[719] | 1 | include "ASM/ASM.ma". |
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| 2 | include "ASM/Arithmetic.ma". |
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| 3 | include "ASM/Fetch.ma". |
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| 4 | include "ASM/Status.ma". |
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[1459] | 5 | include alias "basics/logic.ma". |
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[1528] | 6 | include alias "arithmetics/nat.ma". |
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[1948] | 7 | include "utilities/extralib.ma". |
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[719] | 8 | |
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[989] | 9 | (**************************************** START OF POLICY ABSTRACTION ********************) |
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| 10 | |
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[1925] | 11 | (* definition of & operations on jump length *) |
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[1615] | 12 | inductive jump_length: Type[0] ≝ |
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| 13 | | short_jump: jump_length |
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[2101] | 14 | | absolute_jump: jump_length |
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[1925] | 15 | | long_jump: jump_length. |
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[2048] | 16 | |
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| 17 | (* Functions that define the conditions under which jumps are possible *) |
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| 18 | definition short_jump_cond: Word → Word → (*pseudo_instruction →*) |
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| 19 | bool × (BitVector 8) ≝ |
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| 20 | λpc_plus_jmp_length.λaddr.(*λinstr.*) |
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| 21 | let 〈result, flags〉 ≝ sub_16_with_carry addr pc_plus_jmp_length false in |
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[2051] | 22 | let 〈upper, lower〉 ≝ vsplit ? 9 7 result in |
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[2048] | 23 | if get_index' ? 2 0 flags then |
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[2051] | 24 | 〈eq_bv 9 upper [[true;true;true;true;true;true;true;true;true]], true:::lower〉 |
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[2048] | 25 | else |
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[2051] | 26 | 〈eq_bv 9 upper (zero …), false:::lower〉. |
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[2048] | 27 | |
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[2101] | 28 | definition absolute_jump_cond: Word → Word → (*pseudo_instruction →*) |
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[2048] | 29 | bool × (BitVector 11) ≝ |
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| 30 | λpc_plus_jmp_length.λaddr.(*λinstr.*) |
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| 31 | let 〈fst_5_addr, rest_addr〉 ≝ vsplit bool 5 11 addr in |
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| 32 | let 〈fst_5_pc, rest_pc〉 ≝ vsplit bool 5 11 pc_plus_jmp_length in |
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| 33 | 〈eq_bv 5 fst_5_addr fst_5_pc, rest_addr〉. |
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[1615] | 34 | |
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| 35 | definition assembly_preinstruction ≝ |
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| 36 | λA: Type[0]. |
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| 37 | λaddr_of: A → Byte. (* relative *) |
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| 38 | λpre: preinstruction A. |
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| 39 | match pre with |
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| 40 | [ ADD addr1 addr2 ⇒ |
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| 41 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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| 42 | [ REGISTER r ⇒ λ_.[ ([[false;false;true;false;true]]) @@ r ] |
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| 43 | | DIRECT b1 ⇒ λ_.[ ([[false;false;true;false;false;true;false;true]]); b1 ] |
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| 44 | | INDIRECT i1 ⇒ λ_. [ ([[false;false;true;false;false;true;true;i1]]) ] |
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| 45 | | DATA b1 ⇒ λ_. [ ([[false;false;true;false;false;true;false;false]]) ; b1 ] |
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| 46 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 47 | | ADDC addr1 addr2 ⇒ |
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| 48 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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| 49 | [ REGISTER r ⇒ λ_.[ ([[false;false;true;true;true]]) @@ r ] |
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| 50 | | DIRECT b1 ⇒ λ_.[ ([[false;false;true;true;false;true;false;true]]); b1 ] |
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| 51 | | INDIRECT i1 ⇒ λ_. [ ([[false;false;true;true;false;true;true;i1]]) ] |
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| 52 | | DATA b1 ⇒ λ_. [ ([[false;false;true;true;false;true;false;false]]) ; b1 ] |
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| 53 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 54 | | ANL addrs ⇒ |
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| 55 | match addrs with |
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| 56 | [ inl addrs ⇒ match addrs with |
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| 57 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 58 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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| 59 | [ REGISTER r ⇒ λ_.[ ([[false;true;false;true;true]]) @@ r ] |
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| 60 | | DIRECT b1 ⇒ λ_.[ ([[false;true;false;true;false;true;false;true]]); b1 ] |
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| 61 | | INDIRECT i1 ⇒ λ_. [ ([[false;true;false;true;false;true;true;i1]]) ] |
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| 62 | | DATA b1 ⇒ λ_. [ ([[false;true;false;true;false;true;false;false]]) ; b1 ] |
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| 63 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 64 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 65 | let b1 ≝ |
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| 66 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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| 67 | [ DIRECT b1 ⇒ λ_.b1 |
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| 68 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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| 69 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;data]] x) → ? with |
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| 70 | [ ACC_A ⇒ λ_.[ ([[false;true;false;true;false;false;true;false]]) ; b1 ] |
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| 71 | | DATA b2 ⇒ λ_. [ ([[false;true;false;true;false;false;true;true]]) ; b1 ; b2 ] |
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| 72 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 73 | ] |
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| 74 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 75 | match addr2 return λx. bool_to_Prop (is_in ? [[bit_addr;n_bit_addr]] x) → ? with |
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| 76 | [ BIT_ADDR b1 ⇒ λ_.[ ([[true;false;false;false;false;false;true;false]]) ; b1 ] |
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| 77 | | N_BIT_ADDR b1 ⇒ λ_. [ ([[true;false;true;true;false;false;false;false]]) ; b1 ] |
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| 78 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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| 79 | | CLR addr ⇒ |
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| 80 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;carry;bit_addr]] x) → ? with |
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| 81 | [ ACC_A ⇒ λ_. |
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| 82 | [ ([[true; true; true; false; false; true; false; false]]) ] |
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| 83 | | CARRY ⇒ λ_. |
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| 84 | [ ([[true; true; false; false; false; false; true; true]]) ] |
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| 85 | | BIT_ADDR b1 ⇒ λ_. |
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| 86 | [ ([[true; true; false; false; false; false; true; false]]) ; b1 ] |
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| 87 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 88 | | CPL addr ⇒ |
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| 89 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;carry;bit_addr]] x) → ? with |
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| 90 | [ ACC_A ⇒ λ_. |
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| 91 | [ ([[true; true; true; true; false; true; false; false]]) ] |
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| 92 | | CARRY ⇒ λ_. |
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| 93 | [ ([[true; false; true; true; false; false; true; true]]) ] |
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| 94 | | BIT_ADDR b1 ⇒ λ_. |
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| 95 | [ ([[true; false; true; true; false; false; true; false]]) ; b1 ] |
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| 96 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 97 | | DA addr ⇒ |
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| 98 | [ ([[true; true; false; true; false; true; false; false]]) ] |
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| 99 | | DEC addr ⇒ |
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| 100 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;registr;direct;indirect]] x) → ? with |
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| 101 | [ ACC_A ⇒ λ_. |
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| 102 | [ ([[false; false; false; true; false; true; false; false]]) ] |
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| 103 | | REGISTER r ⇒ λ_. |
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| 104 | [ ([[false; false; false; true; true]]) @@ r ] |
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| 105 | | DIRECT b1 ⇒ λ_. |
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| 106 | [ ([[false; false; false; true; false; true; false; true]]); b1 ] |
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| 107 | | INDIRECT i1 ⇒ λ_. |
