[1537] | 1 | |
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| 2 | include "RTLabs/semantics.ma". |
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| 3 | include "common/StructuredTraces.ma". |
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| 4 | |
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[1552] | 5 | discriminator status_class. |
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[1537] | 6 | |
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[1565] | 7 | (* NB: For RTLabs we only classify branching behaviour as jumps. Other jumps |
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| 8 | will be added later (LTL → LIN). *) |
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[1552] | 9 | |
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[1563] | 10 | definition RTLabs_classify : state → status_class ≝ |
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| 11 | λs. match s with |
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[1565] | 12 | [ State f _ _ ⇒ |
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| 13 | match lookup_present ?? (f_graph (func f)) (next f) (next_ok f) with |
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| 14 | [ St_cond _ _ _ ⇒ cl_jump |
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| 15 | | St_jumptable _ _ ⇒ cl_jump |
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| 16 | | _ ⇒ cl_other |
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| 17 | ] |
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[1563] | 18 | | Callstate _ _ _ _ _ ⇒ cl_call |
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| 19 | | Returnstate _ _ _ _ ⇒ cl_return |
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[1713] | 20 | | Finalstate _ ⇒ cl_other |
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[1563] | 21 | ]. |
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[1552] | 22 | |
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[1705] | 23 | definition is_cost_label : statement → bool ≝ |
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| 24 | λs. match s with [ St_cost _ _ ⇒ true | _ ⇒ false ]. |
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| 25 | |
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[1583] | 26 | definition RTLabs_cost : state → bool ≝ |
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| 27 | λs. match s with |
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| 28 | [ State f fs m ⇒ |
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[1586] | 29 | is_cost_label (lookup_present ?? (f_graph (func f)) (next f) (next_ok f)) |
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[1583] | 30 | | _ ⇒ false |
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| 31 | ]. |
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[1552] | 32 | |
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[1537] | 33 | definition RTLabs_status : genv → abstract_status ≝ |
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| 34 | λge. |
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| 35 | mk_abstract_status |
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| 36 | state |
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| 37 | (λs,s'. ∃t. eval_statement ge s = Value ??? 〈t,s'〉) |
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[1563] | 38 | (λs,c. RTLabs_classify s = c) |
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[1583] | 39 | (λs. RTLabs_cost s = true) |
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[1537] | 40 | (λs,s'. match s with |
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[1601] | 41 | [ mk_Sig s p ⇒ |
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[1563] | 42 | match s return λs. RTLabs_classify s = cl_call → ? with |
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[1537] | 43 | [ Callstate fd args dst stk m ⇒ |
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| 44 | λ_. match s' with |
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[1736] | 45 | [ State f fs m ⇒ match stk with [ nil ⇒ False | cons h t ⇒ next h = next f ∧ f_graph (func h) = f_graph (func f) ] |
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| 46 | | Finalstate r ⇒ stk = [ ] |
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[1537] | 47 | | _ ⇒ False |
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| 48 | ] |
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| 49 | | State f fs m ⇒ λH.⊥ |
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| 50 | | _ ⇒ λH.⊥ |
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| 51 | ] p |
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| 52 | ]). |
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[1563] | 53 | [ normalize in H; destruct |
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[1713] | 54 | | normalize in H; destruct |
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[1565] | 55 | | whd in H:(??%?); |
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| 56 | cases (lookup_present LabelTag statement (f_graph (func f)) (next f) (next_ok f)) in H; |
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| 57 | normalize try #a try #b try #c try #d try #e try #g try #h destruct |
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[1563] | 58 | ] qed. |
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[1559] | 59 | |
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[1670] | 60 | lemma RTLabs_not_cost : ∀ge,s. |
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| 61 | RTLabs_cost s = false → |
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| 62 | ¬ as_costed (RTLabs_status ge) s. |
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| 63 | #ge #s #E % whd in ⊢ (% → ?); >E #E' destruct |
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| 64 | qed. |
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| 65 | |
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[1559] | 66 | (* Before attempting to construct a structured trace, let's show that we can |
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| 67 | form flat traces with evidence that they were constructed from an execution. |
