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83: @p] % |
---|
| 823 | [1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83: @j] |
---|
| 824 | [2,3,26,27,28,29,30,31,32,33,34: >lookup_opt_insert_miss] |
---|
| 825 | [2,4,6,8,10,12,14,16,18,20,22: @bitvector_of_nat_abs @lt_to_not_eq @(le_S_S_to_le … Hi) ] @Hpj |
---|
| 826 | qed. |
---|
| 827 | |
---|
| 828 | (* not really recursive *) |
---|
[1393] | 829 | let rec jump_expansion_step (program: list labelled_instruction) |
---|
[1404] | 830 | (old_policy: Σpolicy.jump_in_policy program policy): |
---|
| 831 | (Σpolicy. |
---|
| 832 | jump_in_policy program policy ∧ |
---|
[1417] | 833 | (jump_in_policy program policy → |
---|
| 834 | (∀i.i < |program| → is_jump (nth i ? program 〈None ?, Comment []〉) → |
---|
| 835 | jmpleq |
---|
| 836 | (\snd (lookup … (bitvector_of_nat 16 i) old_policy 〈0,short_jump〉)) |
---|
| 837 | (\snd (lookup … (bitvector_of_nat 16 i) policy 〈0,short_jump〉))) |
---|
| 838 | ) |
---|
| 839 | ) ≝ |
---|
[1393] | 840 | let labels ≝ create_label_trie program old_policy in |
---|
| 841 | let 〈final_pc, final_policy〉 ≝ |
---|
[1309] | 842 | foldl_strong (option Identifier × pseudo_instruction) |
---|
[1404] | 843 | (λprefix.ℕ × Σpolicy. |
---|
| 844 | jump_in_policy prefix policy ∧ |
---|
[1417] | 845 | (jump_in_policy prefix policy → |
---|
| 846 | (∀i.i < |prefix| → is_jump (nth i ? prefix 〈None ?, Comment []〉) → |
---|
| 847 | jmpleq |
---|
| 848 | (\snd (lookup … (bitvector_of_nat 16 i) old_policy 〈0,short_jump〉)) |
---|
| 849 | (\snd (lookup … (bitvector_of_nat 16 i) policy 〈0,short_jump〉))) |
---|
| 850 | ) |
---|
[1393] | 851 | ) |
---|
[1309] | 852 | program |
---|
| 853 | (λprefix.λx.λtl.λprf.λacc. |
---|
[1393] | 854 | let 〈pc, policy〉 ≝ acc in |
---|
[1309] | 855 | let 〈label,instr〉 ≝ x in |
---|
[1417] | 856 | let old_jump_length ≝ lookup_opt ? ? (bitvector_of_nat 16 (|prefix|)) old_policy in |
---|
[1309] | 857 | let add_instr ≝ |
---|
| 858 | match instr with |
---|
| 859 | [ Instruction i ⇒ jump_expansion_step_instruction labels pc i |
---|
| 860 | | Call c ⇒ Some ? (select_call_length labels pc c) |
---|
| 861 | | Jmp j ⇒ Some ? (select_jump_length labels pc j) |
---|
| 862 | | _ ⇒ None ? |
---|
| 863 | ] in |
---|
| 864 | let 〈new_pc, new_policy〉 ≝ |
---|
[1417] | 865 | let 〈ignore,old_length〉 ≝ lookup … (bitvector_of_nat 16 (|prefix|)) old_policy 〈0, short_jump〉 in |
---|
[1309] | 866 | match add_instr with |
---|
| 867 | [ None ⇒ (* i.e. it's not a jump *) |
---|
| 868 | 〈add_instruction_size pc long_jump instr, policy〉 |
---|
| 869 | | Some ai ⇒ |
---|
| 870 | let new_length ≝ max_length old_length ai in |
---|
[1393] | 871 | 〈add_instruction_size pc new_length instr, insert … (bitvector_of_nat 16 (|prefix|)) 〈pc, new_length〉 policy〉 |
---|
[1309] | 872 | ] in |
---|
[1393] | 873 | 〈new_pc, new_policy〉 |
---|
| 874 | ) 〈0, Stub ? ?〉 in |
---|
[1309] | 875 | final_policy. |
---|
[1404] | 876 | [ @conj |
---|
| 877 | [ #i >append_length <commutative_plus #Hi normalize in Hi; cases (le_to_or_lt_eq … Hi) -Hi; |
---|
| 878 | [ #Hi; >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) #Hj |
---|
| 879 | lapply (sig2 … policy) #Hacc elim ((proj1 … Hacc) i (le_S_S_to_le … Hi) Hj) #h #n elim n |
---|
| 880 | -n #n #Hn |
---|
| 881 | % [ @h | % [ @n | normalize nodelta cases instr normalize nodelta |
---|
[1417] | 882 | [2,3: #x cases (lookup ??? old_policy ?) #y #z @Hn |
---|
| 883 | |6: #x #y cases (lookup ??? old_policy ?) #w #z @Hn |
---|
| 884 | |1: #pi cases (lookup ??? old_policy ?) #x #y cases (jump_expansion_step_instruction ???) |
---|
[1404] | 885 | [ @Hn |
---|
| 886 | | #z normalize nodelta <Hn @lookup_opt_insert_miss |
---|
[1459] | 887 | @bitvector_of_nat_abs @(lt_to_not_eq i (|prefix|)) @(le_S_S_to_le … Hi) |
---|
[1393] | 888 | ] |
---|
[1417] | 889 | |4,5: #id cases (lookup ??? old_policy ?) #x #y normalize nodelta <Hn |
---|
[1459] | 890 | @lookup_opt_insert_miss @bitvector_of_nat_abs |
---|
| 891 | @(lt_to_not_eq i (|prefix|)) @(le_S_S_to_le … Hi) |
---|
[1404] | 892 | ] ] ] |
---|
| 893 | | #Hi >nth_append_second >(injective_S … Hi) |
---|
| 894 | [ <minus_n_n #Hj normalize in Hj; normalize nodelta cases instr in Hj; |
---|
| 895 | [2,3: #x #Hj @⊥ @Hj |
---|
| 896 | |6: #x #y #Hj @⊥ @Hj |
---|
| 897 | |1: #pi cases pi |
---|
| 898 | [35,36,37: #Hj @⊥ @Hj |
---|
| 899 | |4,5,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32: #x #Hj @⊥ @Hj |
---|
| 900 | |1,2,3,6,7,33,34: #x #y #Hj @⊥ @Hj |
---|
| 901 | |9,10,14,15: #j normalize nodelta #_ |
---|
| 902 | % [1,3,5,7: @pc |
---|
[1417] | 903 | |2,4,6,8: lapply (refl ? (lookup … (bitvector_of_nat ? (|prefix|)) old_policy 〈0,short_jump〉)) |
---|
| 904 | cases (lookup … (bitvector_of_nat ? (|prefix|)) old_policy 〈0,short_jump〉) in ⊢ (???% → %) |
---|
[1404] | 905 | #x #y #Hl normalize nodelta |
---|
| 906 | % [1,3,5,7: @(max_length y (select_reljump_length (create_label_trie program old_policy) pc j)) |
---|
| 907 | |2,4,6,8: @lookup_opt_insert_hit |
---|
| 908 | ] ] |
---|
| 909 | |11,12,13,16,17: #x #j normalize nodelta #_ |
---|
| 910 | % [1,3,5,7,9: @pc |
---|
[1417] | 911 | |2,4,6,8,10: lapply (refl ? (lookup … (bitvector_of_nat ? (|prefix|)) old_policy 〈0,short_jump〉)) |
---|
| 912 | cases (lookup … (bitvector_of_nat ? (|prefix|)) old_policy 〈0,short_jump〉) in ⊢ (???% → %) |
---|
[1404] | 913 | #x #y #Hl normalize nodelta |
---|
| 914 | % [1,3,5,7,9: @(max_length y (select_reljump_length (create_label_trie program old_policy) pc j)) |
---|
| 915 | |2,4,6,8,10: @lookup_opt_insert_hit |
---|
| 916 | ] ] |
---|
| 917 | ] |
---|
| 918 | |4,5: #j normalize nodelta #_ |
---|
| 919 | % [1,3: @pc |
---|
[1417] | 920 | |2,4: cases (lookup … (bitvector_of_nat ? (|prefix|)) old_policy 〈0,short_jump〉) |
---|
