[2286] | 1 | include "joint/TranslateUtils.ma". |
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[2200] | 2 | include "utilities/hide.ma". |
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[2182] | 3 | |
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| 4 | definition graph_to_lin_statement : |
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| 5 | ∀p : unserialized_params.∀globals. |
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| 6 | joint_statement (mk_graph_params p) globals → joint_statement (mk_lin_params p) globals ≝ |
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| 7 | λp,globals,stmt.match stmt return λ_.joint_statement (mk_lin_params ?) ? with |
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| 8 | [ sequential c _ ⇒ sequential … c it |
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| 9 | | final c ⇒ final … c |
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| 10 | ]. |
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| 11 | |
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| 12 | lemma graph_to_lin_labels : |
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| 13 | ∀p : unserialized_params.∀globals,s. |
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| 14 | stmt_labels … (graph_to_lin_statement p globals s) = |
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| 15 | stmt_explicit_labels … s. |
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| 16 | #p#globals * [//] * // |
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| 17 | qed. |
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| 18 | |
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| 19 | (* discard all elements passing test, return first element failing it *) |
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| 20 | (* and the rest of the list, if any. *) |
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| 21 | let rec chop A (test : A → bool) (l : list A) on l : option (A × (list A)) ≝ |
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| 22 | match l with |
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| 23 | [ nil ⇒ None ? |
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| 24 | | cons x l' ⇒ if test x then chop A test l' else return 〈x, l'〉 |
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| 25 | ]. |
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| 26 | |
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| 27 | lemma chop_ok : ∀A,f,l. |
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| 28 | match chop A f l with |
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| 29 | [ Some pr ⇒ |
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| 30 | let x ≝ \fst pr in |
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| 31 | let post ≝ \snd pr in |
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| 32 | ∃pre.All ? (λx.bool_to_Prop (f x)) pre ∧ |
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| 33 | l = pre @ x :: post ∧ ¬bool_to_Prop (f x) |
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| 34 | | None ⇒ All A (λx.bool_to_Prop (f x)) l |
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| 35 | ]. |
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| 36 | #A #f #l elim l |
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| 37 | [ % |
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| 38 | | #hd #tl #IH whd in match (chop ???); |
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| 39 | elim (true_or_false_Prop (f hd)) #H >H normalize nodelta |
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| 40 | [ lapply IH elim (chop ???) normalize nodelta |
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| 41 | [ #G %{H G} |
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| 42 | | #pr * #pre ** #H1 #H2 #H3 %{(hd :: pre)} %{H3} % |
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| 43 | [ %{H H1} |
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| 44 | | >H2 % |
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| 45 | ] |
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| 46 | ] |
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| 47 | | %{[ ]} %{H} %% |
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| 48 | ] |
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| 49 | ] |
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| 50 | qed. |
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| 51 | |
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| 52 | unification hint 0 ≔ p, globals; |
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| 53 | lp ≟ lin_params_to_params p, |
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| 54 | sp ≟ stmt_pars lp, |
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| 55 | js ≟ joint_statement sp globals, |
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| 56 | lo ≟ labelled_obj LabelTag js |
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| 57 | (*----------------------------*)⊢ |
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| 58 | list lo ≡ codeT lp globals. |
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| 59 | |
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| 60 | |
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| 61 | definition graph_visit_ret_type ≝ λp,globals.λg : codeT (mk_graph_params p) globals. |
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| 62 | λentry : label. |
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| 63 | (Σ〈visited' : identifier_map LabelTag ℕ, |
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| 64 | required' : identifier_set LabelTag, |
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| 65 | generated' : codeT (mk_lin_params p) globals〉.'hide ( |
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| 66 | And (And (And (And |
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| 67 | (lookup ?? visited' entry = Some ? 0) |
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| 68 | (required' ⊆ visited')) |
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| 69 | (∃last.stmt_at … generated' 0 = Some ? (final … last))) |
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| 70 | (code_forall_labels … (λl.bool_to_Prop (l∈required')) (rev … generated'))) |
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| 71 | (∀l,n.lookup ?? visited' l = Some ? n → |
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| 72 | And (bool_to_Prop (code_has_label … (rev ? generated') l)) |
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| 73 | (∃s.And (And |
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| 74 | (lookup … g l = Some ? s) |
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| 75 | (nth_opt … n (rev … generated') = Some ? 〈Some ? l, graph_to_lin_statement … s〉)) |
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| 76 | (opt_All ? |
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| 77 | (λnext.Or (lookup … visited' next = Some ? (S n)) |
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| 78 | (nth_opt … (S n) (rev … generated') = Some ? 〈None ?, GOTO … next〉)) |
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| 79 | (stmt_implicit_label … s)))))). |
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| 80 | |
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| 81 | unification hint 0 ≔ tag ⊢ identifier_set tag ≡ identifier_map tag unit. |
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| 82 | |
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| 83 | include alias "common/Identifiers.ma". |
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| 84 | |
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| 85 | lemma lookup_safe_elim : ∀tag,A.∀P : A → Prop.∀m,i,prf. |
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| 86 | (∀x.lookup tag A m i = Some ? x → P x) → P (lookup_safe tag A m i prf). |
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| 87 | #tag #A #P #m #i #prf #H @H @lookup_eq_safe qed. |
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| 88 | |
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| 89 | let rec graph_visit |
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| 90 | (p : unserialized_params) |
