[2034] | 1 | include "ASM/PolicyFront.ma". |
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| 2 | include alias "basics/lists/list.ma". |
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| 3 | include alias "arithmetics/nat.ma". |
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| 4 | include alias "basics/logic.ma". |
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| 5 | |
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| 6 | (* One step in the search for a jump expansion fixpoint. *) |
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| 7 | definition jump_expansion_step: ∀program:(Σl:list labelled_instruction.S (|l|) < 2^16). |
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| 8 | ∀labels:(Σlm:label_map.∀l. |
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| 9 | occurs_exactly_once ?? l program → |
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| 10 | bitvector_of_nat ? (lookup_def … lm l 0) = |
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| 11 | address_of_word_labels_code_mem program l). |
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| 12 | ∀old_policy:(Σpolicy:ppc_pc_map. |
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[2059] | 13 | And (And (And (And (out_of_program_none program policy) |
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[2101] | 14 | (not_jump_default program policy)) |
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[2034] | 15 | (\fst (bvt_lookup … (bitvector_of_nat ? 0) (\snd policy) 〈0,short_jump〉) = 0)) |
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[2059] | 16 | (\fst policy = \fst (bvt_lookup … (bitvector_of_nat ? (|program|)) (\snd policy) 〈0,short_jump〉))) |
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[2034] | 17 | (\fst policy < 2^16)). |
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| 18 | (Σx:bool × (option ppc_pc_map). |
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| 19 | let 〈no_ch,y〉 ≝ x in |
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| 20 | match y with |
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| 21 | [ None ⇒ nec_plus_ultra program old_policy |
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[2059] | 22 | | Some p ⇒ And (And (And (And (And (And (And (And (out_of_program_none program p) |
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[2101] | 23 | (not_jump_default program p)) |
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| 24 | (jump_increase program old_policy p)) |
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| 25 | (no_ch = true → sigma_compact program labels p)) |
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[2034] | 26 | (\fst (bvt_lookup … (bitvector_of_nat ? 0) (\snd p) 〈0,short_jump〉) = 0)) |
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| 27 | (\fst p = \fst (bvt_lookup … (bitvector_of_nat ? (|program|)) (\snd p) 〈0,short_jump〉))) |
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[2101] | 28 | (no_ch = true → sigma_pc_equal program old_policy p)) |
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| 29 | (sigma_jump_equal program old_policy p → no_ch = true)) |
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[2034] | 30 | (\fst p < 2^16) |
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| 31 | ]) |
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| 32 | ≝ |
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| 33 | λprogram.λlabels.λold_sigma. |
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| 34 | let 〈final_added, final_policy〉 ≝ |
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| 35 | pi1 ?? (foldl_strong (option Identifier × pseudo_instruction) |
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| 36 | (λprefix.Σx:ℕ × ppc_pc_map. |
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| 37 | let 〈added,policy〉 ≝ x in |
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[2059] | 38 | And (And (And (And (And (And (And (And (out_of_program_none prefix policy) |
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[2101] | 39 | (not_jump_default prefix policy)) |
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| 40 | (jump_increase prefix old_sigma policy)) |
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| 41 | (sigma_compact_unsafe prefix labels policy)) |
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| 42 | (sigma_safe prefix labels added old_sigma policy)) |
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[2034] | 43 | (\fst (bvt_lookup … (bitvector_of_nat ? 0) (\snd policy) 〈0,short_jump〉) = 0)) |
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| 44 | (\fst policy = \fst (bvt_lookup … (bitvector_of_nat ? (|prefix|)) (\snd policy) 〈0,short_jump〉))) |
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[2101] | 45 | (added = 0 → sigma_pc_equal prefix old_sigma policy)) |
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| 46 | (sigma_jump_equal prefix old_sigma policy → added = 0)) |
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[2034] | 47 | program |
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| 48 | (λprefix.λx.λtl.λprf.λacc. |
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| 49 | let 〈inc_added, inc_pc_sigma〉 ≝ (pi1 ?? acc) in |
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| 50 | let 〈label,instr〉 ≝ x in |
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| 51 | (* Now, we must add the current ppc and its pc translation. |
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| 52 | * Three possibilities: |
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| 53 | * - Instruction is not a jump; i.e. constant size whatever the sigma we use; |
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| 54 | * - Instruction is a backward jump; we can use the sigma we're constructing, |
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| 55 | * since it will already know the translation of its destination; |
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| 56 | * - Instruction is a forward jump; we must use the old sigma (the new sigma |
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| 57 | * does not know the translation yet), but compensate for the jumps we |
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| 58 | * have lengthened. |
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| 59 | *) |
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| 60 | let add_instr ≝ match instr with |
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| 61 | [ Jmp j ⇒ Some ? (select_jump_length labels old_sigma inc_pc_sigma inc_added (|prefix|) j) |
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| 62 | | Call c ⇒ Some ? (select_call_length labels old_sigma inc_pc_sigma inc_added (|prefix|) c) |
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| 63 | | Instruction i ⇒ jump_expansion_step_instruction labels old_sigma inc_pc_sigma inc_added (|prefix|) i |
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| 64 | | _ ⇒ None ? |
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| 65 | ] in |
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| 66 | let 〈inc_pc, inc_sigma〉 ≝ inc_pc_sigma in |
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| 67 | let old_length ≝ |
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| 68 | \snd (bvt_lookup … (bitvector_of_nat ? (|prefix|)) (\snd (pi1 ?? old_sigma)) 〈0,short_jump〉) in |
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| 69 | let old_size ≝ instruction_size_jmplen old_length instr in |
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| 70 | let 〈new_length, isize〉 ≝ match add_instr with |
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| 71 | [ None ⇒ 〈short_jump, instruction_size_jmplen short_jump instr〉 |
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| 72 | | Some pl ⇒ 〈max_length old_length pl, instruction_size_jmplen (max_length old_length pl) instr〉 |
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| 73 | ] in |
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| 74 | let new_added ≝ match add_instr with |
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| 75 | [ None ⇒ inc_added |
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| 76 | | Some x ⇒ plus inc_added (minus isize old_size) |
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| 77 | ] in |