[1404] | 921 | #x #y normalize nodelta |
---|
| 922 | % [1: @(max_length y (select_jump_length (create_label_trie program old_policy) pc j)) |
---|
| 923 | |3: @(max_length y (select_call_length (create_label_trie program old_policy) pc j)) |
---|
| 924 | |2,4: @lookup_opt_insert_hit |
---|
| 925 | ] |
---|
[1393] | 926 | ] |
---|
[1404] | 927 | ] |
---|
| 928 | | @le_n |
---|
[1393] | 929 | ] |
---|
| 930 | ] |
---|
[1417] | 931 | | #Hjip #i >append_length <commutative_plus #Hi normalize in Hi; cases (le_to_or_lt_eq … Hi) -Hi; |
---|
| 932 | [ #Hi >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) #Hj |
---|
| 933 | lapply (sig2 … policy) #Hpolicy normalize nodelta cases instr normalize nodelta |
---|
| 934 | [2,3: #x cases (lookup ? 16 (bitvector_of_nat 16 (|prefix|)) old_policy ?) #y #z |
---|
| 935 | normalize nodelta @(proj2 ? ? Hpolicy (proj1 ? ? Hpolicy) ? (le_S_S_to_le … Hi) Hj) |
---|
| 936 | |6: #x #y cases (lookup ? 16 (bitvector_of_nat 16 (|prefix|)) old_policy ?) #w #z |
---|
| 937 | normalize nodelta @(proj2 ? ? Hpolicy (proj1 ? ? Hpolicy) ? (le_S_S_to_le … Hi) Hj) |
---|
| 938 | |1: #pi cases (lookup ? 16 (bitvector_of_nat 16 (|prefix|)) old_policy ?) #x #y |
---|
| 939 | cases (jump_expansion_step_instruction ???) |
---|
| 940 | [ @(proj2 ? ? Hpolicy (proj1 ? ? Hpolicy) ? (le_S_S_to_le … Hi) Hj) |
---|
| 941 | | #z normalize nodelta |
---|
| 942 | whd in match (snd ℕ jump_expansion_policy ?) >lookup_insert_miss |
---|
| 943 | [ @(proj2 … Hpolicy (proj1 ? ? Hpolicy) ? (le_S_S_to_le … Hi) Hj) |
---|
[1459] | 944 | | @bitvector_of_nat_abs @(lt_to_not_eq i (|prefix|)) @(le_S_S_to_le … Hi) |
---|
[1417] | 945 | ] |
---|
| 946 | ] |
---|
| 947 | |4,5: #id cases (lookup ? 16 (bitvector_of_nat 16 (|prefix|)) old_policy ?) #x #y |
---|
| 948 | normalize nodelta >lookup_insert_miss |
---|
| 949 | [1,3: @(proj2 … Hpolicy (proj1 ? ? Hpolicy) ? (le_S_S_to_le … Hi) Hj) |
---|
[1459] | 950 | |2,4: @bitvector_of_nat_abs @(lt_to_not_eq i (|prefix|)) @(le_S_S_to_le … Hi) |
---|
[1417] | 951 | ] |
---|
| 952 | ] |
---|
| 953 | | #Hi >nth_append_second >(injective_S … Hi) |
---|
| 954 | [ <minus_n_n #Hj normalize in Hj; normalize nodelta cases instr in Hj; |
---|
| 955 | [2,3: #x #Hj @⊥ @Hj |
---|
| 956 | |6: #x #y #Hj @⊥ @Hj |
---|
| 957 | |1: #pi cases pi |
---|
| 958 | [35,36,37: #Hj @⊥ @Hj |
---|
| 959 | |4,5,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32: #x #Hj @⊥ @Hj |
---|
| 960 | |1,2,3,6,7,33,34: #x #y #Hj @⊥ @Hj |
---|
| 961 | |9,10,14,15: #j normalize nodelta #_ |
---|
| 962 | cases (lookup ???? 〈0,short_jump〉) #x #y |
---|
| 963 | whd in match (snd ℕ jump_expansion_policy ?) |
---|
| 964 | >lookup_insert_hit normalize @jmpleq_max_length |
---|
| 965 | |11,12,13,16,17: #z #j normalize nodelta #_ |
---|
| 966 | cases (lookup ???? 〈0,short_jump〉) #x #y |
---|
| 967 | whd in match (snd ℕ jump_expansion_policy ?) |
---|
| 968 | >lookup_insert_hit normalize @jmpleq_max_length |
---|
| 969 | ] |
---|
| 970 | |4,5: #id #_ cases (lookup ???? 〈0,short_jump〉) #x #y |
---|
| 971 | whd in match (snd ℕ jump_expansion_policy ?) |
---|
| 972 | >lookup_insert_hit normalize @jmpleq_max_length |
---|
| 973 | ] |
---|
| 974 | | @le_n |
---|
| 975 | ] |
---|
| 976 | ] |
---|
[1393] | 977 | ] |
---|
[1404] | 978 | | @conj |
---|
| 979 | [ #i #Hi @⊥ @(absurd (i<0)) [ @Hi | @(not_le_Sn_O) ] |
---|
[1417] | 980 | | #H #i #Hi @⊥ @(absurd (i<0)) [ @Hi | @(not_le_Sn_O) ] |
---|
[1404] | 981 | ] |
---|
[1393] | 982 | ] |
---|
[1309] | 983 | qed. |
---|
[1103] | 984 | |
---|
[1459] | 985 | let rec jump_expansion_internal (program: list labelled_instruction) |
---|
| 986 | (n: ℕ) on n: (Σpolicy:jump_expansion_policy.jump_in_policy program policy) ≝ |
---|
| 987 | match n with |
---|
| 988 | [ O ⇒ jump_expansion_start program |
---|
| 989 | | S m ⇒ jump_expansion_step program (jump_expansion_internal program m) |
---|
| 990 | ]. |
---|
| 991 | [ @(sig2 … (jump_expansion_start program)) |
---|
| 992 | | @(proj1 … (sig2 … (jump_expansion_step program (jump_expansion_internal program m)))) |
---|
| 993 | ] |
---|
| 994 | qed. |
---|
| 995 | |
---|
[989] | 996 | (**************************************** START OF POLICY ABSTRACTION ********************) |
---|
| 997 | |
---|
[987] | 998 | definition policy_type ≝ Word → jump_length. |
---|
| 999 | |
---|
| 1000 | definition jump_expansion': pseudo_assembly_program → policy_type ≝ |
---|
| 1001 | λprogram.λpc. |
---|
[1479] | 1002 | let policy ≝ jump_expansion_internal (\snd program) (|\snd program|) in |
---|
[1393] | 1003 | let 〈n,j〉 ≝ lookup ? ? pc policy 〈0, long_jump〉 in |
---|
[1309] | 1004 | j. |
---|
| 1005 | |
---|
[993] | 1006 | definition assembly_1_pseudoinstruction_safe ≝ |
---|
[913] | 1007 | λprogram: pseudo_assembly_program. |
---|
[987] | 1008 | λjump_expansion: policy_type. |
---|
[922] | 1009 | λppc: Word. |
---|
[913] | 1010 | λpc: Word. |
---|
[832] | 1011 | λlookup_labels. |
---|
| 1012 | λlookup_datalabels. |
---|
| 1013 | λi. |
---|
[987] | 1014 | let expansion ≝ jump_expansion ppc in |
---|
[1309] | 1015 | match expand_pseudo_instruction_safe lookup_labels lookup_datalabels pc expansion i with |
---|
[832] | 1016 | [ None ⇒ None ? |
---|
[836] | 1017 | | Some pseudos ⇒ |
---|
| 1018 | let mapped ≝ map ? ? assembly1 pseudos in |
---|
| 1019 | let flattened ≝ flatten ? mapped in |
---|
[922] | 1020 | let pc_len ≝ length ? flattened in |
---|
| 1021 | Some ? (〈pc_len, flattened〉) |
---|
[832] | 1022 | ]. |
---|
[913] | 1023 | |
---|
[993] | 1024 | definition construct_costs_safe ≝ |
---|
[832] | 1025 | λprogram. |
---|
[987] | 1026 | λjump_expansion: policy_type. |
---|
[922] | 1027 | λpseudo_program_counter: nat. |
---|
[832] | 1028 | λprogram_counter: nat. |
---|
| 1029 | λcosts: BitVectorTrie Word 16. |
---|
