[2532] | 1 | (* include "joint/TranslateUtils.ma". *) |
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| 2 | include "joint/Joint.ma". |
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[2200] | 3 | include "utilities/hide.ma". |
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[2182] | 4 | |
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| 5 | definition graph_to_lin_statement : |
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| 6 | ∀p : unserialized_params.∀globals. |
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[2532] | 7 | ∀A.identifier_map LabelTag A → |
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[2182] | 8 | joint_statement (mk_graph_params p) globals → joint_statement (mk_lin_params p) globals ≝ |
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[2532] | 9 | λp,globals,A,visited,stmt. |
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| 10 | match stmt return λ_.joint_statement (mk_lin_params ?) ? with |
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| 11 | [ sequential c nxt ⇒ |
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| 12 | match c with |
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| 13 | [ COND a ltrue ⇒ |
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| 14 | if nxt ∈ visited then FCOND … I a ltrue nxt else |
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| 15 | sequential (mk_lin_params p) … c it |
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| 16 | | _ ⇒ sequential … c it |
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| 17 | ] |
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[2182] | 18 | | final c ⇒ final … c |
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[2532] | 19 | | FCOND abs _ _ _ ⇒ Ⓧabs |
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[2182] | 20 | ]. |
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| 21 | |
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[2532] | 22 | (* |
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[2182] | 23 | lemma graph_to_lin_labels : |
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| 24 | ∀p : unserialized_params.∀globals,s. |
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| 25 | stmt_labels … (graph_to_lin_statement p globals s) = |
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| 26 | stmt_explicit_labels … s. |
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| 27 | #p#globals * [//] * // |
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| 28 | qed. |
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[2532] | 29 | *) |
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[2182] | 30 | |
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| 31 | (* discard all elements passing test, return first element failing it *) |
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| 32 | (* and the rest of the list, if any. *) |
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| 33 | let rec chop A (test : A → bool) (l : list A) on l : option (A × (list A)) ≝ |
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| 34 | match l with |
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| 35 | [ nil ⇒ None ? |
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| 36 | | cons x l' ⇒ if test x then chop A test l' else return 〈x, l'〉 |
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| 37 | ]. |
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| 38 | |
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| 39 | lemma chop_ok : ∀A,f,l. |
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| 40 | match chop A f l with |
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| 41 | [ Some pr ⇒ |
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| 42 | let x ≝ \fst pr in |
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| 43 | let post ≝ \snd pr in |
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| 44 | ∃pre.All ? (λx.bool_to_Prop (f x)) pre ∧ |
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| 45 | l = pre @ x :: post ∧ ¬bool_to_Prop (f x) |
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| 46 | | None ⇒ All A (λx.bool_to_Prop (f x)) l |
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| 47 | ]. |
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| 48 | #A #f #l elim l |
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| 49 | [ % |
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| 50 | | #hd #tl #IH whd in match (chop ???); |
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| 51 | elim (true_or_false_Prop (f hd)) #H >H normalize nodelta |
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| 52 | [ lapply IH elim (chop ???) normalize nodelta |
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| 53 | [ #G %{H G} |
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| 54 | | #pr * #pre ** #H1 #H2 #H3 %{(hd :: pre)} %{H3} % |
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| 55 | [ %{H H1} |
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| 56 | | >H2 % |
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| 57 | ] |
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| 58 | ] |
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| 59 | | %{[ ]} %{H} %% |
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| 60 | ] |
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| 61 | ] |
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| 62 | qed. |
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| 63 | |
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| 64 | unification hint 0 ≔ p, globals; |
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| 65 | lp ≟ lin_params_to_params p, |
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| 66 | sp ≟ stmt_pars lp, |
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| 67 | js ≟ joint_statement sp globals, |
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| 68 | lo ≟ labelled_obj LabelTag js |
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| 69 | (*----------------------------*)⊢ |
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| 70 | list lo ≡ codeT lp globals. |
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| 71 | |
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| 72 | definition graph_visit_ret_type ≝ λp,globals.λg : codeT (mk_graph_params p) globals. |
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| 73 | λentry : label. |
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| 74 | (Σ〈visited' : identifier_map LabelTag ℕ, |
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| 75 | required' : identifier_set LabelTag, |
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| 76 | generated' : codeT (mk_lin_params p) globals〉.'hide ( |
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| 77 | And (And (And (And |
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| 78 | (lookup ?? visited' entry = Some ? 0) |
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| 79 | (required' ⊆ visited')) |
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[2532] | 80 | ((∃last.stmt_at … generated' 0 = Some ? (final … last)) ∨ |
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| 81 | (∃a,ltrue,lfalse.stmt_at … generated' 0 = Some ? (FCOND … I a ltrue lfalse)))) |
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[2182] | 82 | (code_forall_labels … (λl.bool_to_Prop (l∈required')) (rev … generated'))) |
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| 83 | (∀l,n.lookup ?? visited' l = Some ? n → |
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| 84 | And (bool_to_Prop (code_has_label … (rev ? generated') l)) |
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[2532] | 85 | (∃s.And |
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[2182] | 86 | (lookup … g l = Some ? s) |
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[2532] | 87 | (match s with |
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| 88 | [ sequential s' nxt ⇒ |
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| 89 | match s' with |
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| 90 | [ step_seq _ ⇒ |
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| 91 | And |
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| 92 | (nth_opt … n (rev … generated') = |
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| 93 | Some ? 〈Some ? l, sequential … s' it〉) |
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| 94 | (Or (lookup … visited' nxt = Some ? (S n)) |
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| 95 | (nth_opt … (S n) (rev … generated') = |
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| 96 | Some ? 〈None ?, GOTO … nxt〉)) |
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| 97 | | COND a ltrue ⇒ |
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| 98 | Or |
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| 99 | (And (nth_opt … n (rev … generated') = |
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| 100 | Some ? 〈Some ? l, sequential … (COND … a ltrue) it〉) |
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| 101 | (lookup … visited' nxt = Some ? (S n))) |
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| 102 | (nth_opt … n (rev … generated') = |