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| 91 | (globals: list ident) |
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| 92 | (g : codeT (mk_graph_params p) globals) |
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| 93 | (* = graph (joint_statement (mk_graph_params p) globals *) |
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| 94 | (required: identifier_set LabelTag) |
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| 95 | (visited: identifier_map LabelTag ℕ) (* the reversed index of labels in the new code *) |
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| 96 | (generated: codeT (mk_lin_params p) globals) |
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| 97 | (* ≝ list ((option label)×(joint_statement (mk_lin_params p) globals)) *) |
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| 98 | (visiting: list label) |
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| 99 | (gen_length : ℕ) |
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| 100 | (n: nat) |
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| 101 | (entry : label) |
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| 102 | (g_prf : code_closed … g) |
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| 103 | (required_prf1 : ∀i.i∈required → Or (In ? visiting i) (bool_to_Prop (i∈visited))) |
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| 104 | (required_prf2 : code_forall_labels … (λl.bool_to_Prop (l ∈ required)) (rev … generated)) |
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| 105 | (generated_prf1 : ∀l,n.lookup … visited l = Some ? n → hide_Prop ( |
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| 106 | And (bool_to_Prop (code_has_label … (rev ? generated) l)) |
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| 107 | (∃s.And (And |
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| 108 | (lookup … g l = Some ? s) |
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| 109 | (nth_opt ? n (rev … generated) = Some ? 〈Some ? l, graph_to_lin_statement … s〉)) |
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| 110 | (opt_All ? (λnext.match lookup … visited next with |
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| 111 | [ Some n' ⇒ Or (n' = S n) (nth_opt ? (S n) (rev ? generated) = Some ? 〈None ?, GOTO … next〉) |
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| 112 | | None ⇒ And (nth_opt ? 0 visiting = Some ? next) (S n = gen_length) ]) (stmt_implicit_label … s))))) |
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| 113 | (generated_prf2 : ∀l.lookup … visited l = None ? → does_not_occur … l (rev ? generated)) |
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| 114 | (generated_prf3 : (∃last.stmt_at … generated 0 = Some ? (final … last)) ∨ |
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| 115 | (¬All ? (λx.bool_to_Prop (x∈visited)) visiting)) |
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| 116 | (visiting_prf : All … (λl.bool_to_Prop (l∈g)) visiting) |
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| 117 | (gen_length_prf : gen_length = length ? generated) |
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| 118 | (entry_prf : Or (And (And (visiting = [entry]) (gen_length = 0)) (Not (bool_to_Prop (entry∈visited)))) |
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| 119 | (lookup … visited entry = Some ? 0)) |
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| 120 | (n_prf : le (id_map_size … g) (plus n (id_map_size … visited))) |
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| 121 | on n |
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| 122 | : graph_visit_ret_type … g entry ≝ |
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| 123 | match chop ? (λx.x∈visited) visiting |
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| 124 | return λx.? → graph_visit_ret_type … g entry with |
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| 125 | [ None ⇒ λH. |
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| 126 | «〈visited, required, generated〉, ?» |
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| 127 | | Some pr ⇒ λH. |
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| 128 | let vis_hd ≝ \fst pr in |
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| 129 | let vis_tl ≝ \snd pr in |
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| 130 | match n return λx.? → graph_visit_ret_type … g entry with |
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| 131 | [ O ⇒ λn_prf'.⊥ |
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| 132 | | S n' ⇒ λn_prf'. |
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| 133 | (* add the label to the visited ones *) |
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| 134 | let visited' ≝ add … visited vis_hd gen_length in |
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| 135 | (* take l's statement *) |
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| 136 | let hd_vis_in_g ≝ (hide_prf ? ?) in |
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| 137 | let statement ≝ lookup_safe ?? g vis_hd hd_vis_in_g in |
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| 138 | (* translate it to its linear version *) |
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| 139 | let translated_statement ≝ graph_to_lin_statement p globals statement in |
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| 140 | (* add the translated statement to the code (which will be reversed later) *) |
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| 141 | let generated' ≝ 〈Some … vis_hd, translated_statement〉 :: generated in |
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| 142 | let required' ≝ union_set ??? (set_from_list … (stmt_explicit_labels … statement)) required in |
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| 143 | (* put successors in the visiting worklist *) |
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| 144 | let visiting' ≝ stmt_labels … statement @ vis_tl in |
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| 145 | (* finally, check the implicit successor *) |
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| 146 | (* if it has been already visited, we need to add a GOTO *) |
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| 147 | let add_req_gen ≝ |
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| 148 | match stmt_implicit_label … statement |
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| 149 | with |
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| 150 | [ Some next ⇒ |
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| 151 | if next ∈ visited' then |
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| 152 | 〈1, {(next)}, [〈None label, (GOTO … next : joint_statement ??)〉]〉 |
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| 153 | else 〈0, ∅, [ ]〉 |
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| 154 | | None ⇒ 〈0, ∅, [ ]〉 |
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| 155 | ] in |
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| 156 | (* prepare a common utility to deal with add_req_gen *) |
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| 157 | let add_req_gen_prf : |
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| 158 | ∀P : (ℕ × (identifier_set LabelTag) × (codeT (mk_lin_params p) globals)) → Prop. |
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| 159 | (opt_All ? (λnext.¬bool_to_Prop (next ∈ visited')) (stmt_implicit_label … statement) → |
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| 160 | P 〈0,∅,[ ]〉) → |
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| 161 | (∀next.stmt_implicit_label … statement = Some ? next → |
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| 162 | next ∈ visited' → |
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| 163 | P 〈1, {(next)}, [〈None ?, GOTO (mk_lin_params p) next〉]〉) → |
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| 164 | P add_req_gen ≝ hide_prf ?? in |
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| 165 | graph_visit ??? |
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| 166 | (union_set ??? (\snd (\fst add_req_gen)) required') |