| 1030 | λi. |
---|
| 1031 | match i with |
---|
[840] | 1032 | [ Cost cost ⇒ |
---|
[832] | 1033 | let program_counter_bv ≝ bitvector_of_nat ? program_counter in |
---|
| 1034 | Some ? 〈program_counter, (insert ? ? program_counter_bv cost costs)〉 |
---|
| 1035 | | _ ⇒ |
---|
| 1036 | let pc_bv ≝ bitvector_of_nat ? program_counter in |
---|
[922] | 1037 | let ppc_bv ≝ bitvector_of_nat ? pseudo_program_counter in |
---|
[993] | 1038 | let lookup_labels ≝ λx.pc_bv in |
---|
| 1039 | let lookup_datalabels ≝ λx.zero ? in |
---|
[836] | 1040 | let pc_delta_assembled ≝ |
---|
[993] | 1041 | assembly_1_pseudoinstruction_safe program jump_expansion ppc_bv pc_bv |
---|
[836] | 1042 | lookup_labels lookup_datalabels i |
---|
| 1043 | in |
---|
| 1044 | match pc_delta_assembled with |
---|
[832] | 1045 | [ None ⇒ None ? |
---|
[836] | 1046 | | Some pc_delta_assembled ⇒ |
---|
| 1047 | let 〈pc_delta, assembled〉 ≝ pc_delta_assembled in |
---|
[837] | 1048 | Some ? 〈pc_delta + program_counter, costs〉 |
---|
[832] | 1049 | ] |
---|
| 1050 | ]. |
---|
| 1051 | |
---|
[985] | 1052 | (* This establishes the correspondence between pseudo program counters and |
---|
| 1053 | program counters. It is at the heart of the proof. *) |
---|
| 1054 | (*CSC: code taken from build_maps *) |
---|
[1309] | 1055 | definition sigma00: pseudo_assembly_program → policy_type → list ? → ? → option (nat × (nat × (BitVectorTrie Word 16))) ≝ |
---|
[987] | 1056 | λinstr_list. |
---|
| 1057 | λjump_expansion: policy_type. |
---|
| 1058 | λl:list labelled_instruction. |
---|
[1309] | 1059 | λacc. |
---|
[985] | 1060 | foldl ?? |
---|
| 1061 | (λt,i. |
---|
[832] | 1062 | match t with |
---|
| 1063 | [ None ⇒ None ? |
---|
[985] | 1064 | | Some ppc_pc_map ⇒ |
---|
| 1065 | let 〈ppc,pc_map〉 ≝ ppc_pc_map in |
---|
| 1066 | let 〈program_counter, sigma_map〉 ≝ pc_map in |
---|
| 1067 | let 〈label, i〉 ≝ i in |
---|
[993] | 1068 | match construct_costs_safe instr_list jump_expansion ppc program_counter (Stub …) i with |
---|
[832] | 1069 | [ None ⇒ None ? |
---|
[985] | 1070 | | Some pc_ignore ⇒ |
---|
| 1071 | let 〈pc,ignore〉 ≝ pc_ignore in |
---|
| 1072 | Some … 〈S ppc,〈pc, insert ? ? (bitvector_of_nat ? ppc) (bitvector_of_nat ? pc) sigma_map〉〉 ] |
---|
[1309] | 1073 | ]) acc l. |
---|
[985] | 1074 | |
---|
[987] | 1075 | definition sigma0: pseudo_assembly_program → policy_type → option (nat × (nat × (BitVectorTrie Word 16))) |
---|
[1309] | 1076 | ≝ λprog.λjump_expansion.sigma00 prog jump_expansion (\snd prog) (Some ? 〈0, 〈0, (Stub ? ?)〉〉). |
---|
[985] | 1077 | |
---|
[987] | 1078 | definition tech_pc_sigma00: pseudo_assembly_program → policy_type → list labelled_instruction → option (nat × nat) ≝ |
---|
| 1079 | λprogram,jump_expansion,instr_list. |
---|
[1309] | 1080 | match sigma00 program jump_expansion instr_list (Some ? 〈0, 〈0, (Stub ? ?)〉〉) (* acc copied from sigma0 *) with |
---|
[985] | 1081 | [ None ⇒ None … |
---|
| 1082 | | Some result ⇒ |
---|
| 1083 | let 〈ppc,pc_sigma_map〉 ≝ result in |
---|
| 1084 | let 〈pc,map〉 ≝ pc_sigma_map in |
---|
| 1085 | Some … 〈ppc,pc〉 ]. |
---|
| 1086 | |
---|
[987] | 1087 | definition sigma_safe: pseudo_assembly_program → policy_type → option (Word → Word) ≝ |
---|
| 1088 | λinstr_list,jump_expansion. |
---|
| 1089 | match sigma0 instr_list jump_expansion with |
---|
[832] | 1090 | [ None ⇒ None ? |
---|
| 1091 | | Some result ⇒ |
---|
[985] | 1092 | let 〈ppc,pc_sigma_map〉 ≝ result in |
---|
| 1093 | let 〈pc, sigma_map〉 ≝ pc_sigma_map in |
---|
[832] | 1094 | if gtb pc (2^16) then |
---|
| 1095 | None ? |
---|
| 1096 | else |
---|
[985] | 1097 | Some ? (λx.lookup ?? x sigma_map (zero …)) ]. |
---|
[832] | 1098 | |
---|
[985] | 1099 | (* stuff about policy *) |
---|
| 1100 | |
---|
[987] | 1101 | definition policy_ok ≝ λjump_expansion,p. sigma_safe p jump_expansion ≠ None …. |
---|
[985] | 1102 | |
---|
[987] | 1103 | definition policy ≝ λp. Σjump_expansion:policy_type. policy_ok jump_expansion p. |
---|
| 1104 | |
---|
| 1105 | lemma eject_policy: ∀p. policy p → policy_type. |
---|
| 1106 | #p #pol @(eject … pol) |
---|
| 1107 | qed. |
---|
| 1108 | |
---|
| 1109 | coercion eject_policy nocomposites: ∀p.∀pol:policy p. policy_type ≝ eject_policy on _pol:(policy ?) to policy_type. |
---|
| 1110 | |
---|
| 1111 | definition sigma: ∀p:pseudo_assembly_program. policy p → Word → Word ≝ |
---|
| 1112 | λp,policy. |
---|
| 1113 | match sigma_safe p (eject … policy) return λr:option (Word → Word). r ≠ None … → Word → Word with |
---|
[985] | 1114 | [ None ⇒ λabs. ⊥ |
---|
[987] | 1115 | | Some r ⇒ λ_.r] (sig2 … policy). |
---|
[993] | 1116 | cases abs /2/ |
---|
[985] | 1117 | qed. |
---|
| 1118 | |
---|
[987] | 1119 | example sigma_0: ∀p,pol. sigma p pol (bitvector_of_nat ? 0) = bitvector_of_nat ? 0. |
---|
[985] | 1120 | cases daemon. |
---|
| 1121 | qed. |
---|
| 1122 | |
---|
[1482] | 1123 | axiom fetch_pseudo_instruction_split: |
---|
| 1124 | ∀instr_list,ppc. |
---|
| 1125 | ∃pre,suff,lbl. |
---|
| 1126 | (pre @ [〈lbl,\fst (fetch_pseudo_instruction instr_list ppc)〉]) @ suff = instr_list. |
---|
| 1127 | |
---|
| 1128 | lemma sigma00_append: |
---|
| 1129 | ∀instr_list,jump_expansion,l1,l2,acc. |
---|
| 1130 | sigma00 instr_list jump_expansion (l1@l2) acc = |
---|
| 1131 | sigma00 instr_list jump_expansion |
---|
| 1132 | l2 (sigma00 instr_list jump_expansion l1 acc). |
---|
| 1133 | whd in match sigma00; normalize nodelta; |
---|
| 1134 | #instr_list #jump_expansion #l1 #l2 #acc @foldl_append |
---|
| 1135 | qed. |
---|
| 1136 | |
---|
| 1137 | lemma sigma00_strict: |
---|
| 1138 | ∀instr_list,jump_expansion,l,acc. acc = None ? → |
---|
| 1139 | sigma00 instr_list jump_expansion l acc = None …. |
---|
| 1140 | #instr_list #jump_expansion #l elim l |
---|
| 1141 | [ #acc #H >H % |