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| 103 | Some ? 〈Some ? l, FCOND … I a ltrue nxt〉) |
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| 104 | ] |
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| 105 | | final s' ⇒ |
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| 106 | nth_opt … n (rev … generated') = |
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| 107 | Some ? 〈Some ? l, final … s'〉 |
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| 108 | | FCOND abs _ _ _ ⇒ Ⓧabs |
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| 109 | ]))))). |
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[2182] | 110 | |
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| 111 | unification hint 0 ≔ tag ⊢ identifier_set tag ≡ identifier_map tag unit. |
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| 112 | |
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| 113 | include alias "common/Identifiers.ma". |
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| 114 | |
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| 115 | lemma lookup_safe_elim : ∀tag,A.∀P : A → Prop.∀m,i,prf. |
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[2532] | 116 | (∀x.lookup tag A m i = Some ? x → P x) → P (lookup_safe tag A m i prf) ≝ |
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| 117 | λtag,A,P,m,i,prf,H.(H … (lookup_eq_safe …)). |
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[2182] | 118 | |
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| 119 | let rec graph_visit |
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| 120 | (p : unserialized_params) |
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| 121 | (globals: list ident) |
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| 122 | (g : codeT (mk_graph_params p) globals) |
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| 123 | (* = graph (joint_statement (mk_graph_params p) globals *) |
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| 124 | (required: identifier_set LabelTag) |
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| 125 | (visited: identifier_map LabelTag ℕ) (* the reversed index of labels in the new code *) |
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| 126 | (generated: codeT (mk_lin_params p) globals) |
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| 127 | (* ≝ list ((option label)×(joint_statement (mk_lin_params p) globals)) *) |
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| 128 | (visiting: list label) |
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| 129 | (gen_length : ℕ) |
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| 130 | (n: nat) |
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| 131 | (entry : label) |
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| 132 | (g_prf : code_closed … g) |
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| 133 | (required_prf1 : ∀i.i∈required → Or (In ? visiting i) (bool_to_Prop (i∈visited))) |
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| 134 | (required_prf2 : code_forall_labels … (λl.bool_to_Prop (l ∈ required)) (rev … generated)) |
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| 135 | (generated_prf1 : ∀l,n.lookup … visited l = Some ? n → hide_Prop ( |
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| 136 | And (bool_to_Prop (code_has_label … (rev ? generated) l)) |
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[2532] | 137 | (∃s.And |
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[2182] | 138 | (lookup … g l = Some ? s) |
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[2532] | 139 | (match s with |
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| 140 | [ sequential s' nxt ⇒ |
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| 141 | match s' with |
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| 142 | [ step_seq _ ⇒ |
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| 143 | And |
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| 144 | (nth_opt … n (rev … generated) = |
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| 145 | Some ? 〈Some ? l, sequential … s' it〉) |
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| 146 | (match lookup … visited nxt with |
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| 147 | [ Some n' ⇒ Or (n' = S n) (nth_opt … (S n) (rev … generated) = Some ? 〈None ?, GOTO … nxt〉) |
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| 148 | | None ⇒ And (nth_opt ? 0 visiting = Some ? nxt) (S n = gen_length) |
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| 149 | ]) |
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| 150 | | COND a ltrue ⇒ |
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| 151 | Or |
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| 152 | (And (nth_opt … n (rev … generated) = |
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| 153 | Some ? 〈Some ? l, sequential … (COND … a ltrue) it〉) |
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| 154 | (match lookup … visited nxt with |
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| 155 | [ Some n' ⇒ n' = S n |
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| 156 | | None ⇒ And (nth_opt ? 0 visiting = Some ? nxt) (S n = gen_length) |
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| 157 | ])) |
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| 158 | (nth_opt … n (rev … generated) = |
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| 159 | Some ? 〈Some ? l, FCOND … I a ltrue nxt〉) |
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| 160 | ] |
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| 161 | | final s' ⇒ |
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| 162 | nth_opt … n (rev … generated) = |
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| 163 | Some ? 〈Some ? l, final … s'〉 |
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| 164 | | FCOND abs _ _ _ ⇒ Ⓧabs |
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| 165 | ])))) |
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[2182] | 166 | (generated_prf2 : ∀l.lookup … visited l = None ? → does_not_occur … l (rev ? generated)) |
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| 167 | (generated_prf3 : (∃last.stmt_at … generated 0 = Some ? (final … last)) ∨ |
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[2532] | 168 | (∃a,ltrue,lfalse.stmt_at … generated 0 = Some ? (FCOND … a ltrue lfalse)) ∨ |
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[2182] | 169 | (¬All ? (λx.bool_to_Prop (x∈visited)) visiting)) |
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| 170 | (visiting_prf : All … (λl.bool_to_Prop (l∈g)) visiting) |
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| 171 | (gen_length_prf : gen_length = length ? generated) |
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| 172 | (entry_prf : Or (And (And (visiting = [entry]) (gen_length = 0)) (Not (bool_to_Prop (entry∈visited)))) |
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| 173 | (lookup … visited entry = Some ? 0)) |
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| 174 | (n_prf : le (id_map_size … g) (plus n (id_map_size … visited))) |
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| 175 | on n |
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| 176 | : graph_visit_ret_type … g entry ≝ |
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| 177 | match chop ? (λx.x∈visited) visiting |
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| 178 | return λx.? → graph_visit_ret_type … g entry with |
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| 179 | [ None ⇒ λH. |
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| 180 | «〈visited, required, generated〉, ?» |
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| 181 | | Some pr ⇒ λH. |
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| 182 | let vis_hd ≝ \fst pr in |
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| 183 | let vis_tl ≝ \snd pr in |
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| 184 | match n return λx.? → graph_visit_ret_type … g entry with |
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| 185 | [ O ⇒ λn_prf'.⊥ |
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| 186 | | S n' ⇒ λn_prf'. |
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| 187 | (* add the label to the visited ones *) |
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| 188 | let visited' ≝ add … visited vis_hd gen_length in |
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| 189 | (* take l's statement *) |
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| 190 | let hd_vis_in_g ≝ (hide_prf ? ?) in |
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| 191 | let statement ≝ lookup_safe ?? g vis_hd hd_vis_in_g in |
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| 192 | (* translate it to its linear version *) |
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[2532] | 193 | let translated_statement ≝ graph_to_lin_statement … visited' statement in |
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[2182] | 194 | (* add the translated statement to the code (which will be reversed later) *) |