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| 167 | visited' |
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| 168 | (\snd add_req_gen @ generated') |
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| 169 | visiting' |
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| 170 | (plus (\fst (\fst add_req_gen)) (S gen_length)) |
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| 171 | n' entry g_prf ????????? |
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| 172 | ] n_prf |
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| 173 | ] (chop_ok ? (λx.x∈visited) visiting). |
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| 174 | whd |
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| 175 | [ (* base case, visiting is all visited *) |
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| 176 | %[%[%[%]]] |
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| 177 | [ elim entry_prf |
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| 178 | [ ** #eq_visiting #gen_length_O #entry_vis >eq_visiting in H; * >entry_vis * |
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| 179 | | // |
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| 180 | ] |
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| 181 | | #l #l_req |
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| 182 | elim (required_prf1 … l_req) #G |
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| 183 | [ @(All_In … H G) |
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| 184 | | assumption |
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| 185 | ] |
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| 186 | | cases generated_prf3 [//] |
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| 187 | * #ABS @⊥ @ABS assumption |
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| 188 | | assumption |
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| 189 | | #l #n #H elim (generated_prf1 … H) |
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| 190 | #H1 * #s ** #H2 #H3 #H4 |
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| 191 | % [assumption] %{s} % |
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| 192 | [% assumption |
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| 193 | | @(opt_All_mp … H4) -l #l |
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| 194 | lapply (in_map_domain … visited l) |
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| 195 | elim (true_or_false_Prop (l∈visited)) #l_vis >l_vis |
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| 196 | normalize nodelta [ * #n' ] #EQlookup >EQlookup |
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| 197 | normalize nodelta * |
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| 198 | [ #EQn' % >EQn' % |
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| 199 | | #H %2{H} |
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| 200 | | #H' lapply (All_nth … H … H') >l_vis * |
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| 201 | ] |
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| 202 | ] |
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| 203 | ] |
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| 204 | (* first, close goals where add_gen_req plays no role *) |
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| 205 | |13: (* vis_hd is in g *) |
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| 206 | elim H #pre ** #_ #H2 #_ |
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| 207 | @(All_split … visiting_prf … H2) |
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| 208 | |2: (* n = 0 absrud *) |
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| 209 | elim H #pre ** #_ #H2 #H3 |
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| 210 | @(absurd … n_prf') |
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| 211 | @lt_to_not_le |
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| 212 | lapply (add_size … visited vis_hd 0 (* dummy value *)) |
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| 213 | >H3 normalize nodelta |
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| 214 | whd in match (lt ? ?); |
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| 215 | whd in match (1 + ?); |
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| 216 | #EQ <EQ @subset_card @add_subset |
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| 217 | [ @(All_split ? (λx.bool_to_Prop (x∈g)) ????? H2) @(All_mp … visiting_prf) |
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| 218 | #a elim g #gm whd in ⊢ (?→?%); #H >(lookup_eq_safe … H) % |
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| 219 | | #l #H |
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| 220 | elim (generated_prf1 … (lookup_eq_safe … H)) #_ * #s ** #s_in_g #_ #_ |
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| 221 | whd in ⊢ (?%); >s_in_g % |
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| 222 | ] |
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| 223 | |8: |
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| 224 | elim H #pre ** #_ #H2 #_ |
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| 225 | @All_append |
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| 226 | [ elim(g_prf … vis_hd statement ?) [2:@lookup_eq_safe] #G1 #G2 |
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| 227 | @(All_append … G1) |
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| 228 | whd in G2; lapply G2 elim statement normalize nodelta #s [2: #_ %] |
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| 229 | #l #G' %{G'} % |
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| 230 | | >H2 in visiting_prf; |
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| 231 | #H' lapply (All_append_r … H') -H' * #_ // |
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| 232 | ] |
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| 233 | |10: |
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| 234 | elim H #pre ** #_ #H2 #H3 -add_req_gen_prf |
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| 235 | %2 elim entry_prf |
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| 236 | [ ** >H2 cases pre |
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| 237 | [2: #x' #pre' #ABS normalize in ABS; destruct(ABS) |
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| 238 | cases pre' in e0; [2: #x'' #pre''] #ABS' normalize in ABS'; destruct(ABS') |
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| 239 | |1: #EQ normalize in EQ; destruct(EQ) |
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| 240 | #eq_gen_length #_ |
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| 241 | >lookup_add_hit >eq_gen_length % |
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| 242 | ] |
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| 243 | | #lookup_entry cut (entry ≠ vis_hd) |
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| 244 | [ % whd in match vis_hd; #entry_vis_hd <entry_vis_hd in H3; |
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| 245 | whd in ⊢ (?(?%)→?); >lookup_entry * #ABS @ABS % ] |
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| 246 | #entry_not_vis_hd >(lookup_add_miss ?????? entry_not_vis_hd) assumption |
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| 247 | ] |
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| 248 | |11: |
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| 249 | elim H #pre ** #_ #_ #H3 |
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| 250 | >commutative_plus |
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| 251 | >add_size >H3 normalize nodelta |
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| 252 | whd in match (1 + ?); |
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| 253 | >commutative_plus |
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| 254 | assumption |
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| 255 | |12: (* add_req_gen utility *) |
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| 256 | #P whd in match add_req_gen; |
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| 257 | elim (stmt_implicit_label ???) |