---|
| 1142 | | #hd #tl #IH #acc #H >H change with (sigma00 ?? tl ? = ?) @IH % ] |
---|
| 1143 | qed. |
---|
| 1144 | |
---|
| 1145 | lemma policy_ok_prefix_ok: |
---|
| 1146 | ∀program.∀pol:policy program.∀suffix,prefix. |
---|
| 1147 | prefix@suffix = \snd program → |
---|
| 1148 | sigma00 program pol prefix (Some … 〈0, 〈0, Stub …〉〉) ≠ None …. |
---|
| 1149 | * #preamble #instr_list #pol #suffix #prefix #prf whd in prf:(???%); |
---|
| 1150 | generalize in match (sig2 ?? pol) whd in prf:(???%) <prf in pol; #pol |
---|
| 1151 | whd in match policy_ok; whd in match sigma_safe; whd in match sigma0 |
---|
| 1152 | normalize nodelta >sigma00_append |
---|
| 1153 | cases (sigma00 ?? prefix ?) |
---|
| 1154 | [2: #x #_ % #abs destruct(abs) |
---|
| 1155 | | * #abs @⊥ @abs >sigma00_strict % ] |
---|
| 1156 | qed. |
---|
| 1157 | |
---|
| 1158 | lemma policy_ok_prefix_hd_ok: |
---|
| 1159 | ∀program.∀pol:policy program.∀suffix,hd,prefix,ppc_pc_map. |
---|
| 1160 | (prefix@[hd])@suffix = \snd program → |
---|
| 1161 | Some ? ppc_pc_map = sigma00 program pol prefix (Some … 〈0, 〈0, Stub …〉〉) → |
---|
| 1162 | let 〈ppc,pc_map〉 ≝ ppc_pc_map in |
---|
| 1163 | let 〈program_counter, sigma_map〉 ≝ pc_map in |
---|
| 1164 | let 〈label, i〉 ≝ hd in |
---|
| 1165 | construct_costs_safe program pol ppc program_counter (Stub …) i ≠ None …. |
---|
| 1166 | * #preamble #instr_list #pol #suffix #hd #prefix #ppc_pc_map #EQ1 #EQ2 |
---|
| 1167 | generalize in match (policy_ok_prefix_ok 〈preamble,instr_list〉 pol suffix |
---|
| 1168 | (prefix@[hd]) EQ1) in ⊢ ? >sigma00_append <EQ2 whd in ⊢ (?(??%?) → ?) |
---|
| 1169 | @pair_elim' #ppc #pc_map #EQ3 normalize nodelta |
---|
| 1170 | @pair_elim' #pc #map #EQ4 normalize nodelta |
---|
| 1171 | @pair_elim' #l' #i' #EQ5 normalize nodelta |
---|
| 1172 | cases (construct_costs_safe ??????) normalize |
---|
| 1173 | [* #abs @⊥ @abs % | #X #_ % #abs destruct(abs)] |
---|
| 1174 | qed. |
---|
| 1175 | |
---|
[998] | 1176 | definition expand_pseudo_instruction: |
---|
| 1177 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
[1014] | 1178 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pc. |
---|
[998] | 1179 | lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) → |
---|
| 1180 | lookup_datalabels = (λx. lookup ?? x (construct_datalabels (\fst program)) (zero ?)) → |
---|
[1014] | 1181 | let pi ≝ \fst (fetch_pseudo_instruction (\snd program) ppc) in |
---|
[998] | 1182 | pc = sigma program pol ppc → |
---|
| 1183 | Σres:list instruction. Some … res = expand_pseudo_instruction_safe lookup_labels lookup_datalabels pc (pol ppc) pi |
---|
[1014] | 1184 | ≝ λprogram,pol,ppc,lookup_labels,lookup_datalabels,pc,prf1,prf2,prf3. |
---|
| 1185 | match expand_pseudo_instruction_safe lookup_labels lookup_datalabels pc (pol ppc) (\fst (fetch_pseudo_instruction (\snd program) ppc)) with |
---|
[998] | 1186 | [ None ⇒ let dummy ≝ [ ] in dummy |
---|
| 1187 | | Some res ⇒ res ]. |
---|
[1482] | 1188 | [ @⊥ whd in p:(??%??); |
---|
| 1189 | generalize in match (sig2 ?? pol) whd in ⊢ (% → ?) |
---|
| 1190 | whd in ⊢ (?(??%?) → ?) change in ⊢ (?(??(match % with [_ ⇒ ? | _ ⇒ ?])?) → ?) with (sigma00 ????) |
---|
| 1191 | generalize in match (refl … (sigma00 program pol (\snd program) (Some ? 〈O,〈O,Stub (BitVector 16) 16〉〉))) |
---|
| 1192 | cases (sigma00 ????) in ⊢ (??%? → %) normalize nodelta [#_ * #abs @abs %] |
---|
| 1193 | #res #K |
---|
| 1194 | cases (fetch_pseudo_instruction_split (\snd program) ppc) #pre * #suff * #lbl #EQ1 |
---|
| 1195 | generalize in match (policy_ok_prefix_hd_ok program pol … EQ1 ?) in ⊢ ? |
---|
| 1196 | cases daemon (* CSC: XXXXXXXX Ero qui |
---|
| 1197 | |
---|
| 1198 | [3: @policy_ok_prefix_ok ] |
---|
| 1199 | | sigma00 program pol pre |
---|
| 1200 | |
---|
| 1201 | |
---|
| 1202 | |
---|
| 1203 | QUA USARE LEMMA policy_ok_prefix_hd_ok combinato a lemma da fare che |
---|
| 1204 | fetch ppc = hd sse program = pre @ [hd] @ tl e |pre| = ppc |
---|
| 1205 | per concludere construct_costs_safe ≠ None *) |
---|
[998] | 1206 | | >p %] |
---|
| 1207 | qed. |
---|
| 1208 | |
---|
[993] | 1209 | (* MAIN AXIOM HERE, HIDDEN USING cases daemon *) |
---|
[1309] | 1210 | definition assembly_1_pseudoinstruction': |
---|
[993] | 1211 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
| 1212 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi. |
---|
| 1213 | lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) → |
---|
| 1214 | lookup_datalabels = (λx. lookup ?? x (construct_datalabels (\fst program)) (zero ?)) → |
---|
| 1215 | \fst (fetch_pseudo_instruction (\snd program) ppc) = pi → |
---|
| 1216 | Σres:(nat × (list Byte)). |
---|
[1309] | 1217 | Some … res = assembly_1_pseudoinstruction_safe program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi ∧ |
---|
[993] | 1218 | let 〈len,code〉 ≝ res in |
---|
| 1219 | sigma program pol (\snd (half_add ? ppc (bitvector_of_nat ? 1))) = |
---|
| 1220 | bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len) |
---|
| 1221 | ≝ λprogram: pseudo_assembly_program. |
---|
| 1222 | λpol: policy program. |
---|
| 1223 | λppc: Word. |
---|
| 1224 | λlookup_labels. |
---|
| 1225 | λlookup_datalabels. |
---|
| 1226 | λpi. |
---|
| 1227 | λprf1,prf2,prf3. |
---|
| 1228 | match assembly_1_pseudoinstruction_safe program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi with |
---|
| 1229 | [ None ⇒ let dummy ≝ 〈0,[ ]〉 in dummy |
---|
| 1230 | | Some res ⇒ res ]. |
---|
| 1231 | [ @⊥ elim pi in p [*] |
---|
| 1232 | try (#ARG1 #ARG2 #ARG3 #abs) try (#ARG1 #ARG2 #abs) try (#ARG1 #abs) try #abs |
---|
| 1233 | generalize in match (jmeq_to_eq ??? abs) -abs; #abs whd in abs:(??%?); try destruct(abs) |
---|
| 1234 | whd in abs:(??match % with [_ ⇒ ? | _ ⇒ ?]?); |
---|