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[2059] | 78 | let old_Slength ≝ |
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| 79 | \snd (bvt_lookup … (bitvector_of_nat ? (S (|prefix|))) (\snd (pi1 ?? old_sigma)) 〈0,short_jump〉) in |
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[2034] | 80 | let updated_sigma ≝ |
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| 81 | (* we add the new PC to the next position and the new jump length to this one *) |
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[2059] | 82 | bvt_insert … (bitvector_of_nat ? (S (|prefix|))) 〈inc_pc + isize, old_Slength〉 |
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[2034] | 83 | (bvt_insert … (bitvector_of_nat ? (|prefix|)) 〈inc_pc, new_length〉 inc_sigma) in |
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| 84 | 〈new_added, 〈plus inc_pc isize, updated_sigma〉〉 |
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[2059] | 85 | ) 〈0, 〈0, |
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| 86 | bvt_insert … |
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| 87 | (bitvector_of_nat ? 0) |
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| 88 | 〈0, \snd (bvt_lookup … (bitvector_of_nat ? 0) (\snd (pi1 ?? old_sigma)) 〈0,short_jump〉)〉 |
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| 89 | (Stub ??)〉〉 |
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| 90 | ) in |
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[2034] | 91 | if geb (\fst final_policy) 2^16 then |
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| 92 | 〈eqb final_added 0, None ?〉 |
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| 93 | else |
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| 94 | 〈eqb final_added 0, Some ? final_policy〉. |
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[2059] | 95 | [ normalize nodelta @nmk #Habs lapply p generalize in match (foldl_strong ?????); * |
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| 96 | #x #H #Heq >Heq in H; normalize nodelta -Heq -x #Hind |
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| 97 | (* USE: inc_pc = fst of policy (from fold) *) |
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| 98 | >(proj2 ?? (proj1 ?? (proj1 ?? Hind))) in p1; |
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| 99 | (* USE: fst policy < 2^16, inc_pc = fst of policy (old_sigma) *) |
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| 100 | lapply (proj2 ?? (pi2 ?? old_sigma)) >(proj2 ?? (proj1 ?? (pi2 ?? old_sigma))) |
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| 101 | #Hsig #Hge |
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| 102 | (* USE: added = 0 → policy_pc_equal (from fold) *) |
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| 103 | lapply ((proj2 ?? (proj1 ?? Hind)) ? (|program|) (le_n (|program|))) |
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| 104 | [ @(proj2 ?? Hind) #i #Hi |
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| 105 | cases (dec_is_jump (\snd (nth i ? program 〈None ?, Comment []〉))) |
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| 106 | [ #Hj |
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| 107 | (* USE: policy_increase (from fold) *) |
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| 108 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hind)))))) i (le_S_to_le … Hi)) |
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| 109 | lapply (Habs i Hi Hj) |
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| 110 | cases (bvt_lookup … (bitvector_of_nat ? i) (\snd old_sigma) 〈0,short_jump〉) |
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| 111 | #opc #oj |
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| 112 | cases (bvt_lookup … (bitvector_of_nat ? i) (\snd final_policy) 〈0,short_jump〉) |
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| 113 | #pc #j normalize nodelta #Heq >Heq cases j |
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| 114 | [1,2: #abs cases abs #abs2 try (cases abs2) @refl |
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| 115 | |3: #_ @refl |
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| 116 | ] |
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| 117 | | #Hnj |
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[2102] | 118 | (* USE: not_jump_default (from fold and old_sigma) *) |
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[2059] | 119 | >(proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hind))))))) i Hi Hnj) |
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| 120 | >(proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (pi2 ?? old_sigma)))) i Hi Hnj) |
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| 121 | @refl |
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| 122 | ] |
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| 123 | | #abs >abs in Hsig; #Hsig |
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| 124 | @(absurd ? Hsig) @le_to_not_lt @leb_true_to_le <geb_to_leb @Hge |
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| 125 | ] |
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[2034] | 126 | | normalize nodelta lapply p generalize in match (foldl_strong ?????); * #x #H #H2 |
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| 127 | >H2 in H; normalize nodelta -H2 -x #H @conj |
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[2059] | 128 | [ @conj [ @conj [ @conj [ @conj [ @conj |
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[2102] | 129 | [ (* USE[pass]: out_of_program_none, not_jump_default, policy_increase (from fold) *) |
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[2059] | 130 | @(proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? H)))))) |
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| 131 | | (* policy_compact_unsafe → policy_compact (in the end) *) |
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| 132 | #Hch #i #Hi |
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| 133 | (* USE: policy_compact_unsafe (from fold) *) |
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| 134 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? H))))) i Hi) |
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| 135 | (* USE: policy_safe (from fold) *) |
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| 136 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? H)))) i Hi) |
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| 137 | lapply (refl ? (lookup_opt … (bitvector_of_nat ? i) (\snd final_policy))) |
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| 138 | cases (lookup_opt … (bitvector_of_nat ? i) (\snd final_policy)) in ⊢ (???% → %); |
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| 139 | [ / by / |
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| 140 | | #x cases x -x #x1 #x2 #EQ |
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| 141 | cases (lookup_opt … (bitvector_of_nat ? (S i)) (\snd final_policy)) |
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| 142 | [ / by / |
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| 143 | | #y cases y -y #y1 #y2 normalize nodelta #H #H2 |
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| 144 | cut (instruction_size_jmplen x2 |
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| 145 | (\snd (nth i ? (pi1 ?? program) 〈None ?, Comment []〉)) = |
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| 146 | instruction_size … (bitvector_of_nat ? i) |
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| 147 | (\snd (nth i ? (pi1 ?? program) 〈None ?, Comment []〉))) |
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| 148 | [5: #H3 <H3 @H2 |
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| 149 | |4: whd in match (instruction_size_jmplen ??); |
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| 150 | whd in match (instruction_size …); |
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| 151 | whd in match (assembly_1_pseudoinstruction …); |
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| 152 | whd in match (expand_pseudo_instruction …); |
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| 153 | normalize nodelta whd in match (append …) in H; |