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| 68 | |
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| 69 | For now we don't consider I/O. *) |
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| 70 | |
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| 71 | |
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| 72 | coinductive exec_no_io (o:Type[0]) (i:o → Type[0]) : execution state o i → Prop ≝ |
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| 73 | | noio_stop : ∀a,b,c. exec_no_io o i (e_stop … a b c) |
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| 74 | | noio_step : ∀a,b,e. exec_no_io o i e → exec_no_io o i (e_step … a b e) |
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| 75 | | noio_wrong : ∀m. exec_no_io o i (e_wrong … m). |
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| 76 | |
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| 77 | (* add I/O? *) |
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| 78 | coinductive flat_trace (o:Type[0]) (i:o → Type[0]) (ge:genv) : state → Type[0] ≝ |
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| 79 | | ft_stop : ∀s. RTLabs_is_final s ≠ None ? → flat_trace o i ge s |
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| 80 | | ft_step : ∀s,tr,s'. eval_statement ge s = Value ??? 〈tr,s'〉 → flat_trace o i ge s' → flat_trace o i ge s |
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| 81 | | ft_wrong : ∀s,m. eval_statement ge s = Wrong ??? m → flat_trace o i ge s. |
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| 82 | |
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[1707] | 83 | coinductive not_wrong (o:Type[0]) (i:o → Type[0]) (ge:genv) : ∀s. flat_trace o i ge s → Type[0] ≝ |
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| 84 | | nw_stop : ∀s,H. not_wrong o i ge s (ft_stop o i ge s H) |
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| 85 | | nw_step : ∀s,tr,s',H,tr'. not_wrong o i ge s' tr' → not_wrong o i ge s (ft_step o i ge s tr s' H tr'). |
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| 86 | |
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| 87 | lemma still_not_wrong : ∀o,i,ge,s,tr,s',H,tr'. |
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| 88 | not_wrong o i ge s (ft_step o i ge s tr s' H tr') → |
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| 89 | not_wrong o i ge s' tr'. |
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| 90 | #o #i #ge #s #tr #s' #H #tr' #NW inversion NW |
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| 91 | [ #H105 #H106 #H107 #H108 #H109 destruct |
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| 92 | | #H111 #H112 #H113 #H114 #H115 #H116 #H117 #H118 #H119 destruct // |
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| 93 | ] qed. |
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| 94 | |
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[1559] | 95 | let corec make_flat_trace ge s |
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| 96 | (H:exec_no_io … (exec_inf_aux … RTLabs_fullexec ge (eval_statement ge s))) : |
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| 97 | flat_trace io_out io_in ge s ≝ |
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| 98 | let e ≝ exec_inf_aux … RTLabs_fullexec ge (eval_statement ge s) in |
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| 99 | match e return λx. e = x → ? with |
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| 100 | [ e_stop tr i s' ⇒ λE. ft_step … s tr s' ? (ft_stop … s' ?) |
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| 101 | | e_step tr s' e' ⇒ λE. ft_step … s tr s' ? (make_flat_trace ge s' ?) |
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| 102 | | e_wrong m ⇒ λE. ft_wrong … s m ? |
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| 103 | | e_interact o f ⇒ λE. ⊥ |
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| 104 | ] (refl ? e). |
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| 105 | [ 1,2: whd in E:(??%?); >exec_inf_aux_unfold in E; |
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| 106 | cases (eval_statement ge s) |
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| 107 | [ 1,4: #O #K whd in ⊢ (??%? → ?); #E destruct |
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| 108 | | 2,5: * #tr #s1 whd in ⊢ (??%? → ?); |
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| 109 | >(?:is_final ????? = RTLabs_is_final s1) // |
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| 110 | lapply (refl ? (RTLabs_is_final s1)) |
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| 111 | cases (RTLabs_is_final s1) in ⊢ (???% → %); |
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| 112 | [ 1,3: #_ whd in ⊢ (??%? → ?); #E destruct |
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| 113 | | #i #_ whd in ⊢ (??%? → ?); #E destruct /2/ @refl |
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| 114 | | #i #E whd in ⊢ (??%? → ?); #E2 destruct >E % #E' destruct |
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| 115 | ] |
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| 116 | | *: #m whd in ⊢ (??%? → ?); #E destruct |
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| 117 | ] |
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| 118 | | whd in E:(??%?); >exec_inf_aux_unfold in E; |
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| 119 | cases (eval_statement ge s) |
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| 120 | [ #O #K whd in ⊢ (??%? → ?); #E destruct |
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| 121 | | * #tr #s1 whd in ⊢ (??%? → ?); |
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| 122 | cases (is_final ?????) |
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| 123 | [ whd in ⊢ (??%? → ?); #E destruct @refl |
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| 124 | | #i whd in ⊢ (??%? → ?); #E destruct |
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| 125 | ] |
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| 126 | | #m whd in ⊢ (??%? → ?); #E destruct |