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| 108 | [ ([[false; false; false; true; false; true; true; i1]]) ] |
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| 109 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 110 | | DJNZ addr1 addr2 ⇒ |
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| 111 | let b2 ≝ addr_of addr2 in |
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| 112 | match addr1 return λx. bool_to_Prop (is_in ? [[registr;direct]] x) → ? with |
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| 113 | [ REGISTER r ⇒ λ_. |
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| 114 | [ ([[true; true; false; true; true]]) @@ r ; b2 ] |
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| 115 | | DIRECT b1 ⇒ λ_. |
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| 116 | [ ([[true; true; false; true; false; true; false; true]]); b1; b2 ] |
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| 117 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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| 118 | | JC addr ⇒ |
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| 119 | let b1 ≝ addr_of addr in |
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| 120 | [ ([[false; true; false; false; false; false; false; false]]); b1 ] |
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| 121 | | JNC addr ⇒ |
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| 122 | let b1 ≝ addr_of addr in |
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| 123 | [ ([[false; true; false; true; false; false; false; false]]); b1 ] |
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| 124 | | JZ addr ⇒ |
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| 125 | let b1 ≝ addr_of addr in |
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| 126 | [ ([[false; true; true; false; false; false; false; false]]); b1 ] |
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| 127 | | JNZ addr ⇒ |
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| 128 | let b1 ≝ addr_of addr in |
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| 129 | [ ([[false; true; true; true; false; false; false; false]]); b1 ] |
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| 130 | | JB addr1 addr2 ⇒ |
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| 131 | let b2 ≝ addr_of addr2 in |
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| 132 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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| 133 | [ BIT_ADDR b1 ⇒ λ_. |
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| 134 | [ ([[false; false; true; false; false; false; false; false]]); b1; b2 ] |
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| 135 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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| 136 | | JNB addr1 addr2 ⇒ |
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| 137 | let b2 ≝ addr_of addr2 in |
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| 138 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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| 139 | [ BIT_ADDR b1 ⇒ λ_. |
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| 140 | [ ([[false; false; true; true; false; false; false; false]]); b1; b2 ] |
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| 141 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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| 142 | | JBC addr1 addr2 ⇒ |
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| 143 | let b2 ≝ addr_of addr2 in |
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| 144 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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| 145 | [ BIT_ADDR b1 ⇒ λ_. |
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| 146 | [ ([[false; false; false; true; false; false; false; false]]); b1; b2 ] |
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| 147 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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| 148 | | CJNE addrs addr3 ⇒ |
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| 149 | let b3 ≝ addr_of addr3 in |
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| 150 | match addrs with |
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| 151 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 152 | match addr2 return λx. bool_to_Prop (is_in ? [[direct;data]] x) → ? with |
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| 153 | [ DIRECT b1 ⇒ λ_. |
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| 154 | [ ([[true; false; true; true; false; true; false; true]]); b1; b3 ] |
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| 155 | | DATA b1 ⇒ λ_. |
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| 156 | [ ([[true; false; true; true; false; true; false; false]]); b1; b3 ] |
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| 157 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 158 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 159 | let b2 ≝ |
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| 160 | match addr2 return λx. bool_to_Prop (is_in ? [[data]] x) → ? with |
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| 161 | [ DATA b2 ⇒ λ_. b2 |
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| 162 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) in |
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| 163 | match addr1 return λx. bool_to_Prop (is_in ? [[registr;indirect]] x) → list Byte with |
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| 164 | [ REGISTER r ⇒ λ_. |
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| 165 | [ ([[true; false; true; true; true]]) @@ r; b2; b3 ] |
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| 166 | | INDIRECT i1 ⇒ λ_. |
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| 167 | [ ([[true; false; true; true; false; true; true; i1]]); b2; b3 ] |
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| 168 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
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| 169 | ] |
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| 170 | | DIV addr1 addr2 ⇒ |
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| 171 | [ ([[true;false;false;false;false;true;false;false]]) ] |
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| 172 | | INC addr ⇒ |
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| 173 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;registr;direct;indirect;dptr]] x) → ? with |
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| 174 | [ ACC_A ⇒ λ_. |
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| 175 | [ ([[false;false;false;false;false;true;false;false]]) ] |
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| 176 | | REGISTER r ⇒ λ_. |
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| 177 | [ ([[false;false;false;false;true]]) @@ r ] |
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| 178 | | DIRECT b1 ⇒ λ_. |
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| 179 | [ ([[false; false; false; false; false; true; false; true]]); b1 ] |
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| 180 | | INDIRECT i1 ⇒ λ_. |
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| 181 | [ ([[false; false; false; false; false; true; true; i1]]) ] |
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| 182 | | DPTR ⇒ λ_. |
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| 183 | [ ([[true;false;true;false;false;false;true;true]]) ] |
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| 184 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 185 | | MOV addrs ⇒ |
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| 186 | match addrs with |
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| 187 | [ inl addrs ⇒ |
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| 188 | match addrs with |
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| 189 | [ inl addrs ⇒ |
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| 190 | match addrs with |
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| 191 | [ inl addrs ⇒ |
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| 192 | match addrs with |
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| 193 | [ inl addrs ⇒ |
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| 194 | match addrs with |
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| 195 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 196 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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| 197 | [ REGISTER r ⇒ λ_.[ ([[true;true;true;false;true]]) @@ r ] |
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| 198 | | DIRECT b1 ⇒ λ_.[ ([[true;true;true;false;false;true;false;true]]); b1 ] |
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| 199 | | INDIRECT i1 ⇒ λ_. [ ([[true;true;true;false;false;true;true;i1]]) ] |
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| 200 | | DATA b1 ⇒ λ_. [ ([[false;true;true;true;false;true;false;false]]) ; b1 ] |
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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 addr1 return λx. bool_to_Prop (is_in ? [[registr;indirect]] x) → ? with |
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| 204 | [ REGISTER r ⇒ λ_. |
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| 205 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;direct;data]] x) → ? with |
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| 206 | [ ACC_A ⇒ λ_.[ ([[true;true;true;true;true]]) @@ r ] |
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| 207 | | DIRECT b1 ⇒ λ_.[ ([[true;false;true;false;true]]) @@ r; b1 ] |
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| 208 | | DATA b1 ⇒ λ_. [ ([[false;true;true;true;true]]) @@ r; b1 ] |
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| 209 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 210 | | INDIRECT i1 ⇒ λ_. |
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| 211 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;direct;data]] x) → ? with |
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| 212 | [ ACC_A ⇒ λ_.[ ([[true;true;true;true;false;true;true;i1]]) ] |
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| 213 | | DIRECT b1 ⇒ λ_.[ ([[true;false;true;false;false;true;true;i1]]); b1 ] |
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| 214 | | DATA b1 ⇒ λ_. [ ([[false;true;true;true;false;true;true;i1]]) ; b1 ] |
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| 215 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 216 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1)] |
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| 217 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 218 | let b1 ≝ |