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| 154 | cases (nth i ? (pi1 ?? program) 〈None ?,Comment []〉) in H; |
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| 155 | #lbl #instr cases instr |
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| 156 | [2,3,6: #x [3: #y] normalize nodelta #H @refl |
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| 157 | |4,5: #x >(lookup_opt_lookup_hit … EQ 〈0,short_jump〉) #Hj |
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| 158 | lapply (Hj x (refl ? x)) -Hj normalize nodelta |
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| 159 | >add_bitvector_of_nat_plus <(plus_n_Sm i 0) <plus_n_O |
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| 160 | cases x2 normalize nodelta |
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| 161 | [1,4: whd in match short_jump_cond; normalize nodelta |
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| 162 | lapply (refl ? (leb i (lookup_def ?? labels x 0))) |
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| 163 | cases (leb i (lookup_def ?? labels x 0)) in ⊢ (???% → %); #Hlab |
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| 164 | normalize nodelta |
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| 165 | (* USE: added = 0 → policy_pc_equal *) |
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| 166 | >(proj2 ?? (proj1 ?? H) (eqb_true_to_eq … Hch) (lookup_def ?? labels x 0) ?) |
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[2102] | 167 | [2,4,6,8: @(label_in_program program labels x) |
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| 168 | cases daemon (* XXX this has to come from somewhere else *) |
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| 169 | ] |
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[2059] | 170 | [1,5: >(eqb_true_to_eq … Hch) <plus_n_O] |
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| 171 | cases (sub_16_with_carry (bitvector_of_nat ??) (bitvector_of_nat ??) false) |
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| 172 | #result #flags normalize nodelta |
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| 173 | cases (vsplit bool 9 7 result) #upper #lower normalize nodelta |
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| 174 | cases (get_index' bool 2 0 flags) normalize nodelta |
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| 175 | [1,3,5,7: cases (eq_bv 9 upper ?) |
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| 176 | |2,4,6,8: cases (eq_bv 9 upper (zero 9))] |
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| 177 | [2,4,6,8,10,12,14,16: #H lapply (proj1 ?? H) #H3 destruct (H3) |
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| 178 | |5,7,13,15: #H @⊥ @(absurd ?? (proj2 ?? H)) / by I/ |
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| 179 | |1,3,9,11: #_ normalize nodelta @refl |
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[2034] | 180 | ] |
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[2101] | 181 | |2,5: whd in match short_jump_cond; whd in match absolute_jump_cond; |
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[2059] | 182 | lapply (refl ? (leb i (lookup_def ?? labels x 0))) |
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| 183 | cases (leb i (lookup_def ?? labels x 0)) in ⊢ (???% → %); #Hlab |
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| 184 | normalize nodelta |
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| 185 | (* USE: added = 0 → policy_pc_equal *) |
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| 186 | >(proj2 ?? (proj1 ?? H) (eqb_true_to_eq … Hch) (lookup_def ?? labels x 0) ?) |
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[2102] | 187 | [2,4,6,8: @(label_in_program program labels x) |
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| 188 | cases daemon (* XXX this has to come from somewhere else *)] |
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[2059] | 189 | [1,3: >(eqb_true_to_eq … Hch) <plus_n_O] |
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| 190 | normalize nodelta cases (vsplit bool 5 11 ?) #addr1 #addr2 |
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| 191 | cases (vsplit bool 5 11 ?) #pc1 #pc2 normalize nodelta |
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| 192 | cases (sub_16_with_carry (bitvector_of_nat ??) (bitvector_of_nat ??) false) |
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| 193 | #result #flags normalize nodelta |
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| 194 | cases (vsplit bool 9 7 result) #upper #lower normalize nodelta |
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| 195 | cases (get_index' bool 2 0 flags) normalize nodelta |
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| 196 | #H >(proj2 ?? (proj1 ?? H)) >(proj1 ?? (proj1 ?? H)) normalize nodelta @refl |
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[2101] | 197 | |3,6: whd in match short_jump_cond; whd in match absolute_jump_cond; |
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[2059] | 198 | lapply (refl ? (leb i (lookup_def ?? labels x 0))) |
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| 199 | cases (leb i (lookup_def ?? labels x 0)) in ⊢ (???% → %); #Hlab |
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| 200 | normalize nodelta |
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| 201 | >(proj2 ?? (proj1 ?? H) (eqb_true_to_eq … Hch) (lookup_def ?? labels x 0) ?) |
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[2102] | 202 | [2,4,6,8: @(label_in_program program labels x) |
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| 203 | cases daemon (* XXX this has to come from somewhere else *)] |
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[2059] | 204 | [1,3: >(eqb_true_to_eq … Hch) <plus_n_O] |
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| 205 | normalize nodelta |
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| 206 | cases (vsplit bool 5 11 ?) #addr1 #addr2 |
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| 207 | cases (vsplit bool 5 11 ?) #pc1 #pc2 normalize nodelta |
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| 208 | cases (sub_16_with_carry (bitvector_of_nat ??) (bitvector_of_nat ??) false) |
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| 209 | #result #flags normalize nodelta |
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| 210 | cases (vsplit bool 9 7 result) #upper #lower normalize nodelta |
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| 211 | cases (get_index' bool 2 0 flags) normalize nodelta |
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| 212 | #H >(proj1 ?? H) >(proj2 ?? H) normalize nodelta @refl |
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| 213 | ] |
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| 214 | |1: #pi cases pi |
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| 215 | [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37: |
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| 216 | [1,2,3,6,7,24,25: #x #y |
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| 217 | |4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23: #x] |
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| 218 | normalize nodelta #H @refl |
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| 219 | |9,10,11,12,13,14,15,16,17: [1,2,6,7: #x |3,4,5,8,9: #y #x] |
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| 220 | normalize nodelta >(lookup_opt_lookup_hit … EQ 〈0,short_jump〉) |
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| 221 | #Hj lapply (Hj x (refl ? x)) -Hj |
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| 222 | whd in match expand_relative_jump; normalize nodelta |
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| 223 | whd in match expand_relative_jump_internal; normalize nodelta |
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| 224 | whd in match expand_relative_jump_unsafe; normalize nodelta |
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| 225 | whd in match expand_relative_jump_internal_unsafe; |
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| 226 | normalize nodelta >(add_bitvector_of_nat_plus ? i 1) |
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| 227 | <(plus_n_Sm i 0) <plus_n_O |