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| 195 | let generated' ≝ 〈Some … vis_hd, translated_statement〉 :: generated in |
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[2532] | 196 | let required' ≝ union_set ??? |
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| 197 | (set_from_list … (stmt_explicit_labels … translated_statement)) |
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| 198 | required in |
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[2182] | 199 | (* put successors in the visiting worklist *) |
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| 200 | let visiting' ≝ stmt_labels … statement @ vis_tl in |
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| 201 | (* finally, check the implicit successor *) |
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| 202 | (* if it has been already visited, we need to add a GOTO *) |
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| 203 | let add_req_gen ≝ |
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[2532] | 204 | match statement with |
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| 205 | [ sequential s nxt ⇒ |
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| 206 | match s with |
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| 207 | [ COND _ _ ⇒ 〈0, ∅, [ ]〉 |
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| 208 | | _ ⇒ |
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| 209 | if nxt ∈ visited' then |
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| 210 | 〈1, {(nxt)}, [〈None label, (GOTO … nxt : joint_statement ??)〉]〉 |
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| 211 | else 〈0, ∅, [ ]〉 |
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| 212 | ] |
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| 213 | | _ ⇒ 〈0, ∅, [ ]〉 |
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[2182] | 214 | ] in |
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| 215 | (* prepare a common utility to deal with add_req_gen *) |
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| 216 | let add_req_gen_prf : |
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| 217 | ∀P : (ℕ × (identifier_set LabelTag) × (codeT (mk_lin_params p) globals)) → Prop. |
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[2532] | 218 | (match statement with |
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| 219 | [ sequential s nxt ⇒ |
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| 220 | match s with [ COND _ _ ⇒ True | _ ⇒ ¬bool_to_Prop (nxt ∈ visited') ] |
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| 221 | | _ ⇒ True] → P 〈0,∅,[ ]〉) → |
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| 222 | (∀s,nxt. |
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| 223 | statement = sequential … (step_seq … s) nxt → |
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| 224 | nxt ∈ visited' → P 〈1, {(nxt)}, [〈None ?, GOTO (mk_lin_params p) nxt〉]〉) → |
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[2182] | 225 | P add_req_gen ≝ hide_prf ?? in |
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| 226 | graph_visit ??? |
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| 227 | (union_set ??? (\snd (\fst add_req_gen)) required') |
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| 228 | visited' |
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| 229 | (\snd add_req_gen @ generated') |
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| 230 | visiting' |
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| 231 | (plus (\fst (\fst add_req_gen)) (S gen_length)) |
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| 232 | n' entry g_prf ????????? |
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| 233 | ] n_prf |
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| 234 | ] (chop_ok ? (λx.x∈visited) visiting). |
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[2529] | 235 | (* cases daemon qed. *) |
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[2182] | 236 | whd |
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| 237 | [ (* base case, visiting is all visited *) |
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| 238 | %[%[%[%]]] |
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| 239 | [ elim entry_prf |
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| 240 | [ ** #eq_visiting #gen_length_O #entry_vis >eq_visiting in H; * >entry_vis * |
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| 241 | | // |
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| 242 | ] |
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| 243 | | #l #l_req |
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| 244 | elim (required_prf1 … l_req) #G |
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| 245 | [ @(All_In … H G) |
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| 246 | | assumption |
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| 247 | ] |
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[2532] | 248 | | cases generated_prf3 [/2 by /] |
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[2182] | 249 | * #ABS @⊥ @ABS assumption |
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| 250 | | assumption |
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[2532] | 251 | | #l #n #H elim (generated_prf1 … H) -H |
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| 252 | #H1 * #s * #H2 #H3 %{H1} %{s} %{H2} |
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| 253 | cases s in H3; [3: *] |
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| 254 | [ * ] normalize nodelta |
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| 255 | [ #s' #nxt * #EQnth_opt #H %{EQnth_opt} |
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| 256 | inversion (lookup … visited nxt) in H; [2: #n'] #EQlookup |
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[2182] | 257 | normalize nodelta * |
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[2532] | 258 | [ #EQn' %1 >EQn' % |
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[2182] | 259 | | #H %2{H} |
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[2532] | 260 | | #H' lapply (All_nth … H … H') |
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| 261 | whd in ⊢ (?%→?); >EQlookup * |
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[2182] | 262 | ] |
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[2532] | 263 | | #a #ltrue #lfalse * [2: #H %2{H} ] * #H1 #H2 %1 %{H1} |
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| 264 | inversion (lookup … visited lfalse) in H2; |
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| 265 | [ #ABS * #H' lapply (All_nth … H … H') |
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| 266 | whd in ⊢ (?%→?); >ABS * |
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| 267 | | #n' #_ normalize nodelta #EQ >EQ % |
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| 268 | ] |
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| 269 | | #s #H @H |
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[2182] | 270 | ] |
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| 271 | ] |
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| 272 | (* first, close goals where add_gen_req plays no role *) |
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| 273 | |13: (* vis_hd is in g *) |
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| 274 | elim H #pre ** #_ #H2 #_ |
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| 275 | @(All_split … visiting_prf … H2) |
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| 276 | |2: (* n = 0 absrud *) |
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| 277 | elim H #pre ** #_ #H2 #H3 |
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| 278 | @(absurd … n_prf') |
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| 279 | @lt_to_not_le |
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| 280 | lapply (add_size … visited vis_hd 0 (* dummy value *)) |
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| 281 | >H3 normalize nodelta |
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| 282 | whd in match (lt ? ?); |
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| 283 | whd in match (1 + ?); |
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| 284 | #EQ <EQ @subset_card @add_subset |
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| 285 | [ @(All_split ? (λx.bool_to_Prop (x∈g)) ????? H2) @(All_mp … visiting_prf) |
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| 286 | #a elim g #gm whd in ⊢ (?→?%); #H >(lookup_eq_safe … H) % |
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| 287 | | #l #H |
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[2532] | 288 | elim (generated_prf1 … (lookup_eq_safe … H)) #_ * #s * #s_in_g #_ |
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[2182] | 289 | whd in ⊢ (?%); >s_in_g % |
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| 290 | ] |
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| 291 | |8: |
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| 292 | elim H #pre ** #_ #H2 #_ |