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| 258 | [ #H #_ @H % ] |
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| 259 | #next normalize nodelta elim (true_or_false_Prop (next ∈ visited')) #next_vis |
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| 260 | >next_vis normalize nodelta |
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| 261 | [ #_ #H @H [% | assumption] |
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| 262 | | #H #_ @H whd >next_vis % * |
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| 263 | ] |
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| 264 | | elim H #pre ** #H1 #H2 #_ |
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| 265 | #i >mem_set_union |
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| 266 | #H elim (orb_Prop_true … H) -H |
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| 267 | [ @add_req_gen_prf [ #_ >mem_set_empty * ] |
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| 268 | #next #_ #next_vis #H >(mem_set_singl_to_eq … H) %2 assumption |
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| 269 | | >mem_set_union |
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| 270 | #H elim (orb_Prop_true … H) -H |
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| 271 | [ #i_expl %1 @Exists_append_l @Exists_append_r |
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| 272 | @mem_list_as_set |
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| 273 | @i_expl |
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| 274 | | (* i was already required *) |
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| 275 | #i_req |
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| 276 | elim (required_prf1 … i_req) |
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| 277 | [ >H2 #H elim (Exists_append … H) -H |
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| 278 | [ (* i in preamble → i∈visited *) |
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| 279 | #i_pre %2 >mem_set_add @orb_Prop_r |
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| 280 | lapply (All_In … H1 i_pre) #H @H |
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| 281 | | * |
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| 282 | [ (* i is vis_hd *) |
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| 283 | #eq_i >eq_i %2 @mem_set_add_id |
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| 284 | | (* i in vis_tl → i∈visiting' *) |
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| 285 | #i_post % @Exists_append_r assumption |
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| 286 | ] |
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| 287 | ] |
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| 288 | | (* i in visited *) |
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| 289 | #i_vis %2 >mem_set_add @orb_Prop_r assumption |
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| 290 | ] |
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| 291 | ] |
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| 292 | ] |
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| 293 | |4,5,6: change with reverse in match rev; |
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| 294 | >reverse_append whd in match (reverse ??); >rev_append_def |
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| 295 | >associative_append |
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| 296 | [ #pt #s |
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| 297 | @(leb_elim (S pt) (|generated|)) #cmp |
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| 298 | whd in match (stmt_at ????); |
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| 299 | [ >nth_opt_append_l [2: >length_reverse assumption ] |
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| 300 | change with (stmt_at ???? = ? → ?) |
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| 301 | #EQ lapply(required_prf2 … EQ) @All_mp |
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| 302 | #l #l_req >mem_set_union @orb_Prop_r |
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| 303 | >mem_set_union @orb_Prop_r @l_req |
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| 304 | | >nth_opt_append_r [2: >length_reverse @not_lt_to_le assumption ] |
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| 305 | cases (pt - ?) |
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| 306 | [ whd in match (nth_opt ???); whd in ⊢ (??%?→?); |
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| 307 | #EQ destruct(EQ) whd |
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| 308 | >graph_to_lin_labels in ⊢ (???%); |
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| 309 | whd in match required'; |
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| 310 | generalize in match (stmt_explicit_labels … statement); |
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| 311 | #l @list_as_set_All |
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| 312 | #i >mem_set_union >mem_set_union |
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| 313 | #i_l @orb_Prop_r @orb_Prop_l @i_l |
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| 314 | | @add_req_gen_prf |
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| 315 | [ #_ | #next #_ #next_vis * |
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| 316 | [ whd in ⊢ (??%?→?); |
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| 317 | #EQ' destruct(EQ') whd %{I} >mem_set_union |
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| 318 | @orb_Prop_l @mem_set_add_id ]] |
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| 319 | #n whd in ⊢ (??%?→?); #ABS destruct(ABS) |
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| 320 | ] |
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| 321 | ] |
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| 322 | | elim H #pre ** #H1 #H2 #H3 |
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| 323 | #i whd in match visited'; |
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| 324 | @(eq_identifier_elim … i vis_hd) |
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| 325 | [ #EQi >EQi -i #pos |
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| 326 | >lookup_add_hit #EQpos (* too slow: destruct(EQpos) *) |
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| 327 | cut (gen_length = pos) |
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| 328 | [1,3,5: (* BUG: -graph_visit *) -visited destruct(EQpos) %] |
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| 329 | -EQpos #EQpos <EQpos -pos |
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| 330 | % |
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| 331 | [ >lin_code_has_label |
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| 332 | @add_req_gen_prf |
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| 333 | [ #_ |
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| 334 | | #next #_ #next_vis |
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| 335 | change with (? @ ([?] @ [?])) in match (? @ [? ; ?]); |
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| 336 | <associative_append >occurs_exactly_once_None |
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| 337 | ] |
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| 338 | >occurs_exactly_once_Some_eq >eq_identifier_refl |
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| 339 | normalize nodelta |
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| 340 | @generated_prf2 |
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| 341 | lapply (in_map_domain … visited vis_hd) |
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| 342 | >H3 normalize nodelta // |
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| 343 | | %{statement} |
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| 344 | % [ % ] |
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| 345 | [ @lookup_eq_safe |
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| 346 | | <associative_append @nth_opt_append_hit_l |
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| 347 | >nth_opt_append_r |