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| 127 | ] |
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| 128 | | whd in E:(??%?); >E in H; #H >exec_inf_aux_unfold in E; |
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| 129 | cases (eval_statement ge s) |
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| 130 | [ #O #K whd in ⊢ (??%? → ?); #E destruct |
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| 131 | | * #tr #s1 whd in ⊢ (??%? → ?); |
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| 132 | cases (is_final ?????) |
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| 133 | [ whd in ⊢ (??%? → ?); #E |
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| 134 | change with (eval_statement ge s1) in E:(??(??????(?????%))?); |
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| 135 | destruct |
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| 136 | inversion H |
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| 137 | [ #a #b #c #E1 destruct |
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| 138 | | #trx #sx #ex #H1 #E2 #E3 destruct @H1 |
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| 139 | | #m #E1 destruct |
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| 140 | ] |
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| 141 | | #i whd in ⊢ (??%? → ?); #E destruct |
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| 142 | ] |
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| 143 | | #m whd in ⊢ (??%? → ?); #E destruct |
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| 144 | ] |
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| 145 | | whd in E:(??%?); >exec_inf_aux_unfold in E; |
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| 146 | cases (eval_statement ge s) |
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| 147 | [ #O #K whd in ⊢ (??%? → ?); #E destruct |
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| 148 | | * #tr1 #s1 whd in ⊢ (??%? → ?); |
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| 149 | cases (is_final ?????) |
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| 150 | [ whd in ⊢ (??%? → ?); #E destruct |
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| 151 | | #i whd in ⊢ (??%? → ?); #E destruct |
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| 152 | ] |
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| 153 | | #m whd in ⊢ (??%? → ?); #E destruct @refl |
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| 154 | ] |
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| 155 | | whd in E:(??%?); >E in H; #H |
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| 156 | inversion H |
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| 157 | [ #a #b #c #E destruct |
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| 158 | | #a #b #c #d #E1 destruct |
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| 159 | | #m #E1 destruct |
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| 160 | ] |
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| 161 | ] qed. |
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| 162 | |
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| 163 | let corec make_whole_flat_trace p s |
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| 164 | (H:exec_no_io … (exec_inf … RTLabs_fullexec p)) |
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| 165 | (I:make_initial_state ??? p = OK ? s) : |
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| 166 | flat_trace io_out io_in (make_global … RTLabs_fullexec p) s ≝ |
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| 167 | let ge ≝ make_global … p in |
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| 168 | let e ≝ exec_inf_aux ?? RTLabs_fullexec ge (Value … 〈E0, s〉) in |
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| 169 | match e return λx. e = x → ? with |
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| 170 | [ e_stop tr i s' ⇒ λE. ft_stop ?? ge s ? |
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| 171 | | e_step _ _ e' ⇒ λE. make_flat_trace ge s ? |
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| 172 | | e_wrong m ⇒ λE. ⊥ |
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| 173 | | e_interact o f ⇒ λE. ⊥ |
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| 174 | ] (refl ? e). |
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| 175 | [ whd in E:(??%?); >exec_inf_aux_unfold in E; |
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| 176 | whd in ⊢ (??%? → ?); |
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| 177 | >(?:is_final ????? = RTLabs_is_final s) // |
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| 178 | lapply (refl ? (RTLabs_is_final s)) |
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| 179 | cases (RTLabs_is_final s) in ⊢ (???% → %); |
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| 180 | [ #_ whd in ⊢ (??%? → ?); #E destruct |
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| 181 | | #i #E whd in ⊢ (??%? → ?); #E2 % #E3 destruct |
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| 182 | ] |
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| 183 | | whd in H:(???%); >I in H; whd in ⊢ (???% → ?); whd in E:(??%?); |
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| 184 | >exec_inf_aux_unfold in E ⊢ %; whd in ⊢ (??%? → ???% → ?); cases (is_final ?????) |
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| 185 | [ whd in ⊢ (??%? → ???% → ?); #E #H inversion H |
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| 186 | [ #a #b #c #E1 destruct |
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| 187 | | #tr1 #s1 #e1 #H1 #E1 #E2 -E2 -I destruct (E1) |
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| 188 | @H1 |
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| 189 | | #m #E1 destruct |
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| 190 | ] |
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| 191 | | #i whd in ⊢ (??%? → ???% → ?); #E destruct |
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| 192 | ] |