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| 293 | @All_append |
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| 294 | [ elim(g_prf … vis_hd statement ?) [2:@lookup_eq_safe] #G1 #G2 |
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| 295 | @(All_append … G1) |
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[2532] | 296 | cases statement in G2; [2: // |3: * ] |
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| 297 | #s #nxt #G' %{G'} % |
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[2182] | 298 | | >H2 in visiting_prf; |
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| 299 | #H' lapply (All_append_r … H') -H' * #_ // |
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| 300 | ] |
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| 301 | |10: |
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| 302 | elim H #pre ** #_ #H2 #H3 -add_req_gen_prf |
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| 303 | %2 elim entry_prf |
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| 304 | [ ** >H2 cases pre |
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| 305 | [2: #x' #pre' #ABS normalize in ABS; destruct(ABS) |
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| 306 | cases pre' in e0; [2: #x'' #pre''] #ABS' normalize in ABS'; destruct(ABS') |
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| 307 | |1: #EQ normalize in EQ; destruct(EQ) |
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| 308 | #eq_gen_length #_ |
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| 309 | >lookup_add_hit >eq_gen_length % |
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| 310 | ] |
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| 311 | | #lookup_entry cut (entry ≠ vis_hd) |
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| 312 | [ % whd in match vis_hd; #entry_vis_hd <entry_vis_hd in H3; |
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| 313 | whd in ⊢ (?(?%)→?); >lookup_entry * #ABS @ABS % ] |
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| 314 | #entry_not_vis_hd >(lookup_add_miss ?????? entry_not_vis_hd) assumption |
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| 315 | ] |
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| 316 | |11: |
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| 317 | elim H #pre ** #_ #_ #H3 |
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| 318 | >commutative_plus |
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| 319 | >add_size >H3 normalize nodelta |
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| 320 | whd in match (1 + ?); |
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| 321 | >commutative_plus |
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| 322 | assumption |
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| 323 | |12: (* add_req_gen utility *) |
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| 324 | #P whd in match add_req_gen; |
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[2532] | 325 | cases statement [ * ] [|*: [ #a #ltrue #lfalse | #s | * ] #H #_ @(H I) ] |
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| 326 | #s #nxt normalize nodelta inversion (nxt ∈ visited') #EQ normalize nodelta |
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| 327 | #H1 #H2 [ @(H2 … (refl …)) >EQ % | @H1 % * ] |
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| 328 | |3: elim H #pre ** #H1 #H2 #_ |
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[2182] | 329 | #i >mem_set_union |
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| 330 | #H elim (orb_Prop_true … H) -H |
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| 331 | [ @add_req_gen_prf [ #_ >mem_set_empty * ] |
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[2532] | 332 | #s #next #_ #next_vis #H >(mem_set_singl_to_eq … H) %2 assumption |
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[2182] | 333 | | >mem_set_union |
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| 334 | #H elim (orb_Prop_true … H) -H |
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[2532] | 335 | [ #i_expl %1 @Exists_append_l |
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| 336 | lapply i_expl whd in match translated_statement; |
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| 337 | cases statement [ * [ #s | #a #ltrue ] #nxt | #s | *] normalize nodelta #H |
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| 338 | lapply (mem_list_as_set … H) -H #H |
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| 339 | [1,3: @Exists_append_r assumption ] |
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| 340 | cases (nxt ∈ visited') in H; normalize nodelta * [2,4: * [2: * ]] |
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| 341 | #EQ destruct(EQ) [ %1 % |*: %2 %1 % ] |
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[2182] | 342 | | (* i was already required *) |
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| 343 | #i_req |
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| 344 | elim (required_prf1 … i_req) |
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| 345 | [ >H2 #H elim (Exists_append … H) -H |
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| 346 | [ (* i in preamble → i∈visited *) |
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| 347 | #i_pre %2 >mem_set_add @orb_Prop_r |
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[2532] | 348 | @(All_In … H1 i_pre) |
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[2182] | 349 | | * |
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| 350 | [ (* i is vis_hd *) |
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| 351 | #eq_i >eq_i %2 @mem_set_add_id |
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| 352 | | (* i in vis_tl → i∈visiting' *) |
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| 353 | #i_post % @Exists_append_r assumption |
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| 354 | ] |
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| 355 | ] |
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| 356 | | (* i in visited *) |
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| 357 | #i_vis %2 >mem_set_add @orb_Prop_r assumption |
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| 358 | ] |
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| 359 | ] |
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| 360 | ] |
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| 361 | |4,5,6: change with reverse in match rev; |
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| 362 | >reverse_append whd in match (reverse ??); >rev_append_def |
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| 363 | >associative_append |
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| 364 | [ #pt #s |
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| 365 | @(leb_elim (S pt) (|generated|)) #cmp |
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| 366 | whd in match (stmt_at ????); |
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| 367 | [ >nth_opt_append_l [2: >length_reverse assumption ] |
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| 368 | change with (stmt_at ???? = ? → ?) |
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| 369 | #EQ lapply(required_prf2 … EQ) @All_mp |
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| 370 | #l #l_req >mem_set_union @orb_Prop_r |
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| 371 | >mem_set_union @orb_Prop_r @l_req |
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| 372 | | >nth_opt_append_r [2: >length_reverse @not_lt_to_le assumption ] |
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| 373 | cases (pt - ?) |
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| 374 | [ whd in match (nth_opt ???); whd in ⊢ (??%?→?); |
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[2532] | 375 | #EQ lapply (sym_eq ??? EQ) -EQ #EQ destruct(EQ) whd |
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[2182] | 376 | whd in match required'; |
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[2532] | 377 | cut (∀p : lin_params.∀globals.∀s. |
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| 378 | stmt_implicit_label p globals s = None ?) |
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| 379 | [ #p #globals * normalize // ] |
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| 380 | #lin_implicit_label change with (? @ ?) in match (stmt_labels ???); |
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| 381 | >lin_implicit_label |
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| 382 | change with (All ?? (stmt_explicit_labels ???)) |
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| 383 | generalize in match (stmt_explicit_labels … translated_statement); |
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[2182] | 384 | #l @list_as_set_All |
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| 385 | #i >mem_set_union >mem_set_union |
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[2532] | 386 | #i_l @orb_Prop_r @orb_Prop_l assumption |
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[2182] | 387 | | @add_req_gen_prf |