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| 348 | >rev_length |
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| 349 | <gen_length_prf |
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| 350 | [<minus_n_n] % |
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| 351 | | @add_req_gen_prf |
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| 352 | [ lapply (refl … (stmt_implicit_label ?? statement)) |
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| 353 | cases (stmt_implicit_label ???) in ⊢ (???%→%); |
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| 354 | [#_ #_ %] |
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| 355 | #next #K change with (¬bool_to_Prop (?∈?)→?) #next_vis |
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| 356 | | #next #K >K #next_vis |
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| 357 | ] whd |
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| 358 | >mem_set_add in next_vis; |
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| 359 | @eq_identifier_elim |
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| 360 | [1,3: #EQnext >EQnext * [#ABS elim(ABS I)] |
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| 361 | >lookup_add_hit |
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| 362 | |*: #NEQ >(lookup_add_miss … visited … NEQ) |
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| 363 | change with (?∈?) in match (false∨?); |
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| 364 | #next_vis lapply(in_map_domain … visited next) >next_vis |
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| 365 | whd in ⊢ (% → ?); [2: * #s ] |
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| 366 | #EQlookup >EQlookup |
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| 367 | ] whd |
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| 368 | [1,2: %2 <associative_append |
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| 369 | >nth_opt_append_r >append_length >rev_length >commutative_plus |
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| 370 | <gen_length_prf |
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| 371 | [1,3: <minus_n_n ] % |
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| 372 | | % [2: %] @nth_opt_append_hit_l whd in match (stmt_labels … statement); |
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| 373 | >K % |
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| 374 | ] |
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| 375 | ] |
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| 376 | ] |
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| 377 | | #NEQ #n_i >(lookup_add_miss … visited … NEQ) |
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| 378 | #Hlookup elim (generated_prf1 … Hlookup) |
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| 379 | #G1 * #s ** #G2 #G3 #G4 |
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| 380 | % |
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| 381 | [ >lin_code_has_label <associative_append |
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| 382 | >occurs_exactly_once_append |
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| 383 | @orb_Prop_l @andb_Prop |
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| 384 | [ >occurs_exactly_once_Some_eq |
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| 385 | >eq_identifier_false [2: % #ABS >ABS in NEQ; * #ABS' @ABS' % ] |
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| 386 | normalize nodelta >lin_code_has_label in G1; #K @K |
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| 387 | | @add_req_gen_prf |
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| 388 | [ #_ % | #next #_ #_ % ] |
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| 389 | ] |
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| 390 | | %{s} |
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| 391 | % [ % ] |
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| 392 | [ assumption |
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| 393 | | @nth_opt_append_hit_l assumption |
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| 394 | | @(opt_All_mp … G4) |
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| 395 | #x |
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| 396 | @(eq_identifier_elim … x vis_hd) #Heq |
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| 397 | [ >Heq |
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| 398 | lapply (in_map_domain … visited vis_hd) |
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| 399 | >H3 normalize nodelta in ⊢ (%→?); |
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| 400 | #EQlookup >EQlookup * #nth_opt_visiting #gen_length_eq |
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| 401 | >lookup_add_hit %1 >gen_length_eq % |
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| 402 | | >(lookup_add_miss ?????? Heq) |
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| 403 | lapply (in_map_domain … visited x) |
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| 404 | elim (true_or_false_Prop (x∈visited)) #x_vis >x_vis |
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| 405 | normalize nodelta in ⊢ (%→?); [ * #n' ] |
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| 406 | #EQlookupx >EQlookupx whd in ⊢ (%→%); * |
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| 407 | [ #G %1{G} |
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| 408 | | #G %2 @nth_opt_append_hit_l |
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| 409 | assumption |
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| 410 | | #G elim(absurd ?? Heq) |
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| 411 | (* BUG (but useless): -required -g -generated *) |
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| 412 | >H2 in G; lapply H1 cases pre |
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| 413 | [ #_ |
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| 414 | | #hd #tl * #hd_vis #_ |
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| 415 | ] normalize #EQ' destruct(EQ') |
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| 416 | [ % |
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| 417 | | >x_vis in hd_vis; * |
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| 418 | ] |
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| 419 | ] |
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| 420 | ] |
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| 421 | ] |
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| 422 | ] |
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| 423 | ] |
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| 424 | | #i whd in match visited'; |
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| 425 | @(eq_identifier_elim … i vis_hd) #Heq |
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| 426 | [ >Heq >lookup_add_hit #ABS destruct(ABS) ] |
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| 427 | >(lookup_add_miss ?????? Heq) |
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| 428 | #i_n_vis |
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| 429 | >does_not_occur_append @andb_Prop |
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| 430 | [ @generated_prf2 assumption |
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| 431 | | change with (bool_to_Prop (¬eq_identifier ??? ∧ ?)) |
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| 432 | >eq_identifier_false [2: % #ABS <ABS in Heq; * #ABS' @ABS' % ] |
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| 433 | @add_req_gen_prf [ #_ | #next #_ #_ ] % |
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| 434 | ] |
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| 435 | ] |
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| 436 | | @add_req_gen_prf |
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| 437 | [ #K | #next #K #next_vis %1 %{(GOTO … next)} % ] |
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| 438 | whd in match generated'; whd in match translated_statement; |