| 1235 | (* WRONG HERE, NEEDS LEMMA SAYING THAT THE POLICY DOES NOT RETURN MEDIUM! *) |
---|
| 1236 | cases daemon |
---|
[1309] | 1237 | | % [ >p %] |
---|
| 1238 | cases res in p ⊢ %; -res; #len #code #EQ normalize nodelta; |
---|
[993] | 1239 | (* THIS SHOULD BE TRUE INSTEAD *) |
---|
| 1240 | cases daemon] |
---|
| 1241 | qed. |
---|
| 1242 | |
---|
[1309] | 1243 | definition assembly_1_pseudoinstruction: |
---|
| 1244 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
| 1245 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi. |
---|
| 1246 | lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) → |
---|
| 1247 | lookup_datalabels = (λx. lookup ?? x (construct_datalabels (\fst program)) (zero ?)) → |
---|
| 1248 | \fst (fetch_pseudo_instruction (\snd program) ppc) = pi → |
---|
| 1249 | nat × (list Byte) |
---|
| 1250 | ≝ λprogram,pol,ppc,lookup_labels,lookup_datalabels,pi,prf1,prf2,prf3. |
---|
| 1251 | assembly_1_pseudoinstruction' program pol ppc lookup_labels lookup_datalabels pi prf1 |
---|
| 1252 | prf2 prf3. |
---|
| 1253 | |
---|
| 1254 | lemma assembly_1_pseudoinstruction_ok1: |
---|
| 1255 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
| 1256 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi. |
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| 1257 | ∀prf1:lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)). |
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| 1258 | ∀prf2:lookup_datalabels = (λx. lookup ?? x (construct_datalabels (\fst program)) (zero ?)). |
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| 1259 | ∀prf3:\fst (fetch_pseudo_instruction (\snd program) ppc) = pi. |
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| 1260 | Some … (assembly_1_pseudoinstruction program pol ppc lookup_labels lookup_datalabels pi prf1 prf2 prf3) |
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| 1261 | = assembly_1_pseudoinstruction_safe program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi. |
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| 1262 | #program #pol #ppc #lookup_labels #lookup_datalabels #pi #prf1 #prf2 #prf3 |
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| 1263 | cases (sig2 … (assembly_1_pseudoinstruction' program pol ppc lookup_labels lookup_datalabels pi prf1 prf2 prf3)) |
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| 1264 | #H1 #_ @H1 |
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| 1265 | qed. |
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| 1266 | |
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[993] | 1267 | (* MAIN AXIOM HERE, HIDDEN USING cases daemon *) |
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| 1268 | definition construct_costs': |
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[1309] | 1269 | ∀program. ∀pol:policy program. ∀ppc,pc,costs,i. |
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| 1270 | Σres:(nat × (BitVectorTrie Word 16)). Some … res = construct_costs_safe program pol ppc pc costs i |
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[993] | 1271 | ≝ |
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| 1272 | λprogram.λpol: policy program.λppc,pc,costs,i. |
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| 1273 | match construct_costs_safe program pol ppc pc costs i with |
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| 1274 | [ None ⇒ let dummy ≝ 〈0, Stub ??〉 in dummy |
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| 1275 | | Some res ⇒ res ]. |
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| 1276 | [ cases daemon |
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[1309] | 1277 | | >p %] |
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[993] | 1278 | qed. |
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| 1279 | |
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| 1280 | definition construct_costs ≝ |
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| 1281 | λprogram,pol,ppc,pc,costs,i. eject … (construct_costs' program pol ppc pc costs i). |
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| 1282 | |
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[1482] | 1283 | (* |
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| 1284 | axiom suffix_of: ∀A:Type[0]. ∀l,prefix:list A. list A. |
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| 1285 | axiom suffix_of_ok: ∀A,l,prefix. prefix @ suffix_of A l prefix = l. |
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[1309] | 1286 | |
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[1482] | 1287 | axiom foldl_strong_step: |
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| 1288 | ∀A:Type[0]. |
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| 1289 | ∀P: list A → Type[0]. |
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| 1290 | ∀l: list A. |
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| 1291 | ∀H: ∀prefix,hd,tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]). |
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| 1292 | ∀acc: P [ ]. |
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| 1293 | ∀Q: ∀prefix. P prefix → Prop. |
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| 1294 | ∀HQ: ∀prefix,hd,tl.∀prf: l = prefix @ [hd] @ tl. |
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| 1295 | ∀acc: P prefix. Q prefix acc → Q (prefix @ [hd]) (H prefix hd tl prf acc). |
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| 1296 | Q [ ] acc → |
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| 1297 | Q l (foldl_strong A P l H acc). |
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| 1298 | (* |
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| 1299 | #A #P #l #H #acc #Q #HQ #Hacc normalize; |
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| 1300 | generalize in match |