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[2102] | 228 | cases daemon (* XXX this needs subadressing mode magic, see above *) |
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[2059] | 229 | ] |
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| 230 | ] |
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| 231 | ] |
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| 232 | ] |
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| 233 | ] |
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| 234 | ] |
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| 235 | | (* USE: 0 ↦ 0 (from fold) *) @(proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? H)))) |
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| 236 | ] |
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| 237 | | (* USE: inc_pc = fst of policy (from fold) *) @(proj2 ?? (proj1 ?? (proj1 ?? H))) |
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| 238 | ] |
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| 239 | | #H2 (* USE: added = 0 → policy_pc_equal (from fold) *) |
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| 240 | @(proj2 ?? (proj1 ?? H)) @eqb_true_to_eq @H2 |
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| 241 | ] |
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| 242 | | #H2 (* USE: policy_jump_equal → added = 0 (from fold *) |
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| 243 | @eq_to_eqb_true @(proj2 ?? H) @H2 |
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| 244 | ] |
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| 245 | | @not_le_to_lt @leb_false_to_not_le <geb_to_leb @p1 |
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| 246 | ] |
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[2034] | 247 | |4: lapply (pi2 ?? acc) >p cases inc_pc_sigma #inc_pc #inc_sigma |
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| 248 | lapply (refl ? (bvt_lookup … (bitvector_of_nat ? (|prefix|)) (\snd (pi1 ?? old_sigma)) 〈0,short_jump〉)) |
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| 249 | cases (bvt_lookup … (bitvector_of_nat ? (|prefix|)) (\snd (pi1 ?? old_sigma)) 〈0,short_jump〉) in ⊢ (???% → %); |
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| 250 | #old_pc #old_length #Holdeq #Hpolicy @pair_elim #added #policy normalize nodelta |
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| 251 | @pair_elim #new_length #isize normalize nodelta #Heq1 #Heq2 |
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[2059] | 252 | @conj [ @conj [ @conj [ @conj [ @conj [ @conj [ @conj [ @conj |
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[2034] | 253 | [ (* out_of_program_none *) #i #Hi2 >append_length <commutative_plus @conj |
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| 254 | [ (* → *) #Hi normalize in Hi; |
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| 255 | cases instr in Heq2; normalize nodelta |
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| 256 | #x [6: #y] #H <(proj2 ?? (pair_destruct ?????? H)) >lookup_opt_insert_miss |
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| 257 | [1,3,5,7,9,11: >lookup_opt_insert_miss |
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[2059] | 258 | [1,3,5,7,9,11: (* USE: out_of_program_none → *) |
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| 259 | @(proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpolicy))))))) i Hi2)) |
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[2034] | 260 | @le_S_to_le @Hi |
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| 261 | |2,4,6,8,10,12: @bitvector_of_nat_abs |
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| 262 | [1,4,7,10,13,16: @Hi2 |
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| 263 | |2,5,8,11,14,17: @(transitive_lt … Hi2) |
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| 264 | |3,6,9,12,15,18: @sym_neq @lt_to_not_eq |
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| 265 | ] @le_S_to_le @Hi |
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| 266 | ] |
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| 267 | |2,4,6,8,10,12: @bitvector_of_nat_abs |
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| 268 | [1,4,7,10,13,16: @Hi2 |
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| 269 | |2,5,8,11,14,17: @(transitive_lt … Hi2) |
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| 270 | |3,6,9,12,15,18: @sym_neq @lt_to_not_eq |
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| 271 | ] @Hi |
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| 272 | ] |
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| 273 | | (* ← *) <(proj2 ?? (pair_destruct ?????? Heq2)) #Hl |
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| 274 | elim (decidable_lt i (|prefix|)) |
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| 275 | [ #Hi |
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| 276 | lapply (proj2 ?? (insert_lookup_opt_miss ?????? (proj2 ?? (insert_lookup_opt_miss ?????? Hl)))) |
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[2059] | 277 | #Hl2 (* USE: out_of_program_none ← *) |
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| 278 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpolicy))))))) i Hi2) Hl2) |
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[2034] | 279 | #Hi3 @⊥ @(absurd ? Hi) @le_to_not_lt @le_S_to_le @Hi3 |
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| 280 | | #Hi elim (le_to_or_lt_eq … (not_lt_to_le … Hi)) |
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| 281 | [ #Hi3 elim (le_to_or_lt_eq … Hi3) |
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| 282 | [ / by / |
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| 283 | | #X lapply (proj1 ?? (insert_lookup_opt_miss ?????? Hl)) >X >eq_bv_refl #H normalize in H; destruct (H) |
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| 284 | ] |
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| 285 | | #X lapply (proj1 ?? (insert_lookup_opt_miss ?????? (proj2 ?? (insert_lookup_opt_miss ?????? Hl)))) |
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| 286 | >X >eq_bv_refl #H normalize in H; destruct (H) |
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| 287 | ] |
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| 288 | ] |
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| 289 | ] |
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[2102] | 290 | | (* not_jump_default *) #i >append_length <commutative_plus #Hi normalize in Hi; |
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[2034] | 291 | cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi |
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| 292 | [ <(proj2 ?? (pair_destruct ?????? Heq2)) >lookup_insert_miss |
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| 293 | [ >lookup_insert_miss |
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| 294 | [ >(nth_append_first ? i prefix ?? Hi) |
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[2102] | 295 | (* USE: not_jump_default *) |
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[2059] | 296 | @(proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpolicy))))))) i Hi) |
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[2034] | 297 | | @bitvector_of_nat_abs |
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| 298 | [ @(transitive_lt … (pi2 ?? program)) >prf >append_length >plus_n_Sm |
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| 299 | >commutative_plus @le_plus_a @Hi |
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| 300 | | @(transitive_lt … (pi2 ?? program)) >prf >append_length @le_S_S |
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| 301 | @le_plus_n_r |
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| 302 | | @lt_to_not_eq @Hi |
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| 303 | ] |
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| 304 | ] |
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| 305 | | @bitvector_of_nat_abs |
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| 306 | [ @(transitive_lt … (pi2 ?? program)) >prf >append_length >plus_n_Sm |
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| 307 | >commutative_plus @le_plus_a @Hi |
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| 308 | | @(transitive_lt … (pi2 ?? program)) @le_S_S >prf >append_length <plus_n_Sm |