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| 439 | lapply K whd in match visiting'; |
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| 440 | whd in match (stmt_labels ???); cases statement #last |
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| 441 | [2: #_ %1 %{last} % ] #next whd in ⊢ (%→?); #next_vis |
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| 442 | %2 % * >next_vis * |
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| 443 | | whd in match generated'; |
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| 444 | @add_req_gen_prf [ #_ | #next #_ #_ ] normalize >gen_length_prf % |
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| 445 | ] |
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| 446 | qed. |
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| 447 | |
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| 448 | (* CSC: The branch compression (aka tunneling) optimization is not implemented |
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| 449 | in Matita *) |
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| 450 | definition branch_compress |
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| 451 | ≝ λp: graph_params.λglobals.λg:codeT p globals. |
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| 452 | λentry : Σl.bool_to_Prop (code_has_label … g l).g. |
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| 453 | |
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| 454 | lemma branch_compress_closed : ∀p,globals,g,l.code_closed ?? g → |
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| 455 | code_closed … (branch_compress p globals g l). |
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| 456 | #p#globals#g#l#H @H qed. |
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| 457 | |
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| 458 | lemma branch_compress_has_entry : ∀p,globals,g,l. |
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| 459 | code_has_label … (branch_compress p globals g l) l. |
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| 460 | #p#globals#g*#l#l_prf @l_prf qed. |
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| 461 | |
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| 462 | definition filter_labels ≝ λtag,A.λtest : identifier tag → bool.λc : list (labelled_obj tag A). |
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| 463 | map ?? |
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| 464 | (λs. let 〈l_opt,x〉 ≝ s in |
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| 465 | 〈! l ← l_opt ; if test l then return l else None ?, x〉) c. |
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| 466 | |
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| 467 | lemma does_not_occur_filter_labels : |
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| 468 | ∀tag,A,test,id,c. |
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| 469 | does_not_occur ?? id (filter_labels tag A test c) = |
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| 470 | (does_not_occur ?? id c ∨ ¬ test id). |
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| 471 | #tag #A #test #id #c elim c |
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| 472 | [ // |
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| 473 | | ** [2: #lbl] #s #tl #IH |
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| 474 | whd in match (filter_labels ????); normalize nodelta |
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| 475 | whd in match (does_not_occur ????) in ⊢ (??%%); |
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| 476 | [2: @IH] |
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| 477 | normalize in match (! l ← ? ; ?); >IH |
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| 478 | @(eq_identifier_elim ?? lbl id) #Heq [<Heq] |
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| 479 | elim (test lbl) normalize nodelta |
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| 480 | change with (eq_identifier ???) in match (instruction_matches_identifier ????); |
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| 481 | [1,2: >eq_identifier_refl [2: >commutative_orb] normalize % |
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| 482 | |*: >(eq_identifier_false ??? Heq) normalize nodelta % |
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| 483 | ] |
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| 484 | ] |
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| 485 | qed. |
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| 486 | |
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| 487 | lemma occurs_exactly_once_filter_labels : |
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| 488 | ∀tag,A,test,id,c. |
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| 489 | occurs_exactly_once ?? id (filter_labels tag A test c) = |
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| 490 | (occurs_exactly_once ?? id c ∧ test id). |
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| 491 | #tag #A #test #id #c elim c |
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| 492 | [ // |
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| 493 | | ** [2: #lbl] #s #tl #IH |
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| 494 | whd in match (filter_labels ????); normalize nodelta |
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| 495 | whd in match (occurs_exactly_once ????) in ⊢ (??%%); |
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| 496 | [2: @IH] |
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| 497 | normalize in match (! l ← ? ; ?); >IH |
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| 498 | >does_not_occur_filter_labels |
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| 499 | @(eq_identifier_elim ?? lbl id) #Heq [<Heq] |
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| 500 | elim (test lbl) normalize nodelta |
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| 501 | change with (eq_identifier ???) in match (instruction_matches_identifier ????); |
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| 502 | [1,2: >eq_identifier_refl >commutative_andb [ >(commutative_andb ? true) >commutative_orb | >(commutative_andb ? false)] normalize % |
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| 503 | |*: >(eq_identifier_false ??? Heq) normalize nodelta % |
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| 504 | ] |
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| 505 | ] |
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| 506 | qed. |
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| 507 | |
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| 508 | lemma nth_opt_filter_labels : ∀tag,A,test,instrs,n. |
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| 509 | nth_opt ? n (filter_labels tag A test instrs) = |
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| 510 | ! 〈lopt, s〉 ← nth_opt ? n instrs ; |
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| 511 | return 〈 ! lbl ← lopt; if test lbl then return lbl else None ?, s〉. |
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| 512 | #tag #A #test #instrs elim instrs |
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| 513 | [ * [2: #n'] % |
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| 514 | | * #lopt #s #tl #IH * [2: #n'] |
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| 515 | whd in match (filter_labels ????); normalize nodelta |
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| 516 | whd in match (nth_opt ???) in ⊢ (??%%); [>IH] % |
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| 517 | ] |
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| 518 | qed. |
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| 519 | |
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| 520 | lemma stmt_at_filter_labels : ∀p : lin_params.∀globals,test.∀c : codeT p globals. |
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| 521 | ∀i.stmt_at p globals (filter_labels ?? test c) i = stmt_at p globals c i. |
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| 522 | #p#globals#test #c#i |