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[2532] | 388 | [ #_ | #s #next #_ #next_vis * |
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[2182] | 389 | [ whd in ⊢ (??%?→?); |
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| 390 | #EQ' destruct(EQ') whd %{I} >mem_set_union |
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| 391 | @orb_Prop_l @mem_set_add_id ]] |
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| 392 | #n whd in ⊢ (??%?→?); #ABS destruct(ABS) |
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| 393 | ] |
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| 394 | ] |
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| 395 | | elim H #pre ** #H1 #H2 #H3 |
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| 396 | #i whd in match visited'; |
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| 397 | @(eq_identifier_elim … i vis_hd) |
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| 398 | [ #EQi >EQi -i #pos |
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[2532] | 399 | >lookup_add_hit in ⊢ (%→?); #EQpos (* too slow: destruct(EQpos) *) |
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[2182] | 400 | cut (gen_length = pos) |
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[2532] | 401 | [ -visited destruct(EQpos) %] |
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[2182] | 402 | -EQpos #EQpos <EQpos -pos |
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| 403 | % |
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| 404 | [ >lin_code_has_label |
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| 405 | @add_req_gen_prf |
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| 406 | [ #_ |
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[2532] | 407 | | #s #next #_ #next_vis |
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[2182] | 408 | change with (? @ ([?] @ [?])) in match (? @ [? ; ?]); |
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| 409 | <associative_append >occurs_exactly_once_None |
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| 410 | ] |
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| 411 | >occurs_exactly_once_Some_eq >eq_identifier_refl |
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| 412 | normalize nodelta |
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| 413 | @generated_prf2 |
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| 414 | lapply (in_map_domain … visited vis_hd) |
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| 415 | >H3 normalize nodelta // |
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| 416 | | %{statement} |
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[2532] | 417 | % |
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[2182] | 418 | [ @lookup_eq_safe |
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[2532] | 419 | | normalize nodelta |
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| 420 | change with statement in match (lookup_safe ?????); |
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| 421 | cases statement; |
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| 422 | [ * [ #s | #a #ltrue ] #nxt | #s | * ] normalize nodelta |
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| 423 | [1,2: inversion (nxt ∈ visited') normalize nodelta #nxt_vis ] |
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| 424 | [1,2: % | %2 | %1 % ] |
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| 425 | [1,3,5,6,8: |
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| 426 | >nth_opt_append_r >rev_length <gen_length_prf try % |
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| 427 | <minus_n_n try % whd in match graph_to_lin_statement; normalize nodelta |
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| 428 | >nxt_vis % |
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| 429 | |*: |
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| 430 | lapply (in_map_domain … visited' nxt) >nxt_vis normalize nodelta |
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| 431 | [ * #n' ] #EQlookup >EQlookup normalize nodelta |
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| 432 | [ %2 >nth_opt_append_r >rev_length <gen_length_prf [2: %2 %1 ] |
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| 433 | <minus_Sn_n % |
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| 434 | |*: %% |
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| 435 | ] |
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[2182] | 436 | ] |
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| 437 | ] |
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| 438 | ] |
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| 439 | | #NEQ #n_i >(lookup_add_miss … visited … NEQ) |
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| 440 | #Hlookup elim (generated_prf1 … Hlookup) |
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[2532] | 441 | #G1 * #s * #G2 #G3 |
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[2182] | 442 | % |
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| 443 | [ >lin_code_has_label <associative_append |
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| 444 | >occurs_exactly_once_append |
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| 445 | @orb_Prop_l @andb_Prop |
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| 446 | [ >occurs_exactly_once_Some_eq |
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| 447 | >eq_identifier_false [2: % #ABS >ABS in NEQ; * #ABS' @ABS' % ] |
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| 448 | normalize nodelta >lin_code_has_label in G1; #K @K |
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| 449 | | @add_req_gen_prf |
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[2532] | 450 | [ #_ % | #s' #next #_ #_ % ] |
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[2182] | 451 | ] |
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| 452 | | %{s} |
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[2532] | 453 | %{G2} |
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| 454 | cases s in G3; |
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| 455 | [ * [ #s | #a #ltrue ] #nxt | #s | * ] |
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| 456 | [ * #EQnth_opt #next_spec % | * [*] #EQnth_opt [#next_spec %1 % | %2] | #EQnth_opt ] |
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| 457 | [1,3,5,6: @nth_opt_append_hit_l assumption |
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| 458 | |*: @(eq_identifier_elim … nxt vis_hd) |
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| 459 | [1,3: #EQ destruct(EQ) >lookup_add_hit whd [ %1 ] |
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[2182] | 460 | lapply (in_map_domain … visited vis_hd) |
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[2532] | 461 | >H3 #EQ >EQ in next_spec; * #_ #OK >OK % |
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| 462 | |*: #NEQ' >(lookup_add_miss … visited … NEQ') |
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| 463 | lapply (in_map_domain … visited nxt) |
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| 464 | inversion (nxt ∈ visited) #nxt_vis [1,3: * #n'' ] #EQlookup' |
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| 465 | >EQlookup' in next_spec; whd in ⊢ (%→%); |
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| 466 | [ * [ #H %1{H} ] #H %2 @nth_opt_append_hit_l assumption |
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| 467 | | #H @H |
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| 468 | |*: * >H2 |
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| 469 | cases pre in H1; |
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| 470 | [1,3: #_ |
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| 471 | |*: #frst #pre_tl * #ABS #_ |
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| 472 | ] whd in ⊢ (??%?→?); #EQ destruct(EQ) |
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| 473 | [1,2: cases NEQ' #ABS cases (ABS ?) % |
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| 474 | |*: >nxt_vis in ABS; * |
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[2182] | 475 | ] |
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| 476 | ] |
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| 477 | ] |
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| 478 | ] |
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| 479 | ] |
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| 480 | ] |
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| 481 | | #i whd in match visited'; |
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| 482 | @(eq_identifier_elim … i vis_hd) #Heq |
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| 483 | [ >Heq >lookup_add_hit #ABS destruct(ABS) ] |
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| 484 | >(lookup_add_miss ?????? Heq) |
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| 485 | #i_n_vis |
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| 486 | >does_not_occur_append @andb_Prop |
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| 487 | [ @generated_prf2 assumption |