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| 1301 | (foldl_strong ? |
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| 1302 | (λpre. Q pre (foldl_strong_internal A P l (suffix_of A l pre) ? [ ] pre acc ?)) |
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| 1303 | l ? Hacc) |
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| 1304 | [3: >suffix_of_ok % | 2: #prefix #hd #tl #EQ @(H prefix hd (tl@suffix_of A l pre) EQ) ] |
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| 1305 | [2: #prefix #hd #tl #prf #X whd in ⊢ (??%) |
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| 1306 | #K |
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[1309] | 1307 | |
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[1482] | 1308 | generalize in match |
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| 1309 | (foldl_strong ? |
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| 1310 | (λpre. Q pre (foldl_strong_internal A P l H pre (suffix_of A l pre) acc (suffix_of_ok A l pre)))) |
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| 1311 | [2: @H |
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| 1312 | *) |
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[1309] | 1313 | |
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[1482] | 1314 | axiom foldl_elim: |
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| 1315 | ∀A:Type[0]. |
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| 1316 | ∀B: Type[0]. |
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| 1317 | ∀H: A → B → A. |
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| 1318 | ∀acc: A. |
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| 1319 | ∀l: list B. |
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| 1320 | ∀Q: A → Prop. |
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| 1321 | (∀acc:A.∀b:B. Q acc → Q (H acc b)) → |
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| 1322 | Q acc → |
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| 1323 | Q (foldl A B H acc l). |
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| 1324 | *) |
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[1309] | 1325 | |
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[1482] | 1326 | lemma option_destruct_Some: ∀A,a,b. Some A a = Some A b → a=b. |
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| 1327 | #A #a #b #EQ destruct // |
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[1309] | 1328 | qed. |
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| 1329 | |
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| 1330 | (* |
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| 1331 | lemma tech_pc_sigma00_append_Some: |
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| 1332 | ∀program.∀pol:policy program.∀prefix,costs,label,i,ppc,pc. |
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| 1333 | tech_pc_sigma00 program pol prefix = Some … 〈ppc,pc〉 → |
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| 1334 | tech_pc_sigma00 program pol (prefix@[〈label,i〉]) = Some … 〈S ppc,\fst (construct_costs program pol … ppc pc costs i)〉. |
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| 1335 | #program #pol #prefix #costs #label #i #ppc #pc #H |
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| 1336 | whd in match tech_pc_sigma00 in ⊢ %; normalize nodelta; |
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| 1337 | whd in match sigma00 in ⊢ %; normalize nodelta in ⊢ %; |
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| 1338 | generalize in match (sig2 … pol) whd in ⊢ (% → ?) whd in ⊢ (?(??%?) → ?) |
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| 1339 | whd in match sigma0; normalize nodelta; |
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| 1340 | >foldl_step |
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| 1341 | change with (? → match match sigma00 program pol prefix with [None ⇒ ? | Some res ⇒ ?] with [ None ⇒ ? | Some res ⇒ ? ] = ?) |
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| 1342 | whd in match tech_pc_sigma00 in H; normalize nodelta in H; |
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| 1343 | cases (sigma00 program pol prefix) in H ⊢ % |
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| 1344 | [ whd in ⊢ (??%% → ?) #abs destruct(abs) |
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| 1345 | | * #ppc' * #pc' #sigma_map normalize nodelta; #H generalize in match (option_destruct_Some ??? H) |
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| 1346 | |
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| 1347 | normalize nodelta; -H; |
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| 1348 | |
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| 1349 | |
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| 1350 | generalize in match H; -H; |
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| 1351 | generalize in match (foldl ?????); in H ⊢ (??match match % with [_ ⇒ ? | _ ⇒ ?] with [_ ⇒ ? | _ ⇒ ?]?) |
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| 1352 | [2: whd in ⊢ (??%%) |
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| 1353 | XXX |
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| 1354 | *) |
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| 1355 | |
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[985] | 1356 | axiom construct_costs_sigma: |
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[993] | 1357 | ∀p.∀pol:policy p.∀ppc,pc,costs,i. |
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[987] | 1358 | bitvector_of_nat ? pc = sigma p pol (bitvector_of_nat ? ppc) → |
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[993] | 1359 | bitvector_of_nat ? (\fst (construct_costs p pol ppc pc costs i)) = sigma p pol (bitvector_of_nat 16 (S ppc)). |