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| 309 | @le_S_S @le_plus_n_r |
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| 310 | | @lt_to_not_eq @le_S @Hi |
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| 311 | ] |
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| 312 | ] |
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| 313 | | <(proj2 ?? (pair_destruct ?????? Heq2)) >Hi >lookup_insert_miss |
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| 314 | [ >lookup_insert_hit cases instr in Heq1; |
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| 315 | [2,3,6: #x [3: #y] normalize nodelta #Heq1 <(proj1 ?? (pair_destruct ?????? Heq1)) #_ @refl |
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| 316 | |4,5: #x normalize nodelta #Heq1 #H @⊥ cases H #H @H >nth_append_second |
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| 317 | [1,3: <minus_n_n whd in match (nth ????); / by I/ |
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| 318 | |2,4: @le_n |
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| 319 | ] |
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| 320 | |1: #pi >nth_append_second [2: @le_n] <minus_n_n whd in match (nth ????); cases pi |
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| 321 | [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37: |
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| 322 | [1,2,3,6,7,24,25: #x #y |
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| 323 | |4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23: #x] normalize nodelta #Heq1 |
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| 324 | <(proj1 ?? (pair_destruct ?????? Heq1)) #_ @refl |
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| 325 | |9,10,11,12,13,14,15,16,17: #x [3,4,5,8,9: #y] normalize nodelta |
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| 326 | #_ #H @⊥ cases H #H2 @H2 / by I/ |
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| 327 | ] |
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| 328 | ] |
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| 329 | | @bitvector_of_nat_abs |
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| 330 | [ @(transitive_lt … (pi2 ?? program)) >prf @le_S_S >append_length @le_plus_n_r |
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| 331 | | @(transitive_lt … (pi2 ?? program)) >prf @le_S_S >append_length <plus_n_Sm |
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| 332 | @le_S_S @le_plus_n_r |
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| 333 | | @lt_to_not_eq @le_n |
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| 334 | ] |
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| 335 | ] |
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| 336 | ] |
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| 337 | ] |
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| 338 | | (* policy_increase *) #i >append_length >commutative_plus #Hi normalize in Hi; |
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[2059] | 339 | cases (le_to_or_lt_eq … Hi) -Hi; #Hi |
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| 340 | [ cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi |
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| 341 | [ (* USE: policy_increase *) |
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| 342 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpolicy)))))) i (le_S_to_le … Hi)) |
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| 343 | <(proj2 ?? (pair_destruct ?????? Heq2)) |
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| 344 | @pair_elim #opc #oj #EQ1 >lookup_insert_miss |
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| 345 | [ >lookup_insert_miss |
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| 346 | [ @pair_elim #pc #j #EQ2 / by / |
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| 347 | | @bitvector_of_nat_abs |
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| 348 | [ @(transitive_lt … (pi2 ?? program)) >prf >append_length @le_S |
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| 349 | >commutative_plus @le_plus_a @Hi |
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| 350 | | @(transitive_lt … (pi2 ?? program)) >prf >append_length @le_S_S @le_plus_n_r |
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| 351 | | @lt_to_not_eq @Hi |
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| 352 | ] |
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| 353 | ] |
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[2034] | 354 | | @bitvector_of_nat_abs |
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[2059] | 355 | [ @(transitive_lt … (pi2 ?? program)) >prf >append_length @le_S >commutative_plus |
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[2034] | 356 | @le_plus_a @Hi |
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[2059] | 357 | | @(transitive_lt … (pi2 ?? program)) >prf >append_length @le_S_S <plus_n_Sm @le_plus_n_r |
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| 358 | | @lt_to_not_eq @le_S @Hi |
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[2034] | 359 | ] |
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| 360 | ] |
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[2059] | 361 | | >Hi <(proj2 ?? (pair_destruct ?????? Heq2)) @pair_elim #opc #oj #EQ1 |
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| 362 | >lookup_insert_miss |
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| 363 | [ >lookup_insert_hit cases (dec_is_jump instr) |
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| 364 | [ cases instr in Heq1; normalize nodelta |
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| 365 | [ #pi cases pi |
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| 366 | [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37: |
---|
| 367 | [1,2,3,6,7,24,25: #x #y |
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| 368 | |4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23: #x] #_ #Hj cases Hj |
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| 369 | |9,10,11,12,13,14,15,16,17: #x [3,4,5,8,9: #y] |
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| 370 | whd in match jump_expansion_step_instruction; normalize nodelta #Heq1 |
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| 371 | #_ <(proj1 ?? (pair_destruct ?????? Heq1)) >Holdeq in EQ1; normalize nodelta |
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| 372 | #H42 >(proj2 ?? (pair_destruct ?????? H42)) @jmpleq_max_length |
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| 373 | ] |
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| 374 | |2,3,6: #x [3: #y] #_ #Hj cases Hj |
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| 375 | |4,5: #x #Heq1 #_ <(proj1 ?? (pair_destruct ?????? Heq1)) >Holdeq in EQ1; |
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| 376 | normalize nodelta #H42 >(proj2 ?? (pair_destruct ?????? H42)) @jmpleq_max_length |
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[2034] | 377 | ] |
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[2059] | 378 | | lapply Heq1 -Heq1; lapply (refl ? instr); cases instr in ⊢ (???% → %); normalize nodelta |
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| 379 | [ #pi cases pi |
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| 380 | [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37: |
---|
| 381 | [1,2,3,6,7,24,25: #x #y |
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| 382 | |4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23: #x] |
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| 383 | whd in match jump_expansion_step_instruction; normalize nodelta #Heqi #Heq1 |
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| 384 | #Hj <(proj1 ?? (pair_destruct ?????? Heq1)) |
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[2102] | 385 | (* USE: not_jump_default (from old_sigma) *) |
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[2059] | 386 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (pi2 ?? old_sigma)))) (|prefix|) ??) |
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| 387 | [1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79,82: |