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| 523 | whd in ⊢ (??%%); >nth_opt_filter_labels |
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| 524 | elim (nth_opt ???); // |
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| 525 | qed. |
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| 526 | |
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| 527 | lemma option_bind_inverse : ∀A,B.∀m : option A.∀f : A → option B.∀r. |
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| 528 | ! x ← m ; f x = return r → |
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| 529 | ∃x.m = return x ∧ f x = return r. |
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| 530 | #A #B * normalize [2:#x] #f #r #EQ destruct |
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| 531 | %{x} %{EQ} % |
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| 532 | qed. |
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| 533 | |
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| 534 | lemma nth_opt_reverse_hit : |
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| 535 | ∀A,l,n.n < |l| → nth_opt A n (reverse ? l) = nth_opt A (|l| - (S n)) l. |
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| 536 | #A #l elim l |
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| 537 | [ #n #ABS normalize in ABS; @⊥ -A /2 by absurd/ |
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| 538 | | #hd #tl #IH #n #lim whd in match (reverse ??); >rev_append_def |
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| 539 | @(leb_elim (S n) (|tl|)) #H |
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| 540 | [ >nth_opt_append_l [2: >length_reverse @H ] |
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| 541 | >(IH … H) >(minus_Sn_m … H) % |
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| 542 | | >(le_to_le_to_eq … (le_S_S_to_le … lim) (not_lt_to_le … H)) |
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| 543 | >nth_opt_append_r >length_reverse [2: % ] |
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| 544 | <minus_n_n <minus_n_n % |
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| 545 | ] |
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| 546 | ] |
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| 547 | qed. |
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| 548 | |
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| 549 | lemma nth_opt_reverse_hit_inv : |
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| 550 | ∀A,l,n.n < |l| → nth_opt A (|l| - (S n)) (reverse ? l) = nth_opt A n l. |
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| 551 | #A #l #n #H <(reverse_reverse ? l) in ⊢ (???%); @sym_eq |
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| 552 | <length_reverse @nth_opt_reverse_hit >length_reverse @H |
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| 553 | qed. |
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| 554 | |
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| 555 | definition linearise_code: |
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| 556 | ∀p : unserialized_params.∀globals. |
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| 557 | ∀g : codeT (mk_graph_params p) globals.code_closed … g → |
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| 558 | ∀entry : (Σl.bool_to_Prop (code_has_label … g l)). |
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| 559 | (Σc : codeT (mk_lin_params p) globals. |
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| 560 | code_closed … c ∧ |
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| 561 | ∃ sigma : identifier_map LabelTag ℕ. |
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| 562 | lookup … sigma entry = Some ? 0 ∧ |
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| 563 | ∀l,n.lookup … sigma l = Some ? n → |
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| 564 | ∃s. lookup … g l = Some ? s ∧ |
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| 565 | opt_Exists ? |
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| 566 | (λls.let 〈lopt, ts〉 ≝ ls in |
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| 567 | opt_All ? (eq ? l) lopt ∧ |
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| 568 | ts = graph_to_lin_statement … s ∧ |
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| 569 | opt_All ? |
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| 570 | (λnext.Or (lookup … sigma next = Some ? (S n)) |
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| 571 | (nth_opt … (S n) c = Some ? 〈None ?, GOTO … next〉)) |
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| 572 | (stmt_implicit_label … s)) (nth_opt … n c)) |
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| 573 | ≝ |
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| 574 | λp,globals,g,g_prf,entry_sig. |
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| 575 | let g ≝ branch_compress (mk_graph_params p) ? g entry_sig in |
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| 576 | let g_prf ≝ branch_compress_closed … g entry_sig g_prf in |
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| 577 | match graph_visit p globals g ∅ (empty_map …) [ ] [entry_sig] 0 (|g|) |
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| 578 | (entry_sig) g_prf ????????? |
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| 579 | with |
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| 580 | [ mk_Sig triple H ⇒ |
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| 581 | let sigma ≝ \fst (\fst triple) in |
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| 582 | let required ≝ \snd (\fst triple) in |
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| 583 | let crev ≝ \snd triple in |
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| 584 | let lbld_code ≝ rev ? crev in |
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| 585 | mk_Sig ?? (filter_labels … (λl.l∈required) lbld_code) ? ]. |
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| 586 | [ (* done later *) |
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| 587 | | #i >mem_set_empty * |
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| 588 | |3,4: #a #b whd in ⊢ (??%?→?); #ABS destruct(ABS) |
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| 589 | | #l #_ % |
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| 590 | | %2 % * >mem_set_empty * |
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| 591 | | % [2: %] @(branch_compress_has_entry … g entry_sig) |
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| 592 | | % |
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| 593 | | % % [% %] cases (pi1 … entry_sig) normalize #_ % // |
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| 594 | | >commutative_plus change with (? ≤ |g|) % |
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| 595 | ] |
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| 596 | lapply (refl … triple) |
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| 597 | generalize in match triple in ⊢ (???%→%); ** |
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| 598 | #visited #required #generated #EQtriple |
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| 599 | >EQtriple in H ⊢ %; normalize nodelta **** |
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| 600 | #entry_O #req_vis #last_fin #labels_in_req #sigma_prop |
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| 601 | % >EQtriple |
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| 602 | [ #i #s |
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| 603 | >stmt_at_filter_labels #EQ |
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| 604 | % |
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| 605 | [ @stmt_forall_labels_explicit |
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| 606 | @(All_mp … (labels_in_req … EQ)) |
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| 607 | #l #l_req >lin_code_has_label |
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| 608 | >occurs_exactly_once_filter_labels >l_req |
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| 609 | >commutative_andb whd in ⊢ (?%); |