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| 488 | | change with (bool_to_Prop (¬eq_identifier ??? ∧ ?)) |
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| 489 | >eq_identifier_false [2: % #ABS <ABS in Heq; * #ABS' @ABS' % ] |
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[2532] | 490 | @add_req_gen_prf [ #_ | #s #next #_ #_ ] % |
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[2182] | 491 | ] |
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| 492 | ] |
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| 493 | | @add_req_gen_prf |
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[2532] | 494 | [ #K | #s #next #K #next_vis %1 %1 %{(GOTO … next)} % ] |
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[2182] | 495 | whd in match generated'; whd in match translated_statement; |
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[2532] | 496 | normalize nodelta |
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| 497 | change with statement in match (lookup_safe ?????); |
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| 498 | cases statement in K; |
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| 499 | [ * [ #s | #a #ltrue ] #nxt | #s #_ %1 %1 %{s} % | * ] normalize nodelta |
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| 500 | [2: #_ cases (true_or_false_Prop (nxt ∈ visited')) #nxt_vis |
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| 501 | [ >nxt_vis normalize nodelta %1 %2 %{a} %{ltrue} %{nxt} % ] |
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| 502 | | #nxt_vis ] |
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| 503 | %2 % * >nxt_vis * |
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[2182] | 504 | | whd in match generated'; |
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[2532] | 505 | @add_req_gen_prf [ #_ | #s #next #_ #_ ] normalize >gen_length_prf % |
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| 506 | | |
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[2182] | 507 | ] |
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| 508 | qed. |
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| 509 | |
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| 510 | (* CSC: The branch compression (aka tunneling) optimization is not implemented |
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| 511 | in Matita *) |
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| 512 | definition branch_compress |
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| 513 | ≝ λp: graph_params.λglobals.λg:codeT p globals. |
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| 514 | λentry : Σl.bool_to_Prop (code_has_label … g l).g. |
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| 515 | |
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| 516 | lemma branch_compress_closed : ∀p,globals,g,l.code_closed ?? g → |
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| 517 | code_closed … (branch_compress p globals g l). |
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| 518 | #p#globals#g#l#H @H qed. |
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| 519 | |
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| 520 | lemma branch_compress_has_entry : ∀p,globals,g,l. |
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| 521 | code_has_label … (branch_compress p globals g l) l. |
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| 522 | #p#globals#g*#l#l_prf @l_prf qed. |
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| 523 | |
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| 524 | definition filter_labels ≝ λtag,A.λtest : identifier tag → bool.λc : list (labelled_obj tag A). |
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| 525 | map ?? |
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| 526 | (λs. let 〈l_opt,x〉 ≝ s in |
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| 527 | 〈! l ← l_opt ; if test l then return l else None ?, x〉) c. |
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| 528 | |
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| 529 | lemma does_not_occur_filter_labels : |
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| 530 | ∀tag,A,test,id,c. |
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| 531 | does_not_occur ?? id (filter_labels tag A test c) = |
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| 532 | (does_not_occur ?? id c ∨ ¬ test id). |
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| 533 | #tag #A #test #id #c elim c |
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| 534 | [ // |
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| 535 | | ** [2: #lbl] #s #tl #IH |
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| 536 | whd in match (filter_labels ????); normalize nodelta |
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| 537 | whd in match (does_not_occur ????) in ⊢ (??%%); |
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| 538 | [2: @IH] |
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| 539 | normalize in match (! l ← ? ; ?); >IH |
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| 540 | @(eq_identifier_elim ?? lbl id) #Heq [<Heq] |
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| 541 | elim (test lbl) normalize nodelta |
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| 542 | change with (eq_identifier ???) in match (instruction_matches_identifier ????); |
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| 543 | [1,2: >eq_identifier_refl [2: >commutative_orb] normalize % |
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| 544 | |*: >(eq_identifier_false ??? Heq) normalize nodelta % |
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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 occurs_exactly_once_filter_labels : |
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| 550 | ∀tag,A,test,id,c. |
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| 551 | occurs_exactly_once ?? id (filter_labels tag A test c) = |
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| 552 | (occurs_exactly_once ?? id c ∧ test id). |
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| 553 | #tag #A #test #id #c elim c |
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| 554 | [ // |
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| 555 | | ** [2: #lbl] #s #tl #IH |
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| 556 | whd in match (filter_labels ????); normalize nodelta |
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| 557 | whd in match (occurs_exactly_once ????) in ⊢ (??%%); |
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| 558 | [2: @IH] |
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| 559 | normalize in match (! l ← ? ; ?); >IH |
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| 560 | >does_not_occur_filter_labels |
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| 561 | @(eq_identifier_elim ?? lbl id) #Heq [<Heq] |
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| 562 | elim (test lbl) normalize nodelta |
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| 563 | change with (eq_identifier ???) in match (instruction_matches_identifier ????); |
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| 564 | [1,2: >eq_identifier_refl >commutative_andb [ >(commutative_andb ? true) >commutative_orb | >(commutative_andb ? false)] normalize % |
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| 565 | |*: >(eq_identifier_false ??? Heq) normalize nodelta % |
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| 566 | ] |
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| 567 | ] |
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| 568 | qed. |
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| 569 | |
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| 570 | lemma nth_opt_filter_labels : ∀tag,A,test,instrs,n. |
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| 571 | nth_opt ? n (filter_labels tag A test instrs) = |
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| 572 | ! 〈lopt, s〉 ← nth_opt ? n instrs ; |
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| 573 | return 〈 ! lbl ← lopt; if test lbl then return lbl else None ?, s〉. |
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| 574 | #tag #A #test #instrs elim instrs |
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| 575 | [ * [2: #n'] % |
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| 576 | | * #lopt #s #tl #IH * [2: #n'] |
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| 577 | whd in match (filter_labels ????); normalize nodelta |
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| 578 | whd in match (nth_opt ???) in ⊢ (??%%); [>IH] % |
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| 579 | ] |
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| 580 | qed. |
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| 581 | |
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| 582 | lemma stmt_at_filter_labels : ∀p : lin_params.∀globals,test.∀c : codeT p globals. |
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| 583 | ∀i.stmt_at p globals (filter_labels ?? test c) i = stmt_at p globals c i. |
---|