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| 193 | | whd in E:(??%?); >exec_inf_aux_unfold in E; whd in ⊢ (??%? → ?); |
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| 194 | cases (is_final ?????) [2:#i] whd in ⊢ (??%? → ?); #E destruct |
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| 195 | | whd in E:(??%?); >exec_inf_aux_unfold in E; whd in ⊢ (??%? → ?); |
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| 196 | cases (is_final ?????) [2:#i] whd in ⊢ (??%? → ?); #E destruct |
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| 197 | ] qed. |
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| 198 | |
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[1563] | 199 | (* Need a way to choose whether a called function terminates. Then, |
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| 200 | if the initial function terminates we generate a purely inductive structured trace, |
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| 201 | otherwise we start generating the coinductive one, and on every function call |
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| 202 | use the choice method again to decide whether to step over or keep going. |
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| 203 | |
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| 204 | Not quite what we need - have to decide on seeing each label whether we will see |
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| 205 | another or hit a non-terminating call? |
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| 206 | |
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| 207 | Also - need the notion of well-labelled in order to break loops. |
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| 208 | |
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| 209 | |
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| 210 | |
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| 211 | outline: |
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| 212 | |
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| 213 | does function terminate? |
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| 214 | - yes, get (bound on the number of steps until return), generate finite |
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| 215 | structure using bound as termination witness |
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| 216 | - no, get (¬ bound on steps to return), start building infinite trace out of |
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| 217 | finite steps. At calls, check for termination, generate appr. form. |
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| 218 | |
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| 219 | generating the finite parts: |
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| 220 | |
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| 221 | We start with the status after the call has been executed; well-labelling tells |
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| 222 | us that this is a labelled state. Now we want to generate a trace_label_return |
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| 223 | and also return the remainder of the flat trace. |
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| 224 | |
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| 225 | *) |
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| 226 | |
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[1595] | 227 | (* [will_return ge depth s trace] says that after a finite number of steps of |
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| 228 | [trace] from [s] we reach the return state for the current function. [depth] |
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| 229 | performs the call/return counting necessary for handling deeper function |
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| 230 | calls. It should be zero at the top level. *) |
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[1637] | 231 | inductive will_return (ge:genv) : nat → ∀s. flat_trace io_out io_in ge s → Type[0] ≝ |
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[1595] | 232 | | wr_step : ∀s,tr,s',depth,EX,trace. |
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[1565] | 233 | RTLabs_classify s = cl_other ∨ RTLabs_classify s = cl_jump → |
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[1595] | 234 | will_return ge depth s' trace → |
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| 235 | will_return ge depth s (ft_step ?? ge s tr s' EX trace) |
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| 236 | | wr_call : ∀s,tr,s',depth,EX,trace. |
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[1563] | 237 | RTLabs_classify s = cl_call → |
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[1595] | 238 | will_return ge (S depth) s' trace → |
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| 239 | will_return ge depth s (ft_step ?? ge s tr s' EX trace) |
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| 240 | | wr_ret : ∀s,tr,s',depth,EX,trace. |
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[1563] | 241 | RTLabs_classify s = cl_return → |
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[1595] | 242 | will_return ge depth s' trace → |
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| 243 | will_return ge (S depth) s (ft_step ?? ge s tr s' EX trace) |
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[1583] | 244 | (* Note that we require the ability to make a step after the return (this |
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| 245 | corresponds to somewhere that will be guaranteed to be a label at the |
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| 246 | end of the compilation chain). *) |
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[1595] | 247 | | wr_base : ∀s,tr,s',EX,trace. |
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[1563] | 248 | RTLabs_classify s = cl_return → |
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[1595] | 249 | will_return ge O s (ft_step ?? ge s tr s' EX trace) |