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| 219 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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| 220 | [ DIRECT b1 ⇒ λ_. b1 |
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| 221 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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| 222 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;registr;direct;indirect;data]] x) → ? with |
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| 223 | [ ACC_A ⇒ λ_.[ ([[true;true;true;true;false;true;false;true]]); b1] |
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| 224 | | REGISTER r ⇒ λ_.[ ([[true;false;false;false;true]]) @@ r; b1 ] |
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| 225 | | DIRECT b2 ⇒ λ_.[ ([[true;false;false;false;false;true;false;true]]); b1; b2 ] |
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| 226 | | INDIRECT i1 ⇒ λ_. [ ([[true;false;false;false;false;true;true;i1]]); b1 ] |
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| 227 | | DATA b2 ⇒ λ_. [ ([[false;true;true;true;false;true;false;true]]); b1; b2 ] |
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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 addr2 return λx. bool_to_Prop (is_in ? [[data16]] x) → ? with |
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| 231 | [ DATA16 w ⇒ λ_. |
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[2032] | 232 | let b1_b2 ≝ vsplit ? 8 8 w in |
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[1615] | 233 | let b1 ≝ \fst b1_b2 in |
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| 234 | let b2 ≝ \snd b1_b2 in |
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| 235 | [ ([[true;false;false;true;false;false;false;false]]); b1; b2] |
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| 236 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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| 237 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 238 | match addr2 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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| 239 | [ BIT_ADDR b1 ⇒ λ_. |
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| 240 | [ ([[true;false;true;false;false;false;true;false]]); b1 ] |
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| 241 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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| 242 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 243 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
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| 244 | [ BIT_ADDR b1 ⇒ λ_. |
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| 245 | [ ([[true;false;false;true;false;false;true;false]]); b1 ] |
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| 246 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1)] |
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| 247 | | MOVX addrs ⇒ |
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| 248 | match addrs with |
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| 249 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 250 | match addr2 return λx. bool_to_Prop (is_in ? [[ext_indirect;ext_indirect_dptr]] x) → ? with |
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| 251 | [ EXT_INDIRECT i1 ⇒ λ_. |
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| 252 | [ ([[true;true;true;false;false;false;true;i1]]) ] |
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| 253 | | EXT_INDIRECT_DPTR ⇒ λ_. |
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| 254 | [ ([[true;true;true;false;false;false;false;false]]) ] |
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| 255 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 256 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 257 | match addr1 return λx. bool_to_Prop (is_in ? [[ext_indirect;ext_indirect_dptr]] x) → ? with |
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| 258 | [ EXT_INDIRECT i1 ⇒ λ_. |
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| 259 | [ ([[true;true;true;true;false;false;true;i1]]) ] |
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| 260 | | EXT_INDIRECT_DPTR ⇒ λ_. |
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| 261 | [ ([[true;true;true;true;false;false;false;false]]) ] |
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| 262 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1)] |
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| 263 | | MUL addr1 addr2 ⇒ |
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| 264 | [ ([[true;false;true;false;false;true;false;false]]) ] |
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| 265 | | NOP ⇒ |
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| 266 | [ ([[false;false;false;false;false;false;false;false]]) ] |
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| 267 | | ORL addrs ⇒ |
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| 268 | match addrs with |
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| 269 | [ inl addrs ⇒ |
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| 270 | match addrs with |
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| 271 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 272 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;data;direct;indirect]] x) → ? with |
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| 273 | [ REGISTER r ⇒ λ_.[ ([[false;true;false;false;true]]) @@ r ] |
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| 274 | | DIRECT b1 ⇒ λ_.[ ([[false;true;false;false;false;true;false;true]]); b1 ] |
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| 275 | | INDIRECT i1 ⇒ λ_. [ ([[false;true;false;false;false;true;true;i1]]) ] |
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| 276 | | DATA b1 ⇒ λ_. [ ([[false;true;false;false;false;true;false;false]]) ; b1 ] |
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| 277 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 278 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 279 | let b1 ≝ |
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| 280 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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| 281 | [ DIRECT b1 ⇒ λ_. b1 |
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| 282 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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| 283 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;data]] x) → ? with |
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| 284 | [ ACC_A ⇒ λ_. |
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| 285 | [ ([[false;true;false;false;false;false;true;false]]); b1 ] |
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| 286 | | DATA b2 ⇒ λ_. |
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| 287 | [ ([[false;true;false;false;false;false;true;true]]); b1; b2 ] |
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| 288 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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| 289 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 290 | match addr2 return λx. bool_to_Prop (is_in ? [[bit_addr;n_bit_addr]] x) → ? with |
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| 291 | [ BIT_ADDR b1 ⇒ λ_. |
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| 292 | [ ([[false;true;true;true;false;false;true;false]]); b1 ] |
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| 293 | | N_BIT_ADDR b1 ⇒ λ_. |
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| 294 | [ ([[true;false;true;false;false;false;false;false]]); b1 ] |
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| 295 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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| 296 | | POP addr ⇒ |
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| 297 | match addr return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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| 298 | [ DIRECT b1 ⇒ λ_. |
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| 299 | [ ([[true;true;false;true;false;false;false;false]]) ; b1 ] |
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| 300 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 301 | | PUSH addr ⇒ |
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| 302 | match addr return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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| 303 | [ DIRECT b1 ⇒ λ_. |
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| 304 | [ ([[true;true;false;false;false;false;false;false]]) ; b1 ] |
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| 305 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 306 | | RET ⇒ |
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| 307 | [ ([[false;false;true;false;false;false;true;false]]) ] |
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| 308 | | RETI ⇒ |
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| 309 | [ ([[false;false;true;true;false;false;true;false]]) ] |
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| 310 | | RL addr ⇒ |
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| 311 | [ ([[false;false;true;false;false;false;true;true]]) ] |
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| 312 | | RLC addr ⇒ |
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| 313 | [ ([[false;false;true;true;false;false;true;true]]) ] |
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| 314 | | RR addr ⇒ |
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| 315 | [ ([[false;false;false;false;false;false;true;true]]) ] |
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| 316 | | RRC addr ⇒ |
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| 317 | [ ([[false;false;false;true;false;false;true;true]]) ] |
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| 318 | | SETB addr ⇒ |
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| 319 | match addr return λx. bool_to_Prop (is_in ? [[carry;bit_addr]] x) → ? with |
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| 320 | [ CARRY ⇒ λ_. |
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| 321 | [ ([[true;true;false;true;false;false;true;true]]) ] |
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| 322 | | BIT_ADDR b1 ⇒ λ_. |
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| 323 | [ ([[true;true;false;true;false;false;true;false]]); b1 ] |
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| 324 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 325 | | SUBB addr1 addr2 ⇒ |
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| 326 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