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| 388 | >prf >nth_append_second |
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| 389 | [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: |
---|
| 390 | <minus_n_n whd in match (nth ????); >p1 >Heqi @Hj |
---|
| 391 | |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: |
---|
| 392 | @le_n |
---|
| 393 | ] |
---|
| 394 | |2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,74,77,80,83: |
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| 395 | >prf >append_length <plus_n_Sm @le_S_S @le_plus_n_r |
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| 396 | |3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81,84: |
---|
| 397 | cases (lookup ?? (bitvector_of_nat ? (|prefix|)) (\snd (pi1 ?? old_sigma)) 〈0,short_jump〉) in EQ1; |
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| 398 | #a #b #EQ1 >(proj2 ?? (pair_destruct ?????? EQ1)) #EQ2 >EQ2 |
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| 399 | %2 @refl |
---|
[2034] | 400 | ] |
---|
[2059] | 401 | |9,10,11,12,13,14,15,16,17: #x [3,4,5,8,9: #y] |
---|
| 402 | #_ #_ #abs cases abs #ABS @⊥ @ABS / by I/ |
---|
[2034] | 403 | ] |
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[2059] | 404 | |2,3,6: #x [3: #y] #Heqi #Heq1 #Hj <(proj1 ?? (pair_destruct ?????? Heq1)) |
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[2102] | 405 | (* USE: not_jump_default (from old_sigma) *) |
---|
[2059] | 406 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (pi2 ?? old_sigma)))) (|prefix|) ??) |
---|
| 407 | [1,4,7: >prf >nth_append_second |
---|
| 408 | [1,3,5: <minus_n_n whd in match (nth ????); >p1 >Heqi @Hj |
---|
| 409 | |2,4,6: @le_n |
---|
| 410 | ] |
---|
| 411 | |2,5,8: >prf >append_length <plus_n_Sm @le_S_S @le_plus_n_r |
---|
| 412 | |3,6,9: |
---|
| 413 | cases (lookup ?? (bitvector_of_nat ? (|prefix|)) (\snd (pi1 ?? old_sigma)) 〈0,short_jump〉) in EQ1; |
---|
| 414 | #a #b #EQ1 >(proj2 ?? (pair_destruct ?????? EQ1)) #EQ2 >EQ2 %2 @refl |
---|
[2034] | 415 | ] |
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[2059] | 416 | |4,5: #x #_ #_ #abs cases abs #ABS @⊥ @ABS / by I/ |
---|
[2034] | 417 | ] |
---|
| 418 | ] |
---|
[2059] | 419 | | @bitvector_of_nat_abs |
---|
| 420 | [ @(transitive_lt … (pi2 … program)) >prf @le_S_S >append_length @le_plus_n_r |
---|
| 421 | | @(transitive_lt … (pi2 … program)) @le_S_S >prf >append_length <plus_n_Sm |
---|
| 422 | @le_S_S @le_plus_n_r |
---|
| 423 | | @lt_to_not_eq @le_n |
---|
| 424 | ] |
---|
[2034] | 425 | ] |
---|
| 426 | ] |
---|
[2059] | 427 | | <(proj2 ?? (pair_destruct ?????? Heq2)) >Hi >lookup_insert_hit |
---|
| 428 | normalize nodelta |
---|
| 429 | cases (bvt_lookup … (bitvector_of_nat ? (S (|prefix|))) (\snd old_sigma) 〈0,short_jump〉) |
---|
| 430 | #a #b normalize nodelta %2 @refl |
---|
[2034] | 431 | ] |
---|
| 432 | ] |
---|
| 433 | | (* policy_compact_unsafe *) #i >append_length <commutative_plus #Hi normalize in Hi; |
---|
| 434 | <(proj2 ?? (pair_destruct ?????? Heq2)) |
---|
| 435 | cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi |
---|
| 436 | [ >lookup_opt_insert_miss |
---|
| 437 | [ >lookup_opt_insert_miss |
---|
| 438 | [ >lookup_opt_insert_miss |
---|
| 439 | [ cases (le_to_or_lt_eq … Hi) -Hi #Hi |
---|
| 440 | [ >lookup_opt_insert_miss |
---|
[2059] | 441 | [ (* USE: policy_compact_unsafe *) |
---|
| 442 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpolicy))))) i ?) |
---|
[2034] | 443 | [ @le_S_to_le @Hi ] |
---|
| 444 | cases (bvt_lookup_opt … (bitvector_of_nat ? i) inc_sigma) |
---|
| 445 | [ normalize nodelta / by / |
---|
| 446 | | #x cases x -x #x1 #x2 |
---|
| 447 | cases (bvt_lookup_opt … (bitvector_of_nat ? (S i)) inc_sigma) |
---|
| 448 | normalize nodelta |
---|
| 449 | [ / by / |
---|
| 450 | | #y cases y -y #y1 #y2 normalize nodelta >nth_append_first |
---|
| 451 | [ / by / |
---|
| 452 | | @le_S_to_le @Hi |
---|
| 453 | ] |
---|
| 454 | ] |
---|
| 455 | ] |
---|
| 456 | | @bitvector_of_nat_abs |
---|
| 457 | [3: @lt_to_not_eq @Hi ] |
---|
| 458 | ] |
---|
| 459 | | >Hi >lookup_opt_insert_hit normalize nodelta |
---|
[2059] | 460 | (* USE: policy_compact_unsafe *) |
---|
| 461 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpolicy))))) i ?) |
---|
[2034] | 462 | [ <Hi @le_n |
---|
| 463 | | cases (bvt_lookup_opt … (bitvector_of_nat ? i) inc_sigma) |
---|
| 464 | [ normalize nodelta / by / |
---|
| 465 | | #x cases x -x #x1 #x2 |
---|
[2059] | 466 | (* USE: inc_pc = fst inc_sigma *) |
---|
| 467 | lapply (proj2 ?? (proj1 ?? (proj1 ?? Hpolicy))) <Hi |
---|
[2034] | 468 | lapply (refl ? (bvt_lookup_opt … (bitvector_of_nat ? (S i)) inc_sigma)) |
---|
| 469 | cases (bvt_lookup_opt … (bitvector_of_nat ? (S i)) inc_sigma) in ⊢ (???% → %); |
---|
| 470 | [ normalize nodelta #_ #_ #H cases H |
---|
| 471 | | #y cases y -y #y1 #y2 #Heq >nth_append_first |
---|
| 472 | [ normalize nodelta >(lookup_opt_lookup_hit … Heq 〈0,short_jump〉) |
---|
| 473 | #Heq2 <Heq2 / by / |
---|
| 474 | | <Hi @le_n |
---|
| 475 | ] |
---|
| 476 | ] |
---|
| 477 | ] |
---|
| 478 | ] |
---|
| 479 | ] |
---|
| 480 | ] |
---|
| 481 | ] |
---|
| 482 | ] |
---|
| 483 | [3,4,5: @bitvector_of_nat_abs] |
---|
| 484 | [1: (* Si < 2^16 *) @(transitive_lt … (pi2 ?? program)) @le_S_S >prf >append_length |
---|
| 485 | >commutative_plus @le_plus_a @Hi |
---|
| 486 | |2,8: (* Spref < 2*16 *) @(transitive_lt … (pi2 ?? program)) @le_S_S >prf >append_length |
---|
| 487 | <plus_n_Sm @le_S_S @le_plus_n_r |
---|
| 488 | | (* Si ≠ Spref *) @lt_to_not_eq @le_S_S @Hi |
---|
| 489 | |4,7: (* i < 2^16 *) @(transitive_lt … (pi2 ?? program)) @le_S_S >prf >append_length |
---|
| 490 | >commutative_plus @le_plus_a @le_S_to_le @Hi |
---|
| 491 | |5,11: (* pref < 2^16 *) @(transitive_lt … (pi2 ?? program)) @le_S_S >prf >append_length |
---|
| 492 | @le_plus_n_r |
---|
| 493 | | @lt_to_not_eq @Hi |
---|
| 494 | | @lt_to_not_eq @le_S_to_le @le_S_S @Hi |
---|
| 495 | | @(transitive_lt … (pi2 ?? program)) >prf >append_length >plus_n_Sm |
---|
| 496 | >commutative_plus @le_plus_a @Hi |
---|
| 497 | ] |
---|
| 498 | | >Hi >lookup_opt_insert_miss |
---|
| 499 | [2: @bitvector_of_nat_abs |
---|
| 500 | [ @(transitive_lt … (pi2 ?? program)) >prf >append_length @le_S_S @le_plus_n_r |
---|
| 501 | | @(transitive_lt … (pi2 ?? program)) >prf >append_length @le_S_S |
---|
| 502 | <plus_n_Sm @le_S_S @le_plus_n_r |
---|
| 503 | | @lt_to_not_eq @le_n |
---|
| 504 | ] |
---|
| 505 | ] |
---|
| 506 | >lookup_opt_insert_hit >lookup_opt_insert_hit normalize nodelta |
---|
[2059] | 507 | (* USE: out_of_program_none ← *) |
---|
| 508 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpolicy))))))) i ?)) |
---|
[2034] | 509 | [ >Hi @(transitive_lt … (pi2 ?? program)) >prf >append_length @le_S_S @le_plus_n_r |
---|
| 510 | | >Hi |
---|
[2059] | 511 | (* USE: inc_pc = fst policy *) |
---|
| 512 | lapply (proj2 ?? (proj1 ?? (proj1 ?? Hpolicy))) |
---|
[2034] | 513 | lapply (refl ? (bvt_lookup_opt … (bitvector_of_nat ? (|prefix|)) inc_sigma)) |
---|
| 514 | cases (bvt_lookup_opt … (bitvector_of_nat ? (|prefix|)) inc_sigma) in ⊢ (???% → %); |
---|
| 515 | [ #Heq #_ #H @⊥ @(absurd (|prefix| > |prefix|)) |
---|
| 516 | [ @H @refl |
---|
| 517 | | @le_to_not_lt @le_n |
---|
| 518 | ] |
---|
| 519 | | #x cases x -x #x1 #x2 #Heq #Hip #_ >nth_append_second |
---|
| 520 | [2: @le_n] <minus_n_n whd in match (nth ????); normalize nodelta |
---|
| 521 | >Hip >(lookup_opt_lookup_hit … Heq 〈0,short_jump〉) |
---|
| 522 | cases instr in Heq1; |
---|
| 523 | [2,3,6: #x [3: #y] #Heq1 <(proj2 ?? (pair_destruct ?????? Heq1)) |
---|
| 524 | / by refl/ |
---|
| 525 | |4,5: #x #Heq1 <(proj2 ?? (pair_destruct ?????? Heq1)) |
---|
| 526 | <(proj1 ?? (pair_destruct ?????? Heq1)) @refl |
---|
| 527 | |1: #pi cases pi normalize nodelta |
---|
| 528 | [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37: |
---|
| 529 | [1,2,3,6,7,24,25: #x #y |
---|
| 530 | |4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23: #x] |
---|
| 531 | #Heq1 <(proj2 ?? (pair_destruct ?????? Heq1)) |
---|
| 532 | / by / |
---|
| 533 | |9,10,11,12,13,14,15,16,17: #x [3,4,5,8,9: #y] #Heq1 |
---|
| 534 | <(proj2 ?? (pair_destruct ?????? Heq1)) |
---|
| 535 | <(proj1 ?? (pair_destruct ?????? Heq1)) |
---|
| 536 | / by / |
---|
| 537 | ] |
---|
| 538 | ] |
---|
| 539 | ] |