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[985] | 1360 | |
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| 1361 | axiom tech_pc_sigma00_append_Some: |
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[993] | 1362 | ∀program.∀pol:policy program.∀prefix,costs,label,i,ppc,pc. |
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[987] | 1363 | tech_pc_sigma00 program pol prefix = Some … 〈ppc,pc〉 → |
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[993] | 1364 | tech_pc_sigma00 program pol (prefix@[〈label,i〉]) = Some … 〈S ppc,\fst (construct_costs program pol … ppc pc costs i)〉. |
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[985] | 1365 | |
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[989] | 1366 | definition build_maps: |
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| 1367 | ∀pseudo_program.∀pol:policy pseudo_program. |
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| 1368 | Σres:((BitVectorTrie Word 16) × (BitVectorTrie Word 16)). |
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| 1369 | let 〈labels,costs〉 ≝ res in |
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| 1370 | ∀id. occurs_exactly_once id (\snd pseudo_program) → |
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| 1371 | lookup ?? id labels (zero …) = sigma pseudo_program pol (address_of_word_labels_code_mem (\snd pseudo_program) id) |
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[993] | 1372 | ≝ |
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[987] | 1373 | λpseudo_program.λpol:policy pseudo_program. |
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[985] | 1374 | let result ≝ |
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| 1375 | foldl_strong |
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| 1376 | (option Identifier × pseudo_instruction) |
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| 1377 | (λpre. Σres:((BitVectorTrie Word 16) × (nat × (nat × (BitVectorTrie Word 16)))). |
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| 1378 | let 〈labels,ppc_pc_costs〉 ≝ res in |
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| 1379 | let 〈ppc',pc_costs〉 ≝ ppc_pc_costs in |
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| 1380 | let 〈pc',costs〉 ≝ pc_costs in |
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[993] | 1381 | And ( And ( |
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| 1382 | And (bitvector_of_nat ? pc' = sigma pseudo_program pol (bitvector_of_nat ? ppc')) (*∧*) |
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| 1383 | (ppc' = length … pre)) (*∧ *) |
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| 1384 | (tech_pc_sigma00 pseudo_program (eject … pol) pre = Some ? 〈ppc',pc'〉)) (*∧*) |
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| 1385 | (∀id. occurs_exactly_once id pre → |
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| 1386 | lookup ?? id labels (zero …) = sigma pseudo_program pol (address_of_word_labels_code_mem pre id))) |
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[985] | 1387 | (\snd pseudo_program) |
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| 1388 | (λprefix,i,tl,prf,t. |
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| 1389 | let 〈labels, ppc_pc_costs〉 ≝ t in |
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| 1390 | let 〈ppc, pc_costs〉 ≝ ppc_pc_costs in |
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| 1391 | let 〈pc,costs〉 ≝ pc_costs in |
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| 1392 | let 〈label, i'〉 ≝ i in |
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| 1393 | let labels ≝ |
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| 1394 | match label with |
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| 1395 | [ None ⇒ labels |
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| 1396 | | Some label ⇒ |
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| 1397 | let program_counter_bv ≝ bitvector_of_nat ? pc in |
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[993] | 1398 | insert ? ? label program_counter_bv labels] |
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[985] | 1399 | in |
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[993] | 1400 | let construct ≝ construct_costs pseudo_program pol ppc pc costs i' in |
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| 1401 | 〈labels, 〈S ppc,construct〉〉 |
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[985] | 1402 | ) 〈(Stub ? ?), 〈0, 〈0, Stub ? ?〉〉〉 |
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| 1403 | in |
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| 1404 | let 〈labels, ppc_pc_costs〉 ≝ result in |
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| 1405 | let 〈ppc,pc_costs〉 ≝ ppc_pc_costs in |
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| 1406 | let 〈pc, costs〉 ≝ pc_costs in |
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| 1407 | 〈labels, costs〉. |
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[993] | 1408 | [2: whd generalize in match (sig2 … t) >p >p1 >p2 >p3 * * * #IHn1 #IH0 #IH1 #IH2 |
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| 1409 | generalize in match (refl … construct); cases construct in ⊢ (???% → %) #PC #CODE #JMEQ % [% [%]] |
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| 1410 | [ <(construct_costs_sigma … costs i1 IHn1) change with (? = ?(\fst construct)) >JMEQ % |
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[1363] | 1411 | | >append_length <IH0 normalize; -IHn1; (*CSC: otherwise it diverges!*) // |
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[993] | 1412 | | >(tech_pc_sigma00_append_Some … costs … IH1) change with (Some … 〈S ppc,\fst construct〉 = ?) >JMEQ % |