| 584 | #p#globals#test #c#i |
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| 585 | whd in ⊢ (??%%); >nth_opt_filter_labels |
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| 586 | elim (nth_opt ???); // |
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| 587 | qed. |
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| 588 | |
---|
| 589 | lemma option_bind_inverse : ∀A,B.∀m : option A.∀f : A → option B.∀r. |
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| 590 | ! x ← m ; f x = return r → |
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| 591 | ∃x.m = return x ∧ f x = return r. |
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| 592 | #A #B * normalize [2:#x] #f #r #EQ destruct |
---|
| 593 | %{x} %{EQ} % |
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| 594 | qed. |
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| 595 | |
---|
| 596 | lemma nth_opt_reverse_hit : |
---|
| 597 | ∀A,l,n.n < |l| → nth_opt A n (reverse ? l) = nth_opt A (|l| - (S n)) l. |
---|
| 598 | #A #l elim l |
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| 599 | [ #n #ABS normalize in ABS; @⊥ -A /2 by absurd/ |
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| 600 | | #hd #tl #IH #n #lim whd in match (reverse ??); >rev_append_def |
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| 601 | @(leb_elim (S n) (|tl|)) #H |
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| 602 | [ >nth_opt_append_l [2: >length_reverse @H ] |
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| 603 | >(IH … H) >(minus_Sn_m … H) % |
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| 604 | | >(le_to_le_to_eq … (le_S_S_to_le … lim) (not_lt_to_le … H)) |
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| 605 | >nth_opt_append_r >length_reverse [2: % ] |
---|
| 606 | <minus_n_n <minus_n_n % |
---|
| 607 | ] |
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| 608 | ] |
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| 609 | qed. |
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| 610 | |
---|
| 611 | lemma nth_opt_reverse_hit_inv : |
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| 612 | ∀A,l,n.n < |l| → nth_opt A (|l| - (S n)) (reverse ? l) = nth_opt A n l. |
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| 613 | #A #l #n #H <(reverse_reverse ? l) in ⊢ (???%); @sym_eq |
---|
| 614 | <length_reverse @nth_opt_reverse_hit >length_reverse @H |
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| 615 | qed. |
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| 616 | |
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[2473] | 617 | definition good_local_sigma : |
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| 618 | ∀p:unserialized_params. |
---|
| 619 | ∀globals. |
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| 620 | ∀g:codeT (mk_graph_params p) globals. |
---|
| 621 | (Σl.bool_to_Prop (code_has_label … g l)) → |
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| 622 | codeT (mk_lin_params p) globals → |
---|
| 623 | (label → option ℕ) → Prop ≝ |
---|
| 624 | λp,globals,g,entry,c,sigma. |
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| 625 | sigma entry = Some ? 0 ∧ |
---|
[2529] | 626 | (∀l,n.point_of_label … c l = Some ? n → sigma l = Some ? n) ∧ |
---|
[2473] | 627 | ∀l,n.sigma l = Some ? n → |
---|
[2529] | 628 | ∃s. stmt_at … g l = Some ? s ∧ |
---|
[2532] | 629 | All ? (λl.sigma l ≠ None ?) (stmt_labels … s) ∧ |
---|
| 630 | (match s with |
---|
| 631 | [ sequential s' nxt ⇒ |
---|
| 632 | match s' with |
---|
| 633 | [ step_seq _ ⇒ |
---|
| 634 | (stmt_at … c n = Some ? (sequential … s' it)) ∧ |
---|
| 635 | ((sigma nxt = Some ? (S n)) ∨ |
---|
| 636 | (stmt_at … c (S n) = Some ? (GOTO … nxt))) |
---|
| 637 | | COND a ltrue ⇒ |
---|
| 638 | (stmt_at … c n = Some ? (sequential … s' it) ∧ sigma nxt = Some ? (S n)) ∨ |
---|
| 639 | (stmt_at … c n = Some ? (FCOND … I a ltrue nxt)) |
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| 640 | ] |
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| 641 | | final s' ⇒ |
---|
| 642 | stmt_at … c n = Some ? (final … s') |
---|
| 643 | | FCOND abs _ _ _ ⇒ Ⓧabs |
---|
| 644 | ]). |
---|
[2473] | 645 | |
---|
[2182] | 646 | definition linearise_code: |
---|
| 647 | ∀p : unserialized_params.∀globals. |
---|
| 648 | ∀g : codeT (mk_graph_params p) globals.code_closed … g → |
---|
[2473] | 649 | ∀entry : (Σl.bool_to_Prop (code_has_label … g l)) |
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| 650 | .Σ〈c, sigma〉. |
---|
| 651 | good_local_sigma … g entry c sigma ∧ |
---|
| 652 | code_closed … c |
---|
| 653 | (* ∧ |
---|
[2182] | 654 | ∃ sigma : identifier_map LabelTag ℕ. |
---|
| 655 | lookup … sigma entry = Some ? 0 ∧ |
---|
| 656 | ∀l,n.lookup … sigma l = Some ? n → |
---|
| 657 | ∃s. lookup … g l = Some ? s ∧ |
---|
| 658 | opt_Exists ? |
---|
| 659 | (λls.let 〈lopt, ts〉 ≝ ls in |
---|
| 660 | opt_All ? (eq ? l) lopt ∧ |
---|
| 661 | ts = graph_to_lin_statement … s ∧ |
---|
| 662 | opt_All ? |
---|
| 663 | (λnext.Or (lookup … sigma next = Some ? (S n)) |
---|
| 664 | (nth_opt … (S n) c = Some ? 〈None ?, GOTO … next〉)) |
---|
[2473] | 665 | (stmt_implicit_label … s)) (nth_opt … n c)*) |
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[2182] | 666 | ≝ |
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| 667 | λp,globals,g,g_prf,entry_sig. |
---|
| 668 | let g ≝ branch_compress (mk_graph_params p) ? g entry_sig in |
---|
| 669 | let g_prf ≝ branch_compress_closed … g entry_sig g_prf in |
---|
| 670 | match graph_visit p globals g ∅ (empty_map …) [ ] [entry_sig] 0 (|g|) |
---|
| 671 | (entry_sig) g_prf ????????? |
---|
| 672 | with |
---|
| 673 | [ mk_Sig triple H ⇒ |
---|
| 674 | let sigma ≝ \fst (\fst triple) in |
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| 675 | let required ≝ \snd (\fst triple) in |
---|
| 676 | let crev ≝ \snd triple in |
---|
| 677 | let lbld_code ≝ rev ? crev in |
---|
[2473] | 678 | 〈filter_labels … (λl.l∈required) lbld_code, lookup … sigma〉 ]. |
---|
| 679 | [ cases (graph_visit ????????????????????) |
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| 680 | (* done later *) |
---|
[2182] | 681 | | #i >mem_set_empty * |
---|
| 682 | |3,4: #a #b whd in ⊢ (??%?→?); #ABS destruct(ABS) |
---|
| 683 | | #l #_ % |
---|
| 684 | | %2 % * >mem_set_empty * |
---|
| 685 | | % [2: %] @(branch_compress_has_entry … g entry_sig) |
---|
| 686 | | % |
---|
| 687 | | % % [% %] cases (pi1 … entry_sig) normalize #_ % // |
---|
| 688 | | >commutative_plus change with (? ≤ |g|) % |
---|
| 689 | ] |
---|
[2473] | 690 | ** |
---|
| 691 | #visited #required #generated normalize nodelta **** |
---|
[2182] | 692 | #entry_O #req_vis #last_fin #labels_in_req #sigma_prop |
---|
[2473] | 693 | % |
---|
[2529] | 694 | [ % [ % [assumption]] |
---|
| 695 | #lbl #n |
---|
| 696 | [ change with (If ? then with prf do ? else ? = ? → ?) |
---|
| 697 | @If_elim [2: #_ #ABS destruct(ABS) ] |
---|
| 698 | #H lapply H |
---|
| 699 | >occurs_exactly_once_filter_labels |
---|
| 700 | elim (true_or_false_Prop … (occurs_exactly_once ?? lbl ?)) |
---|
| 701 | [1,2: #H1 >H1 |*:] [2: * ] |
---|
| 702 | elim (true_or_false_Prop … (lbl ∈ required)) #H2 >H2 * |
---|
| 703 | lapply (in_map_domain … visited lbl) >(req_vis … H2) |
---|
| 704 | * #n_lbl #EQsigma |
---|
[2532] | 705 | elim (sigma_prop … EQsigma) #_ * #stmt * #_ |
---|
| 706 | cases stmt [ * [ #s | #a #ltrue ] #nxt | #s | * ] normalize nodelta |
---|
| 707 | [ * #EQnth_opt #_ | * [ * ] #EQnth_opt [ #_ ] | #EQnth_opt ] |
---|
| 708 | >(nth_opt_index_of_label ???? n_lbl ? H) |
---|
| 709 | [1,4,7,10: normalize in ⊢ (% → ?); #EQ destruct(EQ) assumption |
---|
| 710 | |3: @(sequential … s it) |6: @(sequential … (COND … a ltrue) it) |
---|
| 711 | |9: @(FCOND … I a ltrue nxt) |12: @(final … s) |
---|
| 712 | |2,5,8,11: >nth_opt_filter_labels >EQnth_opt >m_return_bind >m_return_bind |
---|
[2529] | 713 | >H2 % |
---|
| 714 | ] |
---|
| 715 | | #eq_lookup elim (sigma_prop ?? eq_lookup) |
---|
[2532] | 716 | #lbl_in_gen * #stmt * #stmt_in_g #stmt_spec |
---|
| 717 | % [2: % [ % [ assumption ]] |] |
---|
[2529] | 718 | [ @All_append |
---|
[2532] | 719 | [ cases stmt in stmt_spec; |
---|
| 720 | [ * [ #s | #a #ltrue ] #nxt | #s #_ % | * ] |
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| 721 | normalize nodelta * [ #stmt_at_EQ * #nxt_spec | * #stmt_at_EQ #nxt_spec | #stmt_at_EQ ] |
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| 722 | %{I} |
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| 723 | [1,3: >nxt_spec % #ABS destruct(ABS) |
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| 724 | |*: lapply (in_map_domain … visited nxt) |
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| 725 | >req_vis |
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| 726 | [1,3: * #x #EQ >EQ % #ABS destruct(ABS) |
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| 727 | |2: @(proj1 … (labels_in_req (S n) (GOTO … nxt) …)) |
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| 728 | whd in ⊢ (??%?); >nxt_spec % |
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| 729 | |4: @(proj1 … (proj2 … (labels_in_req n (FCOND … I a ltrue nxt) …))) |