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| 327 | [ REGISTER r ⇒ λ_. |
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| 328 | [ ([[true;false;false;true;true]]) @@ r ] |
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| 329 | | DIRECT b1 ⇒ λ_. |
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| 330 | [ ([[true;false;false;true;false;true;false;true]]); b1] |
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| 331 | | INDIRECT i1 ⇒ λ_. |
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| 332 | [ ([[true;false;false;true;false;true;true;i1]]) ] |
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| 333 | | DATA b1 ⇒ λ_. |
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| 334 | [ ([[true;false;false;true;false;true;false;false]]); b1] |
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| 335 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 336 | | SWAP addr ⇒ |
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| 337 | [ ([[true;true;false;false;false;true;false;false]]) ] |
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| 338 | | XCH addr1 addr2 ⇒ |
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| 339 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect]] x) → ? with |
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| 340 | [ REGISTER r ⇒ λ_. |
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| 341 | [ ([[true;true;false;false;true]]) @@ r ] |
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| 342 | | DIRECT b1 ⇒ λ_. |
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| 343 | [ ([[true;true;false;false;false;true;false;true]]); b1] |
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| 344 | | INDIRECT i1 ⇒ λ_. |
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| 345 | [ ([[true;true;false;false;false;true;true;i1]]) ] |
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| 346 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 347 | | XCHD addr1 addr2 ⇒ |
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| 348 | match addr2 return λx. bool_to_Prop (is_in ? [[indirect]] x) → ? with |
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| 349 | [ INDIRECT i1 ⇒ λ_. |
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| 350 | [ ([[true;true;false;true;false;true;true;i1]]) ] |
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| 351 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 352 | | XRL addrs ⇒ |
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| 353 | match addrs with |
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| 354 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 355 | match addr2 return λx. bool_to_Prop (is_in ? [[data;registr;direct;indirect]] x) → ? with |
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| 356 | [ REGISTER r ⇒ λ_. |
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| 357 | [ ([[false;true;true;false;true]]) @@ r ] |
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| 358 | | DIRECT b1 ⇒ λ_. |
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| 359 | [ ([[false;true;true;false;false;true;false;true]]); b1] |
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| 360 | | INDIRECT i1 ⇒ λ_. |
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| 361 | [ ([[false;true;true;false;false;true;true;i1]]) ] |
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| 362 | | DATA b1 ⇒ λ_. |
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| 363 | [ ([[false;true;true;false;false;true;false;false]]); b1] |
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| 364 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 365 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
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| 366 | let b1 ≝ |
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| 367 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
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| 368 | [ DIRECT b1 ⇒ λ_. b1 |
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| 369 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
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| 370 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;data]] x) → ? with |
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| 371 | [ ACC_A ⇒ λ_. |
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| 372 | [ ([[false;true;true;false;false;false;true;false]]); b1 ] |
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| 373 | | DATA b2 ⇒ λ_. |
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| 374 | [ ([[false;true;true;false;false;false;true;true]]); b1; b2 ] |
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| 375 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
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| 376 | ]. |
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| 377 | |
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| 378 | definition assembly1 ≝ |
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| 379 | λi: instruction. |
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| 380 | match i with |
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| 381 | [ ACALL addr ⇒ |
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| 382 | match addr return λx. bool_to_Prop (is_in ? [[addr11]] x) → ? with |
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| 383 | [ ADDR11 w ⇒ λ_. |
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[2032] | 384 | let v1_v2 ≝ vsplit ? 3 8 w in |
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[1615] | 385 | let v1 ≝ \fst v1_v2 in |
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| 386 | let v2 ≝ \snd v1_v2 in |
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| 387 | [ (v1 @@ [[true; false; false; false; true]]) ; v2 ] |
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| 388 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 389 | | AJMP addr ⇒ |
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| 390 | match addr return λx. bool_to_Prop (is_in ? [[addr11]] x) → ? with |
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| 391 | [ ADDR11 w ⇒ λ_. |
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[2032] | 392 | let v1_v2 ≝ vsplit ? 3 8 w in |
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[1615] | 393 | let v1 ≝ \fst v1_v2 in |
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| 394 | let v2 ≝ \snd v1_v2 in |
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| 395 | [ (v1 @@ [[false; false; false; false; true]]) ; v2 ] |
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| 396 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 397 | | JMP adptr ⇒ |
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| 398 | [ ([[false;true;true;true;false;false;true;true]]) ] |
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| 399 | | LCALL addr ⇒ |
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| 400 | match addr return λx. bool_to_Prop (is_in ? [[addr16]] x) → ? with |
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| 401 | [ ADDR16 w ⇒ λ_. |
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[2032] | 402 | let b1_b2 ≝ vsplit ? 8 8 w in |
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[1615] | 403 | let b1 ≝ \fst b1_b2 in |
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| 404 | let b2 ≝ \snd b1_b2 in |
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| 405 | [ ([[false;false;false;true;false;false;true;false]]); b1; b2 ] |
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| 406 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 407 | | LJMP addr ⇒ |
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| 408 | match addr return λx. bool_to_Prop (is_in ? [[addr16]] x) → ? with |
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| 409 | [ ADDR16 w ⇒ λ_. |
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[2032] | 410 | let b1_b2 ≝ vsplit ? 8 8 w in |
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[1615] | 411 | let b1 ≝ \fst b1_b2 in |
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| 412 | let b2 ≝ \snd b1_b2 in |
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| 413 | [ ([[false;false;false;false;false;false;true;false]]); b1; b2 ] |
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| 414 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 415 | | MOVC addr1 addr2 ⇒ |
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| 416 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_dptr;acc_pc]] x) → ? with |
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| 417 | [ ACC_DPTR ⇒ λ_. |
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| 418 | [ ([[true;false;false;true;false;false;true;true]]) ] |
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| 419 | | ACC_PC ⇒ λ_. |
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| 420 | [ ([[true;false;false;false;false;false;true;true]]) ] |
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| 421 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
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| 422 | | SJMP addr ⇒ |
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| 423 | match addr return λx. bool_to_Prop (is_in ? [[relative]] x) → ? with |
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| 424 | [ RELATIVE b1 ⇒ λ_. |
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| 425 | [ ([[true;false;false;false;false;false;false;false]]); b1 ] |
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| 426 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
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| 427 | | RealInstruction instr ⇒ |
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| 428 | assembly_preinstruction [[ relative ]] |
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| 429 | (λx. |
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| 430 | match x return λs. bool_to_Prop (is_in ? [[ relative ]] s) → ? with |
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| 431 | [ RELATIVE r ⇒ λ_. r |
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| 432 | | _ ⇒ λabsd. ⊥ |
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| 433 | ] (subaddressing_modein … x)) instr |
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| 434 | ]. |
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| 435 | cases absd |
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[1592] | 436 | qed. |
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| 437 | |
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[2015] | 438 | (* XXX: pc_plus_sjmp_length used to be just sigma of ppc. This is incorrect |
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| 439 | as relative lengths are computed from the *end* of the SJMP, not from |
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[2022] | 440 | the beginning. |