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[985] | 1413 | | #id normalize nodelta; -labels1; cases label; normalize nodelta |
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| 1414 | [ #K <address_of_word_labels_code_mem_None [2:@K] @IH2 -IHn1; (*CSC: otherwise it diverges!*) // |
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| 1415 | | #l #H generalize in match (occurs_exactly_once_Some ???? H) in ⊢ ?; |
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| 1416 | generalize in match (refl … (eq_bv ? l id)); cases (eq_bv … l id) in ⊢ (???% → %) |
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| 1417 | [ #EQ #_ <(eq_bv_eq … EQ) >lookup_insert_hit >address_of_word_labels_code_mem_Some_hit |
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| 1418 | <IH0 [@IHn1 | <(eq_bv_eq … EQ) in H #H @H] |
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[1482] | 1419 | | #EQ change with (occurs_exactly_once ?? → ?) #K >lookup_insert_miss [2: -IHn1; (*Andrea:XXXX used to work /2/*)>eq_bv_sym >EQ // ] |
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[985] | 1420 | <(address_of_word_labels_code_mem_Some_miss … EQ … H) @IH2 -IHn1; //]]] |
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[993] | 1421 | |3: whd % [% [%]] // [>sigma_0//] #id normalize in ⊢ (% → ?) #abs @⊥ // |
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| 1422 | | generalize in match (sig2 … result) in ⊢ ?; |
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| 1423 | normalize nodelta in p ⊢ %; -result; >p in ⊢ (match % with [_ ⇒ ?] → ?) |
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| 1424 | normalize nodelta; >p1 normalize nodelta; >p2; normalize nodelta; * #_; #H @H] |
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[985] | 1425 | qed. |
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| 1426 | |
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[993] | 1427 | definition assembly: |
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| 1428 | ∀p:pseudo_assembly_program. policy p → list Byte × (BitVectorTrie Identifier 16) ≝ |
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| 1429 | λp.let 〈preamble, instr_list〉 ≝ p in |
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| 1430 | λpol. |
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| 1431 | let 〈labels,costs〉 ≝ build_maps 〈preamble,instr_list〉 pol in |
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[985] | 1432 | let datalabels ≝ construct_datalabels preamble in |
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| 1433 | let lookup_labels ≝ λx. lookup ? ? x labels (zero ?) in |
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| 1434 | let lookup_datalabels ≝ λx. lookup ? ? x datalabels (zero ?) in |
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[993] | 1435 | let result ≝ |
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| 1436 | foldl_strong |
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| 1437 | (option Identifier × pseudo_instruction) |
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| 1438 | (λpre. Σpc_ppc_code:(Word × (Word × (list Byte))). |
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| 1439 | let 〈pc,ppc_code〉 ≝ pc_ppc_code in |
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| 1440 | let 〈ppc,code〉 ≝ ppc_code in |
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| 1441 | ∀ppc'. |
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| 1442 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction instr_list ppc' in |
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| 1443 | let 〈len,assembledi〉 ≝ |
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| 1444 | assembly_1_pseudoinstruction 〈preamble,instr_list〉 pol ppc' lookup_labels lookup_datalabels pi ??? in |
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| 1445 | True) |
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| 1446 | instr_list |
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| 1447 | (λprefix,hd,tl,prf,pc_ppc_code. |
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| 1448 | let 〈pc, ppc_code〉 ≝ pc_ppc_code in |
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| 1449 | let 〈ppc, code〉 ≝ ppc_code in |
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| 1450 | let 〈pc_delta, program〉 ≝ assembly_1_pseudoinstruction 〈preamble,instr_list〉 pol ppc lookup_labels lookup_datalabels (\snd hd) ??? in |
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| 1451 | let 〈new_pc, flags〉 ≝ add_16_with_carry pc (bitvector_of_nat ? pc_delta) false in |
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| 1452 | let new_ppc ≝ \snd (half_add ? ppc (bitvector_of_nat ? 1)) in |
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| 1453 | 〈new_pc, 〈new_ppc, (code @ program)〉〉) |
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| 1454 | 〈(zero ?), 〈(zero ?), [ ]〉〉 |
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[985] | 1455 | in |
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[993] | 1456 | 〈\snd (\snd result), costs〉. |
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| 1457 | [2,5: % |
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| 1458 | |*: cases daemon ] |
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| 1459 | qed. |
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[832] | 1460 | |
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[820] | 1461 | definition assembly_unlabelled_program: assembly_program → option (list Byte × (BitVectorTrie Identifier 16)) ≝ |
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[1014] | 1462 | λp. Some ? (〈foldr ? ? (λi,l. assembly1 i @ l) [ ] p, Stub …〉). |
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