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| 730 | whd in ⊢ (??%?); >stmt_at_EQ % |
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| 731 | ] |
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[2529] | 732 | ] |
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[2532] | 733 | | cases stmt in stmt_spec; |
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| 734 | [ * [ #s | #a #ltrue ] #nxt | #s | * ] normalize nodelta |
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| 735 | [ * #EQ #_ | * [ * #EQ #_ | #EQ ] | #EQ ] |
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| 736 | whd in match stmt_explicit_labels; normalize nodelta |
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| 737 | [ @(All_mp … (labels_in_req n (sequential … s it) ?)) |
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| 738 | | @(All_mp … (labels_in_req n (sequential … (COND … a ltrue) it) ?)) |
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| 739 | | cases (labels_in_req n (FCOND … I a ltrue nxt) ?) |
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| 740 | [ #l_req #_ %{I} lapply (in_map_domain … visited ltrue) |
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| 741 | >(req_vis … l_req) * #n #EQ' >EQ' % #ABS destruct(ABS) ] |
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| 742 | | @(All_mp … (labels_in_req n (final … s) ?)) |
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[2529] | 743 | ] |
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[2532] | 744 | [1,3,6: #l #l_req >(lookup_eq_safe … (req_vis … l_req)) % #ABS destruct(ABS) |
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| 745 | |*: whd in ⊢ (??%?); >EQ % |
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| 746 | ] |
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[2529] | 747 | ] |
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[2532] | 748 | | cases stmt in stmt_spec; |
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| 749 | [ * [ #s | #a #ltrue ] #nxt | #s | * ] normalize nodelta |
---|
| 750 | [ * #EQ #H | * [ * #EQ #H | #EQ ] | #EQ ] |
---|
| 751 | >stmt_at_filter_labels |
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| 752 | whd in match (stmt_at (mk_lin_params p) ?? n); >EQ normalize nodelta |
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| 753 | [ % [%] cases H -H [#H %1{H} | #EQ' %2 >stmt_at_filter_labels whd in ⊢ (??%?); >EQ' % ] |
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| 754 | | %1 %{H} % |
---|
| 755 | | %2 % |
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| 756 | | % |
---|
| 757 | ] |
---|
[2529] | 758 | ] |
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[2473] | 759 | ] |
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| 760 | | #i #s |
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[2182] | 761 | >stmt_at_filter_labels #EQ |
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| 762 | % |
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| 763 | [ @stmt_forall_labels_explicit |
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| 764 | @(All_mp … (labels_in_req … EQ)) |
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| 765 | #l #l_req >lin_code_has_label |
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| 766 | >occurs_exactly_once_filter_labels >l_req |
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| 767 | >commutative_andb whd in ⊢ (?%); |
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| 768 | elim (sigma_prop ?? (lookup_eq_safe … (req_vis … l_req))) |
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| 769 | >lin_code_has_label #H #_ @H |
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[2532] | 770 | | lapply EQ cases s [ #s' * | #s' #_ % | * #a #ltrue #lfalse #_ % ] |
---|
| 771 | -EQ #EQ change with (bool_to_Prop (code_has_point ????)) |
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[2182] | 772 | whd in match (point_of_succ ???); |
---|
| 773 | >lin_code_has_point @leb_elim [ #_ % ] >length_map >length_reverse |
---|
| 774 | #INEQ |
---|
| 775 | cut (|generated| = S i) |
---|
| 776 | [ @(le_to_le_to_eq … (not_lt_to_le … INEQ) ) |
---|
| 777 | elim (option_bind_inverse ????? EQ) #x * #EQ1 #EQ2 |
---|
| 778 | <length_reverse |
---|
| 779 | @(nth_opt_hit_length … EQ1) |
---|
| 780 | ] #EQ_length |
---|
[2532] | 781 | elim last_fin * [ #fin | #a * #ltrue * #lfalse ] #EQ' |
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[2182] | 782 | lapply EQ whd in match (stmt_at ????); |
---|
[2532] | 783 | >nth_opt_reverse_hit >EQ_length [2,4: % ] <minus_n_n |
---|
[2182] | 784 | change with (stmt_at ???? = ? → ?) |
---|
| 785 | >EQ' #ABS destruct(ABS) |
---|
| 786 | ] |
---|
| 787 | ] |
---|
| 788 | qed. |
---|
| 789 | |
---|
| 790 | definition linearise_int_fun : |
---|
| 791 | ∀p : unserialized_params. |
---|
| 792 | ∀globals. |
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[2473] | 793 | ∀fn_in : joint_closed_internal_function (mk_graph_params p) globals |
---|
| 794 | .Σ〈fn_out : joint_closed_internal_function (mk_lin_params p) globals, |
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| 795 | sigma : ?〉. |
---|
| 796 | good_local_sigma … (joint_if_code … fn_in) (joint_if_entry … fn_in) |
---|
| 797 | (joint_if_code … fn_out) sigma |
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[2182] | 798 | (* ∃sigma : identifier_map LabelTag ℕ. |
---|
| 799 | let g ≝ joint_if_code ?? (pi1 … fin) in |
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| 800 | let c ≝ joint_if_code ?? (pi1 … fout) in |
---|
| 801 | let entry ≝ joint_if_entry ?? (pi1 … fin) in |
---|
| 802 | lookup … sigma entry = Some ? 0 ∧ |
---|
| 803 | ∀l,n.lookup … sigma l = Some ? n → |
---|
| 804 | ∃s. lookup … g l = Some ? s ∧ |
---|
| 805 | opt_Exists ? |
---|
| 806 | (λls.let 〈lopt, ts〉 ≝ ls in |
---|
| 807 | opt_All ? (eq ? l) lopt ∧ |
---|
| 808 | ts = graph_to_lin_statement … s ∧ |
---|
| 809 | opt_All ? |
---|
| 810 | (λnext.Or (lookup … sigma next = Some ? (S n)) |
---|
| 811 | (nth_opt … (S n) c = Some ? 〈None ?, GOTO … next〉)) |
---|
| 812 | (stmt_implicit_label … s)) (nth_opt … n c)*) ≝ |
---|
[2473] | 813 | λp,globals,f_sig. |
---|
| 814 | let code_sigma ≝ linearise_code … |
---|
| 815 | (joint_if_code … f_sig) |
---|
| 816 | (pi2 … f_sig) |
---|
| 817 | (joint_if_entry … f_sig) in |
---|
| 818 | let code ≝ \fst code_sigma in |
---|
| 819 | let sigma ≝ \snd code_sigma in |
---|
| 820 | let entry : Σpt.bool_to_Prop (code_has_point … code pt) ≝ «0, hide_prf ??» in |
---|
| 821 | 〈«mk_joint_internal_function (mk_lin_params p) globals |
---|
| 822 | (joint_if_luniverse ?? f_sig) (joint_if_runiverse ?? f_sig) |
---|
| 823 | (joint_if_result ?? f_sig) (joint_if_params ?? f_sig) (joint_if_locals ?? f_sig) |
---|
| 824 | (joint_if_stacksize ?? f_sig) code entry entry (* exit is dummy! *), ?», |
---|
| 825 | sigma〉. |
---|
| 826 | normalize nodelta |
---|
| 827 | cases (linearise_code ?????) * #code #sigma normalize nodelta * #H1 #H2 |
---|
| 828 | [ @H2 |
---|
| 829 | | @H1 |
---|
[2529] | 830 | | cases H1 * #H3 #H4 #H5 elim (H5 … H3) |
---|
[2532] | 831 | #s ** #_ #_ >lin_code_has_point cases code |
---|
| 832 | [ cases s [ * [ #s' | #a #ltrue ] #nxt | #s' | * ] |
---|
| 833 | [ * #ABS #_ | * [ * #ABS #_ | #ABS ] | #ABS ] normalize in ABS; destruct(ABS) |
---|
| 834 | | #hd #tl #_ % |
---|
| 835 | ] |
---|
[2182] | 836 | ] |
---|
| 837 | qed. |
---|
[2422] | 838 | |
---|
| 839 | definition linearise : ∀p : unserialized_params. |
---|
| 840 | program (joint_function (mk_graph_params p)) ℕ → |
---|
[2473] | 841 | program (joint_function (mk_lin_params p)) ℕ |
---|
| 842 | ≝ |
---|
[2422] | 843 | λp,pr.transform_program ??? pr |
---|
[2473] | 844 | (λglobals.transf_fundef ?? (λf_in.\fst (linearise_int_fun p globals f_in))). |
---|
| 845 | |
---|
[2529] | 846 | (* |
---|
[2473] | 847 | definition good_sigma : |
---|
| 848 | ∀p:unserialized_params. |
---|
| 849 | ∀prog_in : joint_program (mk_graph_params p). |
---|
| 850 | ((Σi.is_internal_function_of_program … prog_in i) → label → option ℕ) → Prop ≝ |
---|
| 851 | λp,prog_in,sigma. |
---|
| 852 | let prog_out ≝ linearise … prog_in in |
---|
[2481] | 853 | ∀i : Σi.is_internal_function_of_program … prog_in i.∀prf. |
---|
| 854 | let fn_in ≝ if_of_function … i in |
---|
| 855 | let i' : Σi.is_internal_function_of_program … prog_out i ≝ «pi1 ?? i, prf» in |
---|
| 856 | let fn_out ≝ if_of_function … i' in |
---|
[2473] | 857 | let sigma_local ≝ sigma i in |
---|
| 858 | good_local_sigma ?? (joint_if_code ?? fn_in) (joint_if_entry … fn_in) |
---|
| 859 | (joint_if_code ?? fn_out) sigma_local. |
---|
| 860 | |
---|
| 861 | lemma linearise_spec : ∀p,prog.∃sigma.good_sigma p prog sigma. |
---|
| 862 | #p #prog |
---|
| 863 | letin sigma ≝ |
---|
[2529] | 864 | (λi : Σi.is_internal_function_of_program … prog i. let fn_in ≝ if_of_function … i in |
---|
[2473] | 865 | \snd (linearise_int_fun … fn_in)) |
---|
| 866 | %{sigma} |
---|
[2481] | 867 | #i #prf >(prog_if_of_function_transform … i) [2: % ] |
---|
[2473] | 868 | normalize nodelta |
---|
| 869 | cases (linearise_int_fun ???) * #fn_out #sigma_loc |
---|
| 870 | normalize nodelta #prf @prf |
---|
| 871 | qed. |
---|
[2529] | 872 | *) |
---|