---|
| 540 | ] |
---|
| 541 | ] |
---|
| 542 | ] |
---|
[2059] | 543 | | (* policy_safe *) cases daemon |
---|
| 544 | (* #i >append_length >commutative_plus #Hi normalize in Hi; |
---|
[2034] | 545 | elim (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi |
---|
| 546 | [ >nth_append_first |
---|
| 547 | [2: @Hi] |
---|
| 548 | <(proj2 ?? (pair_destruct ?????? Heq2)) |
---|
| 549 | >lookup_insert_miss |
---|
[2059] | 550 | [ >lookup_insert_miss |
---|
| 551 | [ >lookup_insert_miss |
---|
| 552 | [ lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpolicy))) i Hi) |
---|
| 553 | cases (bvt_lookup … (bitvector_of_nat ? i) inc_sigma 〈0,short_jump〉) |
---|
| 554 | #pc #j cases (nth i ? prefix 〈None ?, Comment []〉) #label #instr cases j |
---|
| 555 | normalize nodelta |
---|
| 556 | [ #H cases (le_to_or_lt_eq … Hi) -Hi #Hi |
---|
| 557 | [ >lookup_insert_miss |
---|
| 558 | [ #dest #Hjmp <(proj1 ?? (pair_destruct ?????? Heq2)) |
---|
| 559 | |
---|
[2034] | 560 | | |
---|
| 561 | ] |
---|
| 562 | | |
---|
| 563 | ] *) |
---|
| 564 | ] |
---|
| 565 | | (* 0 ↦ 0 *) <(proj2 ?? (pair_destruct ?????? Heq2)) >lookup_insert_miss |
---|
| 566 | [ cases (decidable_eq_nat 0 (|prefix|)) |
---|
| 567 | [ #Heq <Heq >lookup_insert_hit |
---|
[2059] | 568 | (* USE: inc_pc = fst policy *) |
---|
| 569 | lapply (proj2 ?? (proj1 ?? (proj1 ?? Hpolicy))) |
---|
| 570 | (* USE: 0 ↦ 0 *) |
---|
| 571 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpolicy)))) <Heq |
---|
[2034] | 572 | #ONE #TWO >TWO >ONE @refl |
---|
| 573 | | #Hneq >lookup_insert_miss |
---|
[2059] | 574 | [ (* USE: 0 ↦ 0 *) @(proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hpolicy)))) |
---|
[2034] | 575 | | @bitvector_of_nat_abs |
---|
| 576 | [3: @Hneq] |
---|
| 577 | ] |
---|
| 578 | ] |
---|
| 579 | | @bitvector_of_nat_abs |
---|
| 580 | [3: @lt_to_not_eq / by / ] |
---|
| 581 | ] |
---|
| 582 | [1,3: / by / |
---|
| 583 | |2,4: @(transitive_lt … (pi2 ?? program)) >prf >append_length @le_S_S |
---|
| 584 | [2: <plus_n_Sm @le_S_S] |
---|
| 585 | @le_plus_n_r |
---|
| 586 | ] |
---|
| 587 | ] |
---|
| 588 | | (* inc_pc = fst of policy *) <(proj2 ?? (pair_destruct ?????? Heq2)) |
---|
| 589 | >append_length >(commutative_plus (|prefix|)) >lookup_insert_hit @refl |
---|
| 590 | ] |
---|
[2059] | 591 | | (* added = 0 → policy_pc_equal *) |
---|
| 592 | (* USE: added = 0 → policy_pc_equal *) |
---|
| 593 | lapply (proj2 ?? (proj1 ?? Hpolicy)) |
---|
[2034] | 594 | lapply Heq2 -Heq2 lapply Heq1 -Heq1 lapply (refl ? instr) |
---|
[2059] | 595 | cases daemon |
---|
| 596 | (*cases instr in ⊢ (???% → % → % → %); normalize nodelta |
---|
[2034] | 597 | [ #pi cases pi normalize nodelta |
---|
| 598 | [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37: |
---|
| 599 | [1,2,3,6,7,24,25: #x #y |
---|
| 600 | |4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23: #x] |
---|
| 601 | #Hins #Heq1 #Heq2 #Hold <(proj1 ?? (pair_destruct ?????? Heq2)) #Hadded |
---|
| 602 | #i >append_length >commutative_plus #Hi normalize in Hi; |
---|
| 603 | cases (le_to_or_lt_eq … Hi) -Hi #Hi |
---|
| 604 | [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: |
---|
| 605 | <(proj2 ?? (pair_destruct ?????? Heq2)) |
---|
| 606 | >(lookup_insert_miss ? 〈0,short_jump〉 〈inc_pc+isize,short_jump〉 16 |
---|
| 607 | (bitvector_of_nat 16 (S (|prefix|))) |
---|
| 608 | (bitvector_of_nat 16 i)) |
---|
| 609 | [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: |
---|
[2059] | 610 | (* le_to_or_lt_eq, part 2 *) |
---|
[2034] | 611 | >(lookup_insert_miss ? 〈0,short_jump〉 〈inc_pc,new_length〉 16 |
---|
| 612 | (bitvector_of_nat 16 (|prefix|)) (bitvector_of_nat 16 i)) |
---|
| 613 | [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: |
---|
| 614 | @(Hold Hadded i (le_S_S_to_le … Hi)) |
---|
| 615 | |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: |
---|
| 616 | @bitvector_of_nat_abs |
---|
| 617 | [1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79,82: |
---|
| 618 | @(transitive_lt … (pi2 ?? program)) >prf >append_length >commutative_plus |
---|
[2059] | 619 | @le_S_S @le_plus_a @(le_S_S_to_le … Hi) |
---|
[2034] | 620 | |2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,74,77,80,83: |
---|
| 621 | @(transitive_lt … (pi2 ?? program)) >prf >append_length @le_S_S |
---|
| 622 | @le_plus_n_r |
---|
| 623 | |3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81,84: |
---|
[2059] | 624 | @lt_to_not_eq @(le_S_to_le … Hi) |
---|
[2034] | 625 | ] |
---|
| 626 | ] |
---|
| 627 | |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: |
---|
| 628 | @bitvector_of_nat_abs |
---|
| 629 | [1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79,82: |
---|
| 630 | @(transitive_lt … (pi2 ?? program)) >prf >append_length >commutative_plus |
---|
| 631 | @le_S_S @le_plus_a @le_S_to_le @(le_S_S_to_le … Hi) |
---|
| 632 | |2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,74,77,80,83: |
---|
| 633 | @(transitive_lt … (pi2 ?? program)) >prf >append_length @le_S_S <plus_n_Sm |
---|
| 634 | @le_S_S @le_plus_n_r |
---|
| 635 | |3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81,84: |
---|
| 636 | @lt_to_not_eq @le_S_to_le @Hi |
---|
| 637 | ] |
---|
| 638 | ] |
---|
| 639 | |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: |
---|
| 640 | <(proj2 ?? (pair_destruct ?????? Heq2)) >(injective_S … Hi) |
---|
| 641 | >(lookup_insert_miss ? 〈0,short_jump〉 〈inc_pc+isize,short_jump〉 16 |
---|
| 642 | (bitvector_of_nat 16 (S (|prefix|))) (bitvector_of_nat ? (|prefix|))) |
---|
| 643 | [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: |
---|
| 644 | >lookup_insert_hit |
---|
| 645 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (pi2 ?? old_sigma))) (|prefix|) ??) |
---|
| 646 | [1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79,82: |
---|
| 647 | >prf >nth_append_second |
---|
| 648 | [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: |
---|
| 649 | <minus_n_n whd in match (nth ????); >p1 >Hins @nmk #H @H |
---|
| 650 | |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: |
---|
| 651 | @le_n |
---|
| 652 | ] |
---|
| 653 | |2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,74,77,80,83: |
---|
| 654 | >prf >append_length <plus_n_Sm @le_S_S @le_plus_n_r |
---|
| 655 | |3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81,84: |
---|
| 656 | cases (bvt_lookup … (bitvector_of_nat ? (|prefix|)) (\snd (pi1 ?? old_sigma)) 〈0,short_jump〉) |
---|
| 657 | #a #b #H >H <(proj1 ?? (pair_destruct ?????? Heq1)) normalize nodelta @refl |
---|
| 658 | ] |
---|
| 659 | |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: |
---|
| 660 | @bitvector_of_nat_abs |
---|
| 661 | [1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79,82: |
---|
| 662 | @(transitive_lt … (pi2 ?? program)) >prf >append_length @le_S_S @le_plus_n_r |
---|
| 663 | |2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,74,77,80,83: |
---|
| 664 | @(transitive_lt … (pi2 ?? program)) >prf >append_length @le_S_S <plus_n_Sm |
---|
| 665 | @le_S_S @le_plus_n_r |
---|
| 666 | |3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81,84: |
---|
| 667 | @lt_to_not_eq @le_n |
---|
| 668 | ] |
---|
| 669 | ] |
---|
| 670 | ] |
---|
| 671 | |9,10,11,12,13,14,15,16,17: #x [3,4,5,8,9: #y] #Hins #Heq1 #Heq2 #Hold |
---|
| 672 | <(proj1 ?? (pair_destruct ?????? Heq2)) <(proj2 ?? (pair_destruct ?????? Heq1)) |
---|
| 673 | #H #i >append_length >commutative_plus #Hi normalize in Hi; |
---|
| 674 | cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi |
---|
| 675 | [1,3,5,7,9,11,13,15,17: <(proj2 ?? (pair_destruct ?????? Heq2)) |
---|
| 676 | >(lookup_insert_miss ? 〈0,short_jump〉 〈inc_pc+isize,short_jump〉 16 |
---|
| 677 | (bitvector_of_nat 16 (S (|prefix|))) (bitvector_of_nat 16 i)) |
---|
| 678 | [1,3,5,7,9,11,13,15,17: |
---|
| 679 | >(lookup_insert_miss ? 〈0,short_jump〉 〈inc_pc,new_length〉 16 |
---|
| 680 | (bitvector_of_nat 16 (|prefix|)) (bitvector_of_nat 16 i)) |
---|
| 681 | [1,3,5,7,9,11,13,15,17: @(Hold ? i Hi) |
---|
| 682 | [1,2,3,4,5,6,7,8,9: @sym_eq @le_n_O_to_eq <H @le_plus_n_r] |
---|
| 683 | ] |
---|
| 684 | @bitvector_of_nat_abs |
---|
| 685 | [1,4,7,10,13,16,19,22,25: @(transitive_lt … (pi2 ?? program)) >prf |
---|
| 686 | >append_length >commutative_plus @le_S @le_plus_a @Hi |
---|