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[2015] | 441 | *) |
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[1905] | 442 | definition expand_relative_jump_internal: |
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[1948] | 443 | ∀lookup_labels:Identifier → Word.∀sigma:Word → Word. |
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[1905] | 444 | Identifier → Word → ([[relative]] → preinstruction [[relative]]) → |
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| 445 | list instruction |
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[1615] | 446 | ≝ |
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[1934] | 447 | λlookup_labels.λsigma.λlbl.λppc,i. |
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[1948] | 448 | let lookup_address ≝ sigma (lookup_labels lbl) in |
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[2022] | 449 | let pc_plus_jmp_length ≝ sigma (add … ppc (bitvector_of_nat … 1)) in |
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[2048] | 450 | let 〈sj_possible, disp〉 ≝ short_jump_cond pc_plus_jmp_length lookup_address in |
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| 451 | if sj_possible |
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| 452 | then |
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| 453 | let address ≝ RELATIVE disp in |
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[1905] | 454 | [ RealInstruction (i address) ] |
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| 455 | else |
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[1615] | 456 | [ RealInstruction (i (RELATIVE (bitvector_of_nat ? 2))); |
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| 457 | SJMP (RELATIVE (bitvector_of_nat ? 3)); (* LJMP size? *) |
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[1649] | 458 | LJMP (ADDR16 lookup_address) |
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[1905] | 459 | ]. |
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[1948] | 460 | % |
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[1615] | 461 | qed. |
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| 462 | |
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[1905] | 463 | definition expand_relative_jump: |
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| 464 | ∀lookup_labels.∀sigma. |
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| 465 | Word → (*jump_length →*) |
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| 466 | preinstruction Identifier → list instruction ≝ |
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| 467 | λlookup_labels: Identifier → Word. |
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[1948] | 468 | λsigma:Word → Word. |
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[1934] | 469 | λppc: Word. |
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[1905] | 470 | (*λjmp_len: jump_length.*) |
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[1615] | 471 | λi: preinstruction Identifier. |
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[1934] | 472 | (*let rel_jmp ≝ RELATIVE (bitvector_of_nat ? 2) in*) |
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[1615] | 473 | match i with |
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[1934] | 474 | [ JC jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JC ?) |
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| 475 | | JNC jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JNC ?) |
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| 476 | | JB baddr jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JB ? baddr) |
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| 477 | | JZ jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JZ ?) |
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| 478 | | JNZ jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JNZ ?) |
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| 479 | | JBC baddr jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JBC ? baddr) |
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| 480 | | JNB baddr jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JNB ? baddr) |
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| 481 | | CJNE addr jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (CJNE ? addr) |
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| 482 | | DJNZ addr jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (DJNZ ? addr) |
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[1905] | 483 | | ADD arg1 arg2 ⇒ [ ADD ? arg1 arg2 ] |
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| 484 | | ADDC arg1 arg2 ⇒ [ ADDC ? arg1 arg2 ] |
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| 485 | | SUBB arg1 arg2 ⇒ [ SUBB ? arg1 arg2 ] |
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| 486 | | INC arg ⇒ [ INC ? arg ] |
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| 487 | | DEC arg ⇒ [ DEC ? arg ] |
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| 488 | | MUL arg1 arg2 ⇒ [ MUL ? arg1 arg2 ] |
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| 489 | | DIV arg1 arg2 ⇒ [ DIV ? arg1 arg2 ] |
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| 490 | | DA arg ⇒ [ DA ? arg ] |
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| 491 | | ANL arg ⇒ [ ANL ? arg ] |
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| 492 | | ORL arg ⇒ [ ORL ? arg ] |
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| 493 | | XRL arg ⇒ [ XRL ? arg ] |
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| 494 | | CLR arg ⇒ [ CLR ? arg ] |
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| 495 | | CPL arg ⇒ [ CPL ? arg ] |
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| 496 | | RL arg ⇒ [ RL ? arg ] |
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| 497 | | RR arg ⇒ [ RR ? arg ] |
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| 498 | | RLC arg ⇒ [ RLC ? arg ] |
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| 499 | | RRC arg ⇒ [ RRC ? arg ] |
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| 500 | | SWAP arg ⇒ [ SWAP ? arg ] |
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| 501 | | MOV arg ⇒ [ MOV ? arg ] |
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| 502 | | MOVX arg ⇒ [ MOVX ? arg ] |
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| 503 | | SETB arg ⇒ [ SETB ? arg ] |
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| 504 | | PUSH arg ⇒ [ PUSH ? arg ] |
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| 505 | | POP arg ⇒ [ POP ? arg ] |
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| 506 | | XCH arg1 arg2 ⇒ [ XCH ? arg1 arg2 ] |
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| 507 | | XCHD arg1 arg2 ⇒ [ XCHD ? arg1 arg2 ] |
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| 508 | | RET ⇒ [ RET ? ] |
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| 509 | | RETI ⇒ [ RETI ? ] |
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| 510 | | NOP ⇒ [ RealInstruction (NOP ?) ] |
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[1615] | 511 | ]. |
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| 512 | |
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[1905] | 513 | definition expand_pseudo_instruction: |
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[1948] | 514 | ∀lookup_labels. |
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| 515 | ∀sigma: Word → Word. |
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| 516 | ∀policy: Word → bool. |
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| 517 | Word → ? → pseudo_instruction → list instruction ≝ |
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| 518 | λlookup_labels: Identifier → Word. |
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| 519 | λsigma: Word → Word. |
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| 520 | λpolicy: Word → bool. |
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[1934] | 521 | λppc. |
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[1905] | 522 | λlookup_datalabels:Identifier → Word. |
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[1615] | 523 | λi. |
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| 524 | match i with |
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[1905] | 525 | [ Cost cost ⇒ [ ] |
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| 526 | | Comment comment ⇒ [ ] |
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[1615] | 527 | | Call call ⇒ |
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[2022] | 528 | let pc_plus_jmp_length ≝ sigma (add … ppc (bitvector_of_nat … 1)) in |
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[2048] | 529 | let lookup_address ≝ sigma (lookup_labels call) in |
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[2101] | 530 | let 〈mj_possible, disp〉 ≝ absolute_jump_cond pc_plus_jmp_length lookup_address in |
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[1948] | 531 | let do_a_long ≝ policy ppc in |
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[2048] | 532 | if mj_possible ∧ ¬ do_a_long then |
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| 533 | let address ≝ ADDR11 disp in |
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[1905] | 534 | [ ACALL address ] |
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| 535 | else |
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[2048] | 536 | let address ≝ ADDR16 lookup_address in |
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[1905] | 537 | [ LCALL address ] |
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[1615] | 538 | | Mov d trgt ⇒ |
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| 539 | let address ≝ DATA16 (lookup_datalabels trgt) in |
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[1905] | 540 | [ RealInstruction (MOV ? (inl ? ? (inl ? ? (inr ? ? 〈DPTR, address〉))))] |
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[1934] | 541 | | Instruction instr ⇒ expand_relative_jump lookup_labels sigma ppc instr |
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[1615] | 542 | | Jmp jmp ⇒ |
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[2022] | 543 | let pc_plus_jmp_length ≝ sigma (add … ppc (bitvector_of_nat … 1)) in |
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[1948] | 544 | let do_a_long ≝ policy ppc in |
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| 545 | let lookup_address ≝ sigma (lookup_labels jmp) in |
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[2048] | 546 | let 〈sj_possible, disp〉 ≝ short_jump_cond pc_plus_jmp_length lookup_address in |
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| 547 | if sj_possible ∧ ¬ do_a_long then |
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| 548 | let address ≝ RELATIVE disp in |
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[1905] | 549 | [ SJMP address ] |