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[1563] | 250 | . |
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| 251 | |
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[1638] | 252 | (* The way we will use [will_return] won't satisfy Matita's guardedness check, |
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| 253 | so we will measure the length of these termination proofs and use an upper |
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| 254 | bound to show termination of the finite structured trace construction |
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| 255 | functions. *) |
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| 256 | |
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[1637] | 257 | let rec will_return_length ge d s tr (T:will_return ge d s tr) on T : nat ≝ |
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| 258 | match T with |
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| 259 | [ wr_step _ _ _ _ _ _ _ T' ⇒ S (will_return_length … T') |
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| 260 | | wr_call _ _ _ _ _ _ _ T' ⇒ S (will_return_length … T') |
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| 261 | | wr_ret _ _ _ _ _ _ _ T' ⇒ S (will_return_length … T') |
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| 262 | | wr_base _ _ _ _ _ _ ⇒ S O |
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| 263 | ]. |
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[1638] | 264 | |
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[1637] | 265 | include alias "arithmetics/nat.ma". |
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| 266 | |
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[1638] | 267 | (* Specialised to the particular situation it is used in. *) |
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[1637] | 268 | lemma wrl_nonzero : ∀ge,d,s,tr,T. O ≥ 3 * (will_return_length ge d s tr T) → False. |
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| 269 | #ge #d #s #tr * #s1 #tr1 #s2 [ 1,2,3: #d ] #EX #tr' #CL [1,2,3:#IH] |
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| 270 | whd in ⊢ (??(??%) → ?); |
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| 271 | >commutative_times |
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| 272 | #H lapply (le_plus_b … H) |
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| 273 | #H lapply (le_to_leb_true … H) |
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| 274 | normalize #E destruct |
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| 275 | qed. |
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[1719] | 276 | |
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| 277 | let rec will_return_end ge d s tr (T:will_return ge d s tr) on T : 𝚺s'.flat_trace io_out io_in ge s' ≝ |
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| 278 | match T with |
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| 279 | [ wr_step _ _ _ _ _ _ _ T' ⇒ will_return_end … T' |
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| 280 | | wr_call _ _ _ _ _ _ _ T' ⇒ will_return_end … T' |
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| 281 | | wr_ret _ _ _ _ _ _ _ T' ⇒ will_return_end … T' |
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| 282 | | wr_base _ _ _ _ tr' _ ⇒ mk_DPair ? (λs.flat_trace io_out io_in ge s) ? tr' |
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| 283 | ]. |
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[1563] | 284 | |
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[1638] | 285 | (* Inversion lemmas on [will_return] that also note the effect on the length |
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| 286 | of the proof. *) |
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| 287 | lemma will_return_call : ∀ge,d,s,tr,s',EX,trace. |
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[1637] | 288 | RTLabs_classify s = cl_call → |
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| 289 | ∀TM:will_return ge d s (ft_step ?? ge s tr s' EX trace). |
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[1719] | 290 | ΣTM':will_return ge (S d) s' trace. will_return_length … TM > will_return_length … TM' ∧ will_return_end … TM = will_return_end … TM'. |
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[1637] | 291 | #ge #d #s #tr #s' #EX #trace #CL #TERM inversion TERM |
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| 292 | [ #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 #H29 #H30 #H31 @⊥ destruct >CL in H25; * #E destruct |
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[1719] | 293 | | #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 #H42 #H43 #H44 #H45 destruct % /2/ |
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[1637] | 294 | | #H47 #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 #H56 #H57 #H58 #H59 @⊥ destruct >CL in H53; #E destruct |
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| 295 | | #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 #H70 @⊥ destruct >CL in H66; #E destruct |
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| 296 | ] qed. |
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[1595] | 297 | |
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[1637] | 298 | lemma will_return_return : ∀ge,d,s,tr,s',EX,trace. |
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| 299 | RTLabs_classify s = cl_return → |
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| 300 | ∀TM:will_return ge d s (ft_step ?? ge s tr s' EX trace). |
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| 301 | match d with |
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[1719] | 302 | [ O ⇒ will_return_end … TM = ❬s', trace❭ |
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[1637] | 303 | | S d' ⇒ |
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[1719] | 304 | ΣTM':will_return ge d' s' trace. will_return_length … TM > will_return_length … TM' ∧ will_return_end … TM = will_return_end … TM' |
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[1637] | 305 | ]. |