| 687 | |2,5,8,11,14,17,20,23,26: @(transitive_lt … (pi2 ?? program)) >prf |
---|
| 688 | >append_length @le_S <plus_n_Sm @le_S_S @le_plus_n_r |
---|
| 689 | |3,6,9,12,15,18,21,24,27: @lt_to_not_eq @Hi |
---|
| 690 | ] |
---|
| 691 | |2,4,6,8,10,12,14,16,18: @bitvector_of_nat_abs |
---|
| 692 | [1,4,7,10,13,16,19,22,25: @(transitive_lt … (pi2 ?? program)) >prf |
---|
| 693 | >append_length >commutative_plus @le_S @le_plus_a @Hi |
---|
| 694 | |2,5,8,11,14,17,20,23,26: @(transitive_lt … (pi2 ?? program)) >prf |
---|
| 695 | >append_length @le_S_S <plus_n_Sm @le_S_S @le_plus_n_r |
---|
| 696 | |3,6,9,12,15,18,21,24,27: @lt_to_not_eq @le_S @Hi |
---|
| 697 | ] |
---|
| 698 | ] |
---|
| 699 | |2,4,6,8,10,12,14,16,18: <(proj2 ?? (pair_destruct ?????? Heq2)) >Hi |
---|
| 700 | >(lookup_insert_miss ? 〈0,short_jump〉 〈inc_pc+isize,short_jump〉 16 |
---|
| 701 | (bitvector_of_nat 16 (S (|prefix|))) (bitvector_of_nat 16 (|prefix|))) |
---|
| 702 | [1,3,5,7,9,11,13,15,17: >lookup_insert_hit |
---|
| 703 | <(proj1 ?? (pair_destruct ?????? Heq1)) |
---|
| 704 | >Holdeq normalize nodelta @sym_eq @blerpque |
---|
| 705 | [3,6,9,12,15,18,21,24,27: |
---|
| 706 | elim (le_to_or_lt_eq … (minus_zero_to_le … (plus_zero_zero … H))) |
---|
| 707 | [1,3,5,7,9,11,13,15,17: #H @⊥ @(absurd ? H) @le_to_not_lt @etblorp |
---|
| 708 | |2,4,6,8,10,12,14,16,18: #H @H |
---|
| 709 | ] |
---|
| 710 | / by I/ |
---|
| 711 | |2,5,8,11,14,17,20,23,26: / by I/ |
---|
| 712 | ] |
---|
| 713 | |2,4,6,8,10,12,14,16,18: @bitvector_of_nat_abs |
---|
| 714 | [1,4,7,10,13,16,19,22,25: @(transitive_lt … (pi2 ?? program)) >prf |
---|
| 715 | >append_length @le_S_S @le_plus_n_r |
---|
| 716 | |2,5,8,11,14,17,20,23,26: @(transitive_lt … (pi2 ?? program)) >prf |
---|
| 717 | >append_length @le_S_S <plus_n_Sm @le_S_S @le_plus_n_r |
---|
| 718 | |3,6,9,12,15,18,21,24,27: @lt_to_not_eq @le_S_S @le_n |
---|
| 719 | ] |
---|
| 720 | ] |
---|
| 721 | ] |
---|
| 722 | ] |
---|
| 723 | |2,3,6: #x [3: #y] #Hins #Heq1 #Heq2 #Hold <(proj1 ?? (pair_destruct ?????? Heq2)) |
---|
| 724 | #Hadded #i >append_length >commutative_plus #Hi normalize in Hi; |
---|
| 725 | cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi |
---|
| 726 | [1,3,5: <(proj2 ?? (pair_destruct ?????? Heq2)) >lookup_insert_miss |
---|
| 727 | [1,3,5: >lookup_insert_miss |
---|
| 728 | [1,3,5: @(Hold Hadded i Hi) |
---|
| 729 | |2,4,6: @bitvector_of_nat_abs |
---|
| 730 | [1,4,7: @(transitive_lt … (pi2 ?? program)) >prf >append_length |
---|
| 731 | >commutative_plus @le_S @le_plus_a @Hi |
---|
| 732 | |2,5,8: @(transitive_lt … (pi2 ?? program)) >prf >append_length |
---|
| 733 | @le_S <plus_n_Sm @le_S_S @le_plus_n_r |
---|
| 734 | |3,6,9: @lt_to_not_eq @Hi |
---|
| 735 | ] |
---|
| 736 | ] |
---|
| 737 | |2,4,6: @bitvector_of_nat_abs |
---|
| 738 | [1,4,7: @(transitive_lt … (pi2 ?? program)) >prf >append_length |
---|
| 739 | >commutative_plus @le_S @le_plus_a @Hi |
---|
| 740 | |2,5,8: @(transitive_lt … (pi2 ?? program)) >prf >append_length |
---|
| 741 | @le_S_S <plus_n_Sm @le_S_S @le_plus_n_r |
---|
| 742 | |3,6,9: @lt_to_not_eq @le_S @Hi |
---|
| 743 | ] |
---|
| 744 | ] |
---|
| 745 | |2,4,6: <(proj2 ?? (pair_destruct ?????? Heq2)) >Hi >lookup_insert_miss |
---|
| 746 | [1,3,5: >lookup_insert_hit |
---|
| 747 | lapply (proj2 ?? (proj1 ?? (proj1 ?? (pi2 ?? old_sigma))) (|prefix|) ??) |
---|
| 748 | [1,4,7: >prf >nth_append_second |
---|
| 749 | [1,3,5: <minus_n_n whd in match (nth ????); >p1 >Hins @nmk #H @H |
---|
| 750 | |2,4,6: @le_n |
---|
| 751 | ] |
---|
| 752 | |2,5,8: >prf >append_length <plus_n_Sm @le_S_S @le_plus_n_r |
---|
| 753 | |3,6,9: |
---|
| 754 | cases (bvt_lookup … (bitvector_of_nat ? (|prefix|)) (\snd (pi1 ?? old_sigma)) 〈0,short_jump〉) |
---|
| 755 | #a #b #H >H <(proj1 ?? (pair_destruct ?????? Heq1)) normalize nodelta @refl |
---|
| 756 | ] |
---|
| 757 | |2,4,6: @bitvector_of_nat_abs |
---|
| 758 | [1,4,7: @(transitive_lt … (pi2 ?? program)) >prf |
---|
| 759 | >append_length @le_S_S @le_plus_n_r |
---|
| 760 | |2,5,8: @(transitive_lt … (pi2 ?? program)) >prf |
---|
| 761 | >append_length @le_S_S <plus_n_Sm @le_S_S @le_plus_n_r |
---|
| 762 | |3,6,9: @lt_to_not_eq @le_S_S @le_n |
---|
| 763 | ] |
---|
| 764 | ] |
---|
| 765 | ] |
---|
| 766 | |4,5: #x #Hins #Heq1 #Heq2 #Hold |
---|
| 767 | <(proj1 ?? (pair_destruct ?????? Heq2)) <(proj2 ?? (pair_destruct ?????? Heq1)) |
---|
| 768 | #H #i >append_length >commutative_plus #Hi normalize in Hi; |
---|
| 769 | cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi |
---|
| 770 | [1,3: <(proj2 ?? (pair_destruct ?????? Heq2)) >lookup_insert_miss |
---|
| 771 | [1,3: >lookup_insert_miss |
---|
| 772 | [1,3: @(Hold ? i Hi) |
---|
| 773 | [1,2: @sym_eq @le_n_O_to_eq <H @le_plus_n_r] |
---|
| 774 | ] |
---|
| 775 | @bitvector_of_nat_abs |
---|
| 776 | [1,4: @(transitive_lt … (pi2 ?? program)) >prf >append_length |
---|
| 777 | >commutative_plus @le_S @le_plus_a @Hi |
---|
| 778 | |2,5: @(transitive_lt … (pi2 ?? program)) >prf >append_length |
---|
| 779 | @le_S_S @le_plus_n_r |
---|
| 780 | |3,6: @lt_to_not_eq @Hi |
---|
| 781 | ] |
---|
| 782 | |2,4: @bitvector_of_nat_abs |
---|
| 783 | [1,4: @(transitive_lt … (pi2 ?? program)) >prf >append_length |
---|
| 784 | >commutative_plus @le_S @le_plus_a @Hi |
---|
| 785 | |2,5: @(transitive_lt … (pi2 ?? program)) >prf >append_length |
---|
| 786 | @le_S_S <plus_n_Sm @le_S_S @le_plus_n_r |
---|
| 787 | |3,6: @lt_to_not_eq @le_S @Hi |
---|
| 788 | ] |
---|
| 789 | ] |
---|
| 790 | |2,4: <(proj2 ?? (pair_destruct ?????? Heq2)) >Hi >lookup_insert_miss |
---|
| 791 | [1,3: >lookup_insert_hit <(proj1 ?? (pair_destruct ?????? Heq1)) |
---|
| 792 | >Holdeq normalize nodelta @sym_eq @blerpque |
---|
| 793 | [3,6: elim (le_to_or_lt_eq … (minus_zero_to_le … (plus_zero_zero … H))) |
---|
| 794 | [1,3: #H @⊥ @(absurd ? H) @le_to_not_lt @etblorp |
---|
| 795 | |2,4: #H @H |
---|
| 796 | ] |
---|
| 797 | / by I/ |
---|
| 798 | |2,5: / by I/ |
---|
| 799 | ] |
---|
| 800 | |2,4: @bitvector_of_nat_abs |
---|
| 801 | [1,4: @(transitive_lt … (pi2 ?? program)) >prf >append_length |
---|
| 802 | @le_S_S @le_plus_n_r |
---|
| 803 | |2,5,8: @(transitive_lt … (pi2 ?? program)) >prf >append_length |
---|
| 804 | @le_S_S <plus_n_Sm @le_S_S @le_plus_n_r |
---|
| 805 | |3,6,9: @lt_to_not_eq @le_S_S @le_n |
---|
| 806 | ] |
---|
| 807 | ] |
---|
| 808 | ] |
---|
[2059] | 809 | ]*) |
---|
[2034] | 810 | ] |
---|
[2059] | 811 | | (* policy_jump_equal → added = 0 *) |
---|
| 812 | cases daemon |
---|
| 813 | ] |
---|
| 814 | | normalize nodelta @conj [ @conj [ @conj [ @conj [ @conj [ @conj [ @conj [ @conj |
---|
[2034] | 815 | [ #i cases i |
---|
| 816 | [ #Hi2 @conj |
---|
| 817 | [ #Hi @⊥ @(absurd ? Hi) @le_to_not_lt / by / |
---|
| 818 | | >lookup_opt_insert_hit #H destruct (H) |
---|
| 819 | ] |
---|
| 820 | | -i #i #Hi2 @conj |
---|
| 821 | [ #Hi >lookup_opt_insert_miss |
---|
| 822 | [ / by refl/ |
---|
| 823 | | @bitvector_of_nat_abs |
---|
| 824 | [ @Hi2 |
---|
| 825 | | / by / |
---|
| 826 | | @sym_neq @lt_to_not_eq / by / |
---|
| 827 | ] |
---|
| 828 | ] |
---|
| 829 | | #_ @le_S_S @le_O_n |
---|
| 830 | ] |
---|
| 831 | ] |
---|
| 832 | | #i cases i |
---|
| 833 | [ #Hi @⊥ @(absurd … Hi) @not_le_Sn_O |
---|
| 834 | | -i #i #Hi #Hj @⊥ @(absurd … Hi) @not_le_Sn_O |
---|
| 835 | ] |
---|
| 836 | ] |
---|
[2059] | 837 | | #i #Hi <(le_n_O_to_eq … Hi) |
---|
| 838 | >lookup_insert_hit cases (bvt_lookup … (bitvector_of_nat ? 0) (\snd old_sigma) 〈0,short_jump〉) |
---|
| 839 | #a #b normalize nodelta %2 @refl |
---|
[2034] | 840 | ] |
---|
| 841 | | #i cases i |
---|
| 842 | [ #Hi @⊥ @(absurd … Hi) @not_le_Sn_O |
---|
| 843 | | -i #i #Hi @⊥ @(absurd … Hi) @not_le_Sn_O |
---|
| 844 | ] |
---|
| 845 | ] |
---|
| 846 | | #i cases i |
---|
| 847 | [ #Hi @⊥ @(absurd … Hi) @not_le_Sn_O |
---|
| 848 | | -i #i #Hi @⊥ @(absurd … Hi) @not_le_Sn_O |
---|
| 849 | ] |
---|
| 850 | ] |
---|
| 851 | | >lookup_insert_hit @refl |
---|
| 852 | ] |
---|
| 853 | | >lookup_insert_hit @refl |
---|
| 854 | ] |
---|
[2059] | 855 | | #_ #i #Hi <(le_n_O_to_eq … Hi) >lookup_insert_hit |
---|
| 856 | (* USE: 0 ↦ 0 (from old_sigma *) |
---|
| 857 | @(proj2 ?? (proj1 ?? (proj1 ?? (pi2 ?? old_sigma)))) |
---|
[2034] | 858 | ] |
---|
[2059] | 859 | | #_ @refl |
---|
| 860 | ] |
---|
[2034] | 861 | ] |
---|
[2038] | 862 | qed. |
---|