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| 550 | else |
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[2101] | 551 | let 〈mj_possible, disp2〉 ≝ absolute_jump_cond pc_plus_jmp_length lookup_address in |
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[2048] | 552 | if mj_possible ∧ ¬ do_a_long then |
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| 553 | let address ≝ ADDR11 disp2 in |
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[1905] | 554 | [ AJMP address ] |
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| 555 | else |
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[1934] | 556 | let address ≝ ADDR16 lookup_address in |
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[1905] | 557 | [ LJMP address ] |
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[1615] | 558 | ]. |
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[1948] | 559 | % |
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[1615] | 560 | qed. |
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[1649] | 561 | |
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| 562 | definition assembly_1_pseudoinstruction ≝ |
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| 563 | λlookup_labels. |
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[1948] | 564 | λsigma: Word → Word. |
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| 565 | λpolicy: Word → bool. |
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[1934] | 566 | λppc: Word. |
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[832] | 567 | λlookup_datalabels. |
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| 568 | λi. |
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[1948] | 569 | let pseudos ≝ expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels i in |
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[1649] | 570 | let mapped ≝ map ? ? assembly1 pseudos in |
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| 571 | let flattened ≝ flatten ? mapped in |
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| 572 | let pc_len ≝ length ? flattened in |
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| 573 | 〈pc_len, flattened〉. |
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[1667] | 574 | |
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[1934] | 575 | definition instruction_size ≝ |
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[1948] | 576 | λlookup_labels. |
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[2268] | 577 | λlookup_datalabels. |
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[1948] | 578 | λsigma: Word → Word. |
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| 579 | λpolicy: Word → bool. |
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| 580 | λppc. |
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| 581 | λi. |
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[2268] | 582 | \fst (assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels i). |
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[2108] | 583 | |
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[1934] | 584 | (* Labels *) |
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| 585 | definition is_label ≝ |
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| 586 | λx:labelled_instruction.λl:Identifier. |
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| 587 | let 〈lbl,instr〉 ≝ x in |
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| 588 | match lbl with |
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| 589 | [ Some l' ⇒ l' = l |
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| 590 | | _ ⇒ False |
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| 591 | ]. |
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| 592 | |
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| 593 | lemma label_does_not_occur: |
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| 594 | ∀i:ℕ.∀p:list labelled_instruction.∀l:Identifier. |
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| 595 | is_label (nth i ? p 〈None ?, Comment [ ]〉) l → does_not_occur ?? l p = false. |
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| 596 | #i #p #l generalize in match i; elim p |
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| 597 | [ #i >nth_nil #H cases H |
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| 598 | | #h #t #IH #i cases i -i |
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| 599 | [ cases h #hi #hp cases hi |
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| 600 | [ normalize #H cases H |
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| 601 | | #l' #Heq whd in ⊢ (??%?); change with (eq_identifier ? l' l) in match (instruction_matches_identifier ????); |
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| 602 | whd in Heq; >Heq |
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| 603 | >eq_identifier_refl / by refl/ |
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| 604 | ] |
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| 605 | | #i #H whd in match (does_not_occur ????); |
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| 606 | whd in match (instruction_matches_identifier ????); |
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| 607 | cases h #hi #hp cases hi normalize nodelta |
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| 608 | [ @(IH i) @H |
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| 609 | | #l' @eq_identifier_elim |
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| 610 | [ normalize / by / |
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| 611 | | normalize #_ @(IH i) @H |
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| 612 | ] |
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| 613 | ] |
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| 614 | ] |
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| 615 | ] |
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[1556] | 616 | qed. |
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| 617 | |
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[2078] | 618 | definition sigma_policy_specification ≝ |
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| 619 | λprogram: pseudo_assembly_program. |
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| 620 | λsigma: Word → Word. |
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| 621 | λpolicy: Word → bool. |
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[2079] | 622 | sigma (zero …) = zero … ∧ |
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[2144] | 623 | let instr_list ≝ \snd program in |
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[2268] | 624 | let preamble ≝ \fst program in |
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[2144] | 625 | ∀ppc: Word. ∀ppc_ok: nat_of_bitvector … ppc < |instr_list|. |
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[2078] | 626 | let pc ≝ sigma ppc in |
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| 627 | let labels ≝ \fst (create_label_cost_map instr_list) in |
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| 628 | let lookup_labels ≝ λx. bitvector_of_nat 16 (lookup_def … labels x 0) in |
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[2268] | 629 | let lookup_datalabels ≝ λx. lookup_def … (construct_datalabels preamble) x (zero …) in |
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[2078] | 630 | let instruction ≝ \fst (fetch_pseudo_instruction instr_list ppc ppc_ok) in |
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| 631 | let next_pc ≝ sigma (add 16 ppc (bitvector_of_nat 16 1)) in |
---|
[2268] | 632 | next_pc = add 16 pc (bitvector_of_nat … (instruction_size lookup_labels lookup_datalabels sigma policy ppc instruction)) |
---|
[2144] | 633 | ∧ |
---|
[2268] | 634 | (nat_of_bitvector … pc + instruction_size lookup_labels lookup_datalabels sigma policy ppc instruction < 2^16 |
---|
[2151] | 635 | ∨ |
---|
[2188] | 636 | (∀ppc'. ∀ppc_ok':nat_of_bitvector … ppc' < |instr_list|. nat_of_bitvector … ppc < nat_of_bitvector … ppc' → |
---|
| 637 | let instruction' ≝ \fst (fetch_pseudo_instruction instr_list ppc' ppc_ok') in |
---|
[2268] | 638 | instruction_size lookup_labels lookup_datalabels sigma policy ppc' instruction' = 0) |
---|
[2188] | 639 | ∧ |
---|
[2268] | 640 | nat_of_bitvector … pc + instruction_size lookup_labels lookup_datalabels sigma policy ppc instruction = 2^16). |
---|
[2078] | 641 | |
---|
[2151] | 642 | |
---|
| 643 | lemma fst_assembly_1_pseudoinstruction_insensible_to_lookup_datalabels: |
---|
| 644 | ∀lookup_labels,sigma,policy,ppc,pi. |
---|
| 645 | ∀lookup_datalabels1,lookup_datalabels2. |
---|
| 646 | \fst (assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels1 pi) = |
---|
| 647 | \fst (assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels2 pi). |
---|
| 648 | #lookup_labels #sigma #policy #ppc #pi #lookup_datalabels1 #lookup_datalabels2 |
---|
| 649 | cases pi // |
---|
| 650 | qed. |
---|
| 651 | |
---|
| 652 | lemma fst_snd_assembly_1_pseudoinstruction: |
---|
| 653 | ∀lookup_labels,sigma,policy,ppc,pi,lookup_datalabels,len,assembled. |
---|
| 654 | assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi |
---|
| 655 | = 〈len,assembled〉 → |
---|
| 656 | len = |assembled|. |
---|
| 657 | #lookup #sigma #policy #ppc #pi #lookup_datalabels #len #assembled |
---|
| 658 | inversion (assembly_1_pseudoinstruction ??????) #len' #assembled' |
---|
| 659 | whd in ⊢ (??%? → ?); #EQ1 #EQ2 destruct % |
---|
| 660 | qed. |
---|
| 661 | |
---|
[2157] | 662 | (* XXX: easy but tedious *) |
---|
| 663 | lemma assembly1_lt_128: |
---|
| 664 | ∀i: instruction. |
---|
| 665 | |(assembly1 i)| < 128. |
---|
| 666 | cases daemon |
---|
| 667 | (* XXX: commented out as takes ages to type check |
---|
| 668 | #i cases i |
---|
| 669 | try (#assm1 #assm2) try #assm1 |
---|
| 670 | [8: |
---|
| 671 | cases assm1 |
---|
| 672 | try (#assm1 #assm2) try #assm1 |
---|
| 673 | whd in match assembly1; normalize nodelta |
---|
| 674 | whd in match assembly_preinstruction; normalize nodelta |
---|
| 675 | try @(subaddressing_mode_elim … assm2) |
---|
| 676 | try @(subaddressing_mode_elim … assm1) try #w try #w' normalize nodelta |
---|
| 677 | [32: |
---|
| 678 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
| 679 | cases assm1 #addr1 #addr2 normalize nodelta |
---|
| 680 | [1: |
---|
| 681 | @(subaddressing_mode_elim … addr2) |
---|
| 682 | |2: |
---|
| 683 | @(subaddressing_mode_elim … addr1) |
---|
| 684 | ] |
---|
| 685 | #w |
---|
| 686 | |35,36,37: |
---|
| 687 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
| 688 | [1,3: |
---|
| 689 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
| 690 | ] |
---|
| 691 | cases assm1 #addr1 #addr2 normalize nodelta |