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| 610 | elim (sigma_prop ?? (lookup_eq_safe … (req_vis … l_req))) |
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| 611 | >lin_code_has_label #H #_ @H |
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| 612 | | lapply EQ cases s #s' [2: #_ % ] |
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| 613 | * -EQ #EQ change with (bool_to_Prop (code_has_point ????)) |
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| 614 | whd in match (point_of_succ ???); |
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| 615 | >lin_code_has_point @leb_elim [ #_ % ] >length_map >length_reverse |
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| 616 | #INEQ |
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| 617 | cut (|generated| = S i) |
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| 618 | [ @(le_to_le_to_eq … (not_lt_to_le … INEQ) ) |
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| 619 | elim (option_bind_inverse ????? EQ) #x * #EQ1 #EQ2 |
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| 620 | <length_reverse |
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| 621 | @(nth_opt_hit_length … EQ1) |
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| 622 | ] #EQ_length |
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| 623 | elim last_fin #fin #EQ' |
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| 624 | lapply EQ whd in match (stmt_at ????); |
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| 625 | >nth_opt_reverse_hit >EQ_length [2: % ] <minus_n_n |
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| 626 | change with (stmt_at ???? = ? → ?) |
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| 627 | >EQ' #ABS destruct(ABS) |
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| 628 | ] |
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| 629 | ] |
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| 630 | %{visited} % [assumption] |
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| 631 | #lbl #n #eq_lookup elim (sigma_prop ?? eq_lookup) |
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| 632 | #lbl_in_gen * #stmt ** #stmt_in_g #nth_opt_is_stmt #succ_is_in |
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| 633 | % [2: % [ assumption ] |] |
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| 634 | >nth_opt_filter_labels in ⊢ (???%); |
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| 635 | >nth_opt_is_stmt >m_return_bind whd >m_return_bind |
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| 636 | % [ % ] |
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| 637 | [ elim (lbl ∈ required) % |
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| 638 | | % |
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| 639 | | lapply succ_is_in |
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| 640 | lapply (refl … (stmt_implicit_label … stmt)) |
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| 641 | cases (stmt_implicit_label … stmt) in ⊢ (???%→%); [#_ #_ %] |
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| 642 | #next #EQ_next * |
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| 643 | [ #H %1{H} ] |
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| 644 | #H %2 |
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| 645 | >nth_opt_filter_labels >H % |
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| 646 | ] |
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| 647 | qed. |
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| 648 | |
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| 649 | (* Paolo: for now I'm leaving the closed graph stuff out, waiting for it to |
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| 650 | be done on all passes *) |
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| 651 | definition linearise_int_fun : |
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| 652 | ∀p : unserialized_params. |
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| 653 | ∀globals. |
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[2422] | 654 | joint_closed_internal_function (mk_graph_params p) globals → |
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| 655 | joint_closed_internal_function (mk_lin_params p) globals |
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[2182] | 656 | (* ∃sigma : identifier_map LabelTag ℕ. |
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| 657 | let g ≝ joint_if_code ?? (pi1 … fin) in |
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| 658 | let c ≝ joint_if_code ?? (pi1 … fout) in |
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| 659 | let entry ≝ joint_if_entry ?? (pi1 … fin) in |
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| 660 | lookup … sigma entry = Some ? 0 ∧ |
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| 661 | ∀l,n.lookup … sigma l = Some ? n → |
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| 662 | ∃s. lookup … g l = Some ? s ∧ |
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| 663 | opt_Exists ? |
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| 664 | (λls.let 〈lopt, ts〉 ≝ ls in |
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| 665 | opt_All ? (eq ? l) lopt ∧ |
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| 666 | ts = graph_to_lin_statement … s ∧ |
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| 667 | opt_All ? |
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| 668 | (λnext.Or (lookup … sigma next = Some ? (S n)) |
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| 669 | (nth_opt … (S n) c = Some ? 〈None ?, GOTO … next〉)) |
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| 670 | (stmt_implicit_label … s)) (nth_opt … n c)*) ≝ |
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[2422] | 671 | λp,globals,f_sig.let f_in ≝ pi1 … f_sig in |
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| 672 | «mk_joint_internal_function (mk_lin_params p) globals |
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[2182] | 673 | (joint_if_luniverse ?? f_in) (joint_if_runiverse ?? f_in) |
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| 674 | (joint_if_result ?? f_in) (joint_if_params ?? f_in) (joint_if_locals ?? f_in) |
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| 675 | (joint_if_stacksize ?? f_in) |
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[2422] | 676 | (linearise_code p globals (joint_if_code … f_in) (pi2 … f_sig) (joint_if_entry … f_in)) |
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[2440] | 677 | 0 0 (* exit is dummy! *), ?». [2,4:// ] |
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[2182] | 678 | elim (linearise_code ?????) #c * #code_closed |
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[2422] | 679 | [3: #_ assumption ] |
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[2182] | 680 | @hide_prf |
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| 681 | * #sigma * #O_in #sigma_prop |
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| 682 | >lin_code_has_point |
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| 683 | elim (sigma_prop … O_in) #s * #_ |
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| 684 | cases c |
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| 685 | [1,3: * |
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| 686 | |*: #hd #tl #_ % |
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| 687 | ] |
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| 688 | qed. |
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[2422] | 689 | |
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| 690 | definition linearise : ∀p : unserialized_params. |
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| 691 | program (joint_function (mk_graph_params p)) ℕ → |
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| 692 | program (joint_function (mk_lin_params p)) ℕ ≝ |
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| 693 | λp,pr.transform_program ??? pr |
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| 694 | (λglobals.transf_fundef ?? (linearise_int_fun p globals)). |
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