---|
| 692 | @(subaddressing_mode_elim … addr2) try #w |
---|
| 693 | |49: |
---|
| 694 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
| 695 | [1: |
---|
| 696 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
| 697 | [1: |
---|
| 698 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
| 699 | [1: |
---|
| 700 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
| 701 | [1: |
---|
| 702 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
| 703 | ] |
---|
| 704 | ] |
---|
| 705 | ] |
---|
| 706 | ] |
---|
| 707 | cases assm1 #addr1 #addr2 normalize nodelta |
---|
| 708 | [1,3,4,5: |
---|
| 709 | @(subaddressing_mode_elim … addr2) try #w |
---|
| 710 | |*: |
---|
| 711 | @(subaddressing_mode_elim … addr1) try #w |
---|
| 712 | normalize nodelta |
---|
| 713 | [1,2: |
---|
| 714 | @(subaddressing_mode_elim … addr2) try #w |
---|
| 715 | ] |
---|
| 716 | ] |
---|
| 717 | |50: |
---|
| 718 | cases assm1 -assm1 #assm1 normalize nodelta |
---|
| 719 | cases assm1 #addr1 #addr2 normalize nodelta |
---|
| 720 | [1: |
---|
| 721 | @(subaddressing_mode_elim … addr2) try #w |
---|
| 722 | |2: |
---|
| 723 | @(subaddressing_mode_elim … addr1) try #w |
---|
| 724 | ] |
---|
| 725 | ] |
---|
| 726 | normalize repeat @le_S_S @le_O_n |
---|
| 727 | ] |
---|
| 728 | whd in match assembly1; normalize nodelta |
---|
| 729 | [6: |
---|
| 730 | normalize repeat @le_S_S @le_O_n |
---|
| 731 | |7: |
---|
| 732 | @(subaddressing_mode_elim … assm2) normalize repeat @le_S_S @le_O_n |
---|
| 733 | |*: |
---|
| 734 | @(subaddressing_mode_elim … assm1) #w normalize nodelta repeat @le_S_S @le_O_n |
---|
| 735 | ] |
---|
| 736 | *) |
---|
| 737 | qed. |
---|
| 738 | |
---|
| 739 | lemma assembly1_pseudoinstruction_lt_2_to_16: |
---|
| 740 | ∀lookup_labels,sigma,policy,ppc,lookup_datalabels,pi. |
---|
| 741 | |\snd (assembly_1_pseudoinstruction |
---|
| 742 | lookup_labels sigma policy ppc lookup_datalabels pi)| |
---|
| 743 | < 2^16. |
---|
| 744 | #lookup_labels #sigma #policy #ppc #lookup_datalabels * |
---|
| 745 | [ cut (128 < 2^16) [@leb_true_to_le %] #LT |
---|
| 746 | * whd in match (assembly_1_pseudoinstruction ??????); |
---|
| 747 | whd in match (expand_pseudo_instruction ??????); |
---|
| 748 | whd in match assembly_1_pseudoinstruction; normalize nodelta |
---|
| 749 | try (#arg1 #arg2 #arg3) try (#arg1 #arg2) try #arg1 |
---|
| 750 | whd in match (expand_pseudo_instruction ??????); |
---|
| 751 | try |
---|
| 752 | (change with (|flatten ? [assembly1 ?]| < ?) |
---|
| 753 | >flatten_singleton |
---|
| 754 | @(transitive_lt … (assembly1_lt_128 ?)) |
---|
| 755 | @LT) |
---|
| 756 | @pair_elim #x #y #_ cases x normalize nodelta |
---|
| 757 | try |
---|
| 758 | (change with (|flatten ? [assembly1 ?]| < ?) |
---|
| 759 | >flatten_singleton |
---|
| 760 | @(transitive_lt … (assembly1_lt_128 ?)) |
---|
| 761 | @LT) |
---|
| 762 | change with (|flatten ? [assembly1 ?; assembly1 ?; assembly1 ?]| < ?) |
---|
| 763 | >length_flatten_cons >length_flatten_cons >length_flatten_cons <plus_n_O |
---|
| 764 | <associative_plus @(transitive_lt … (tech_transitive_lt_3 … (2^7) ???)) |
---|
| 765 | try @assembly1_lt_128 @leb_true_to_le % |
---|
| 766 | |2,3: #msg normalize in ⊢ (?%?); // |
---|
| 767 | | #label whd in match (assembly_1_pseudoinstruction ??????); |
---|
| 768 | whd in match (expand_pseudo_instruction ??????); |
---|
| 769 | @pair_elim #sj_poss #disp cases (?∧?) normalize nodelta #_ |
---|
| 770 | [2: @pair_elim #x #y #_ cases (?∧?)] |
---|
| 771 | normalize in ⊢ (?%?); // |
---|
| 772 | | #label whd in match (assembly_1_pseudoinstruction ??????); |
---|
| 773 | whd in match (expand_pseudo_instruction ??????); |
---|
| 774 | @pair_elim #sj_poss #disp cases (?∧?) normalize nodelta #_ |
---|
| 775 | normalize in ⊢ (?%?); // |
---|
| 776 | | #dptr #id normalize in ⊢ (?%?); // |
---|
| 777 | ] |
---|
| 778 | qed. |
---|
| 779 | |
---|
[993] | 780 | definition assembly: |
---|
[1948] | 781 | ∀p: pseudo_assembly_program. |
---|
| 782 | ∀sigma: Word → Word. |
---|
| 783 | ∀policy: Word → bool. |
---|
[2021] | 784 | Σres:list Byte × (BitVectorTrie costlabel 16). |
---|
[2081] | 785 | sigma_policy_specification p sigma policy → |
---|
[2021] | 786 | let 〈preamble,instr_list〉 ≝ p in |
---|
[2110] | 787 | |instr_list| ≤ 2^16 → |
---|
[2021] | 788 | let 〈assembled,costs〉 ≝ res in |
---|
[2110] | 789 | |assembled| ≤ 2^16 ∧ |
---|
[2146] | 790 | (nat_of_bitvector … (sigma (bitvector_of_nat … (|instr_list|))) = |assembled| ∨ |
---|
| 791 | sigma (bitvector_of_nat … (|instr_list|)) = zero … ∧ |assembled| = 2^16) ∧ |
---|
[2021] | 792 | let 〈labels_to_ppc,ppc_to_costs〉 ≝ create_label_cost_map instr_list in |
---|
| 793 | let datalabels ≝ construct_datalabels preamble in |
---|
[2110] | 794 | let lookup_labels ≝ λx. bitvector_of_nat ? (lookup_def … labels_to_ppc x 0) in |
---|
[2021] | 795 | let lookup_datalabels ≝ λx. lookup_def … datalabels x (zero ?) in |
---|
[2055] | 796 | ∀ppc. ∀ppc_ok:nat_of_bitvector … ppc < |instr_list|. |
---|
| 797 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction instr_list ppc ppc_ok in |
---|
[2021] | 798 | let 〈len,assembledi〉 ≝ |
---|
| 799 | assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi in |
---|
[2147] | 800 | |assembledi| ≤ |assembled| ∧ |
---|
[2197] | 801 | ∀j:nat. ∀H: j < |assembledi|. ∃K. |
---|
[2021] | 802 | nth_safe ? j assembledi H = |
---|
| 803 | nth_safe ? (nat_of_bitvector … (add … (sigma ppc) (bitvector_of_nat ? j))) |
---|
| 804 | assembled K |
---|
[2151] | 805 | ≝ |
---|
[2081] | 806 | λp. |
---|
[1948] | 807 | λsigma. |
---|
| 808 | λpolicy. |
---|
[2021] | 809 | deplet 〈preamble, instr_list〉 as p_refl ≝ p in |
---|
[2151] | 810 | deplet 〈labels_to_ppc,ppc_to_costs〉 as eq_create_label_cost_map ≝ create_label_cost_map instr_list in |
---|
[1948] | 811 | let datalabels ≝ construct_datalabels preamble in |
---|
[2110] | 812 | let lookup_labels ≝ λx. bitvector_of_nat ? (lookup_def … labels_to_ppc x 0) in |
---|
[1948] | 813 | let lookup_datalabels ≝ λx. lookup_def … datalabels x (zero ?) in |
---|
[2138] | 814 | let 〈next_pc,revcode〉 ≝ pi1 … ( |
---|
[2021] | 815 | foldl_strong |
---|
[993] | 816 | (option Identifier × pseudo_instruction) |
---|
[2021] | 817 | (λpre. Σppc_code:(Word × (list Byte)). |
---|
[2081] | 818 | sigma_policy_specification 〈preamble,instr_list〉 sigma policy → |
---|
[2110] | 819 | |instr_list| ≤ 2^16 → |
---|
[993] | 820 | let 〈ppc,code〉 ≝ ppc_code in |
---|
[2021] | 821 | ppc = bitvector_of_nat … (|pre|) ∧ |
---|
[2151] | 822 | |code| ≤ 2^16 ∧ |
---|
[2146] | 823 | (nat_of_bitvector … (sigma ppc) = |code| ∨ |
---|
[2188] | 824 | sigma ppc = zero … ∧ |code| = 2^16 ∧ |
---|
| 825 | (|pre| < 2^16 → ∀ppc'. ∀ppc_ok':nat_of_bitvector … ppc' < |instr_list|. nat_of_bitvector … ppc ≤ nat_of_bitvector … ppc' → |
---|
| 826 | let instruction' ≝ \fst (fetch_pseudo_instruction instr_list ppc' ppc_ok') in |
---|
[2268] | 827 | instruction_size lookup_labels lookup_datalabels sigma policy ppc' instruction' = 0) |
---|
[2188] | 828 | ) ∧ |
---|
[2055] | 829 | ∀ppc'.∀ppc_ok'. |
---|
[2110] | 830 | (nat_of_bitvector … ppc' < nat_of_bitvector … ppc ∨ |pre| = 2^16) → |
---|
[2055] | 831 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction instr_list ppc' ppc_ok' in |
---|
[2021] | 832 | let 〈len,assembledi〉 ≝ |
---|
| 833 | assembly_1_pseudoinstruction lookup_labels sigma policy ppc' lookup_datalabels pi in |
---|
[2151] | 834 | |assembledi| ≤ |reverse … code| ∧ |
---|
[2197] | 835 | ∀j:nat. ∀H: j < |assembledi|. ∃K. |
---|
[2021] | 836 | nth_safe ? j assembledi H = |
---|
| 837 | nth_safe ? (nat_of_bitvector … (add … (sigma ppc') (bitvector_of_nat ? j))) (reverse … code) K) |
---|
[993] | 838 | instr_list |
---|
[2021] | 839 | (λprefix,hd,tl,prf,ppc_code. |
---|
| 840 | let 〈ppc, code〉 ≝ pi1 … ppc_code in |
---|
[1948] | 841 | let 〈pc_delta, program〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels (\snd hd) in |
---|
[1946] | 842 | let new_ppc ≝ add ? ppc (bitvector_of_nat ? 1) in |
---|
[2021] | 843 | 〈new_ppc, (reverse … program @ code)〉) |
---|
| 844 | 〈(zero ?), [ ]〉) |
---|
[985] | 845 | in |
---|
[2021] | 846 | 〈reverse … revcode, |
---|
[1948] | 847 | fold … (λppc.λcost.λpc_to_costs. insert … (sigma ppc) cost pc_to_costs) ppc_to_costs (Stub ??)〉. |
---|
[2151] | 848 | [ cases (foldl_strong ? (λx.Σy.?) ???) in p1; #ignore_revcode #Hfold #EQignore_revcode |
---|
[2081] | 849 | >EQignore_revcode in Hfold; #Hfold #sigma_pol_ok #instr_list_ok |
---|
[2193] | 850 | cases (Hfold sigma_pol_ok instr_list_ok) -Hfold * * #Hfold1 #Hfold4 #Hfold5 #Hfold3 whd |
---|
[2151] | 851 | <eq_create_label_cost_map whd % |
---|
[2188] | 852 | [2: #ppc #LTppc @Hfold3 >Hfold1 @(eqb_elim (|instr_list|) 2^16) |
---|
[2110] | 853 | [ #limit %2 @limit |
---|
| 854 | | #nlimit %1 >nat_of_bitvector_bitvector_of_nat_inverse try assumption |
---|
| 855 | @not_eq_to_le_to_lt assumption ] |
---|
[2188] | 856 | | >length_reverse % try assumption cases Hfold5 -Hfold5 |
---|
| 857 | [ #Hfold5 <Hfold5 % >Hfold1 % |
---|
| 858 | | * #Hfold51 #Hfold52 %2 <Hfold1 assumption ]] |
---|
[2081] | 859 | | * #sigma_pol_ok1 #_ #instr_list_ok % |
---|
[2188] | 860 | [ % % [%] // >sigma_pol_ok1 % ] |
---|
[2110] | 861 | #ppc' #ppc_ok' #abs @⊥ cases abs |
---|
| 862 | [#abs2 cases (not_le_Sn_O ?) [#H @(H abs2) | skip] |
---|
| 863 | |#abs2 change with (0 = S ?) in abs2; destruct(abs2) ] |
---|
[2194] | 864 | | #sigma_pol_ok cases sigma_pol_ok #sigma_pol_ok1 #sigma_pol_ok2 #instr_list_ok cases ppc_code in p1; -ppc_code #ppc_code #IH #EQppc_code >EQppc_code in IH; -EQppc_code |
---|
[2151] | 865 | #IH cases (IH ? instr_list_ok) [2: % assumption ] -IH |
---|
[2193] | 866 | * * #IH1 #IH2 #IH3 #IH4 |
---|
[2151] | 867 | cut (|prefix| < |instr_list|) |
---|
| 868 | [ >prf >length_append normalize <plus_n_Sm @le_S_S // ] #LT_prefix_instr_list |
---|
| 869 | cut (|prefix| < 2^16) |
---|
| 870 | [ @(lt_to_le_to_lt … (|instr_list|)) assumption ] #prefix_ok |
---|
| 871 | cut (nat_of_bitvector … ppc < |instr_list|) |
---|
| 872 | [ >IH1 >nat_of_bitvector_bitvector_of_nat_inverse assumption ] #ppc_ok |
---|
| 873 | cut (\snd hd = \fst (fetch_pseudo_instruction instr_list ppc ppc_ok)) |
---|
| 874 | [ >prf in ppc_ok; >IH1 >(add_zero … (bitvector_of_nat … (|prefix|))) |
---|
| 875 | >fetch_pseudo_instruction_append |
---|
| 876 | [ #ppc_ok whd in match fetch_pseudo_instruction; normalize nodelta |
---|
| 877 | whd in match (nth_safe ????); [ cases hd // | normalize // ] |
---|
| 878 | | <add_zero >nat_of_bitvector_bitvector_of_nat_inverse |
---|
| 879 | [ <prf assumption | assumption ] |
---|
| 880 | | skip |
---|
| 881 | | <prf assumption |
---|
| 882 | ]] #eq_fetch_pseudo_instruction |
---|
| 883 | lapply (fst_snd_assembly_1_pseudoinstruction … p2) #EQpc_delta |
---|
| 884 | cut (pc_delta < 2^16) |
---|
[2158] | 885 | [ >EQpc_delta |
---|
| 886 | @(eq_ind ?? (λp.λ_. |\snd p| < 2^16) ?? p2) |
---|
| 887 | @assembly1_pseudoinstruction_lt_2_to_16 ] #pc_delta_ok |
---|
[2268] | 888 | cut (pc_delta = instruction_size lookup_labels lookup_datalabels sigma policy ppc (\snd hd)) |
---|
[2156] | 889 | [ whd in match instruction_size; normalize nodelta |
---|
| 890 | >fst_assembly_1_pseudoinstruction_insensible_to_lookup_datalabels [ >p2 | skip] % ] |
---|
| 891 | #EQpc_delta2 |
---|
[2188] | 892 | cases (sigma_pol_ok2 … ppc_ok) |
---|
[2159] | 893 | <eq_fetch_pseudo_instruction <eq_create_label_cost_map <EQpc_delta2 |
---|
| 894 | #sigma_pol3 #sigma_pol4 |
---|
[2193] | 895 | % [ % [% ] ] |
---|
[2151] | 896 | [ >length_append normalize nodelta >IH1 @sym_eq @add_bitvector_of_nat |
---|
[2188] | 897 | | >length_append >length_reverse <EQpc_delta |
---|
| 898 | cases IH3 -IH3 |
---|
| 899 | [ #IH3 <IH3 >commutative_plus |
---|
| 900 | cases sigma_pol4 [ #LT @(transitive_le … LT) // | * #_ #EQ >EQ % ] |
---|
| 901 | | * * #IH3a #IH3b #IH3c >IH3b <EQpc_delta >EQpc_delta2 >eq_fetch_pseudo_instruction |
---|
| 902 | >IH3c try % assumption ] |
---|
| 903 | | >length_append >length_reverse |
---|
| 904 | cases IH3 -IH3 |
---|
| 905 | [ #IH3 <IH3 <EQpc_delta cases sigma_pol4 |
---|
| 906 | [ #LT %1 >sigma_pol3 >nat_of_bitvector_add |
---|
| 907 | [2: >nat_of_bitvector_bitvector_of_nat_inverse assumption] |
---|
| 908 | >nat_of_bitvector_bitvector_of_nat_inverse try assumption // |
---|
| 909 | | * #EQ1 #EQ2 %2 % |
---|
| 910 | [ lapply (eq_f … (bitvector_of_nat 16) … EQ2) <add_bitvector_of_nat_plus |
---|
| 911 | >bitvector_of_nat_inverse_nat_of_bitvector <sigma_pol3 #X >X % // |
---|
| 912 | | #LLT_prefix |
---|
| 913 | cut (S (nat_of_bitvector … ppc) < 2^16) |
---|
| 914 | [ >length_append in LLT_prefix; <plus_n_Sm <plus_n_O #LLT_prefix |
---|
| 915 | >IH1 >nat_of_bitvector_bitvector_of_nat_inverse assumption ] |
---|
| 916 | -LLT_prefix #LLT_prefix |
---|
| 917 | #ppc' #ppc_ok' #LEQ_newppc_ppc' whd >EQ1 try % |
---|
| 918 | @(lt_to_le_to_lt … LEQ_newppc_ppc') normalize nodelta |
---|
| 919 | >nat_of_bitvector_add >nat_of_bitvector_bitvector_of_nat_inverse // ]] |
---|
| 920 | | * * #IH5 #IH6 #IH7 %2 % [% ] |
---|
| 921 | [ normalize nodelta >sigma_pol3 >IH5 |
---|
| 922 | >add_commutative <add_zero >nat_of_bitvector_bitvector_of_nat_inverse try assumption |
---|
| 923 | >EQpc_delta2 >eq_fetch_pseudo_instruction >IH7 try % assumption |
---|
| 924 | | >IH6 <EQpc_delta >EQpc_delta2 >eq_fetch_pseudo_instruction >IH7 try % |
---|
| 925 | assumption |
---|
| 926 | | #LLT_prefix |
---|
| 927 | cut (S (nat_of_bitvector … ppc) < 2^16) |
---|
| 928 | [ >length_append in LLT_prefix; <plus_n_Sm <plus_n_O #LLT_prefix |
---|
| 929 | >IH1 >nat_of_bitvector_bitvector_of_nat_inverse assumption ] |
---|
| 930 | -LLT_prefix #LLT_prefix |
---|
| 931 | #ppc' #ppc_ok' #LEQ_newppc_ppc' whd @IH7 try assumption |
---|
| 932 | @(transitive_le … LEQ_newppc_ppc') normalize nodelta |
---|
| 933 | >nat_of_bitvector_add >nat_of_bitvector_bitvector_of_nat_inverse // ]] |
---|
[2161] | 934 | | #ppc' #LTppc' cases hd in prf p2 EQpc_delta2 eq_fetch_pseudo_instruction; #label #pi #prf #p2 |
---|
[2194] | 935 | #EQpc_delta2 #eq_fetch_pseudo_instruction #OR_lt_eq @(eq_bv_elim … ppc' ppc) |
---|
[2189] | 936 | [ #EQppc' >EQppc' in LTppc'; -ppc' >prf |
---|
| 937 | >IH1 #LTppc lapply LTppc |
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| 938 | >(add_zero … (bitvector_of_nat 16 (|prefix|))) in ⊢ (% → match % with [_ ⇒ ?]); |
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| 939 | >fetch_pseudo_instruction_append |
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| 940 | [3: @le_S_S @le_O_n |
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| 941 | |2: lapply LTppc; >(add_zero … (bitvector_of_nat 16 (|prefix|))) in ⊢ (% → ?); #H @H |
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| 942 | |4: <prf <p_refl in instr_list_ok; #H @H ] |
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| 943 | #LTppc' @pair_elim #pi' #newppc' #EQpair destruct(EQpair) <IH1 >p2 % |
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| 944 | [ >length_reverse >length_append >length_reverse // ] |
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| 945 | #j #LTj >nat_of_bitvector_add |
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| 946 | >nat_of_bitvector_bitvector_of_nat_inverse |
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| 947 | [2,4: @(lt_to_le_to_lt … LTj) <EQpc_delta @(transitive_le … pc_delta_ok) %2 % |
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| 948 | |3: @(lt_to_le_to_lt … (nat_of_bitvector … (sigma ppc) + pc_delta)) |
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| 949 | [ >EQpc_delta @monotonic_lt_plus_r assumption |
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| 950 | | cases sigma_pol4 |
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| 951 | [ #H @(transitive_le … H) %2 % |
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| 952 | | * #_ #EQ >EQ % ]]] |
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| 953 | >reverse_append >reverse_reverse |
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| 954 | cases IH3 -IH3 |
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[2197] | 955 | [ #IH3 >IH3 <(length_reverse … code) % |
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| 956 | [ >length_append @monotonic_lt_plus_r assumption |
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| 957 | | @nth_safe_prepend ] |
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[2189] | 958 | | * * #IH3a #IH3b #IH3c >IH3a @⊥ |
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| 959 | cut (|program| = 0) |
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| 960 | [ <EQpc_delta >EQpc_delta2 >eq_fetch_pseudo_instruction @IH3c // ] #EQprogram |
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| 961 | @(absurd ?? (not_le_Sn_O j)) <EQprogram assumption ] |
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| 962 | | #NEQppc' |
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| 963 | lapply (IH4 … LTppc') |
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[2021] | 964 | @pair_elim #pi' #newppc' #eq_fetch_pseudoinstruction |
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| 965 | @pair_elim #len' #assembledi' #eq_assembly_1_pseudoinstruction #IH |
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[2190] | 966 | cases (IH ?) -IH |
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| 967 | [2: %1 cases OR_lt_eq |
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[2191] | 968 | [ normalize nodelta #LT lapply LT >nat_of_bitvector_add |
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| 969 | [2: >nat_of_bitvector_bitvector_of_nat_inverse [2: //] |
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| 970 | cases (le_to_or_lt_eq … (? : nat_of_bitvector … ppc < 2^16)) |
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| 971 | [ #X <plus_n_Sm <plus_n_O @X |
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| 972 | | #abs @⊥ |
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| 973 | <(bitvector_of_nat_inverse_nat_of_bitvector … ppc) in LT; |
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| 974 | >add_overflow [2: <plus_n_Sm <plus_n_O assumption ] |
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| 975 | #abs' @(absurd … abs') normalize in ⊢ (? (??%)); |
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| 976 | @not_le_Sn_O |
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| 977 | | @(lt_to_le_to_lt … ppc_ok) assumption ]] |
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[2190] | 978 | >nat_of_bitvector_bitvector_of_nat_inverse [2: // ] |
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| 979 | <plus_n_Sm <plus_n_O #X lapply (le_S_S_to_le … X) -X #X |
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| 980 | cases (le_to_or_lt_eq … X) [//] #abs @⊥ |
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| 981 | lapply (eq_f … (bitvector_of_nat 16) … abs) |
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| 982 | >bitvector_of_nat_inverse_nat_of_bitvector |
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| 983 | >bitvector_of_nat_inverse_nat_of_bitvector #EQ |
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| 984 | @(absurd … EQ NEQppc') |
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| 985 | | >length_append <plus_n_Sm <plus_n_O #EQ @le_S_S_to_le >IH1 |
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| 986 | >nat_of_bitvector_bitvector_of_nat_inverse try assumption |
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| 987 | cases (le_to_or_lt_eq … (lt_nat_of_bitvector 16 ppc')) [#X >EQ @X] |
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| 988 | #abs @⊥ <EQ in abs; #X lapply (injective_S … X) #abs |
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| 989 | lapply (eq_f … (bitvector_of_nat 16) … abs) |
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| 990 | >bitvector_of_nat_inverse_nat_of_bitvector <IH1 #EQ |
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| 991 | @(absurd … EQ NEQppc') ]] |
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| 992 | #IH6 #IH |
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[2161] | 993 | change with (let 〈len,assembledi〉 ≝ assembly_1_pseudoinstruction ????? pi' in ? ∧ ∀j:ℕ. ∀H:j<|assembledi|.?) |
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| 994 | >eq_assembly_1_pseudoinstruction % |
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[2190] | 995 | [ >reverse_append >length_append |
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| 996 | >(fst_snd_assembly_1_pseudoinstruction … eq_assembly_1_pseudoinstruction) |
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| 997 | @(transitive_le … IH6) // |
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[2197] | 998 | | #j #LTj >reverse_append >reverse_reverse cases (IH … LTj) -IH #K #IH % |
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| 999 | [ >length_append @(lt_to_le_to_lt … K) // |
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| 1000 | | >IH @shift_nth_prefix ]]]]] |
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[2161] | 1001 | qed. |
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[832] | 1002 | |
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[1948] | 1003 | definition assembly_unlabelled_program: |
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| 1004 | assembly_program → option (list Byte × (BitVectorTrie Identifier 16)) ≝ |
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| 1005 | λp. |
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[2264] | 1006 | Some … (〈foldr … (λi,l. assembly1 i @ l) [ ] p, Stub …〉). |
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