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| 306 | #ge #d #s #tr #s' #EX #trace #CL #TERM inversion TERM |
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| 307 | [ #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 #H29 #H30 #H31 @⊥ destruct >CL in H25; * #E destruct |
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| 308 | | #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 #H42 #H43 #H44 #H45 @⊥ destruct >CL in H39; #E destruct |
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[1719] | 309 | | #H47 #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 #H56 #H57 #H58 #H59 destruct % /2/ |
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| 310 | | #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 #H70 destruct @refl |
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[1637] | 311 | ] qed. |
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| 312 | |
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[1596] | 313 | lemma will_return_notfn : ∀ge,d,s,tr,s',EX,trace. |
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[1637] | 314 | (RTLabs_classify s = cl_other) ⊎ (RTLabs_classify s = cl_jump) → |
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| 315 | ∀TM:will_return ge d s (ft_step ?? ge s tr s' EX trace). |
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[1719] | 316 | ΣTM':will_return ge d s' trace. will_return_length … TM > will_return_length … TM' ∧ will_return_end … TM = will_return_end … TM'. |
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[1596] | 317 | #ge #d #s #tr #s' #EX #trace * #CL #TERM inversion TERM |
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[1719] | 318 | [ #H290 #H291 #H292 #H293 #H294 #H295 #H296 #H297 #H298 #H299 #H300 #H301 #H302 destruct % /2/ |
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[1637] | 319 | | #H304 #H305 #H306 #H307 #H308 #H309 #H310 #H311 #H312 #H313 #H314 #H315 #H316 @⊥ destruct >CL in H310; #E destruct |
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| 320 | | #H318 #H319 #H320 #H321 #H322 #H323 #H324 #H325 #H326 #H327 #H328 #H329 #H330 @⊥ destruct >CL in H324; #E destruct |
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| 321 | | #H332 #H333 #H334 #H335 #H336 #H337 #H338 #H339 #H340 #H341 @⊥ destruct >CL in H337; #E destruct |
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[1719] | 322 | | #H343 #H344 #H345 #H346 #H347 #H348 #H349 #H350 #H351 #H352 #H353 #H354 #H355 destruct % /2/ |
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[1637] | 323 | | #H357 #H358 #H359 #H360 #H361 #H362 #H363 #H364 #H365 #H366 #H367 #H368 #H369 @⊥ destruct >CL in H363; #E destruct |
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| 324 | | #H371 #H372 #H373 #H374 #H375 #H376 #H377 #H378 #H379 #H380 #H381 #H382 #H383 @⊥ destruct >CL in H377; #E destruct |
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| 325 | | #H385 #H386 #H387 #H388 #H389 #H390 #H391 #H392 #H393 #H394 @⊥ destruct >CL in H390; #E destruct |
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[1595] | 326 | ] qed. |
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| 327 | |
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[1719] | 328 | (* When it comes to building bits of nonterminating executions we'll need to be |
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| 329 | able to glue termination proofs together. *) |
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| 330 | |
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| 331 | lemma will_return_prepend : ∀ge,d1,s1,t1. |
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| 332 | ∀T1:will_return ge d1 s1 t1. |
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| 333 | ∀d2,s2,t2. |
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| 334 | will_return ge d2 s2 t2 → |
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| 335 | will_return_end … T1 = ❬s2, t2❭ → |
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| 336 | will_return ge (d1 + S d2) s1 t1. |
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| 337 | #ge #d1 #s1 #tr1 #T1 elim T1 |
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| 338 | [ #s #tr #s' #depth #EX #t #CL #T #IH #d2 #s2 #t2 #T2 #E |
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| 339 | %1 // @(IH … T2) @E |
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| 340 | | #s #tr #s' #depth #EX #t #CL #T #IH #d2 #s2 #t2 #T2 #E %2 // @(IH … T2) @E |
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| 341 | | #s #tr #s' #depth #EX #t #CL #T #IH #s2 #s2 #t2 #T2 #E %3 // @(IH … T2) @E |
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| 342 | | #s #tr #s' #EX #t #CL #d2 #s2 #t2 #T2 #E normalize in E; destruct %3 // |
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| 343 | ] qed. |
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| 344 | |
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| 345 | discriminator nat. |
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| 346 | |
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| 347 | lemma will_return_remove_call : ∀ge,d1,s1,t1. |
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| 348 | ∀T1:will_return ge d1 s1 t1. |
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| 349 | ∀d2. |
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| 350 | will_return ge (d1 + S d2) s1 t1 → |
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| 351 | ∀s2,t2. |
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| 352 | will_return_end … T1 = ❬s2, t2❭ → |
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| 353 | will_return ge d2 s2 t2. |
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| 354 | (* The key part of the proof is to show that the two termination proofs follow |
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| 355 | the same pattern. *) |
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| 356 | #ge #d1x #s1x #t1x #T1 elim T1 |
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| 357 | [ #s #tr #s' #d1 #EX #t #CL #T #IH #d2 #T2 #s2 #t2 #E @IH |
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| 358 | [ inversion T2 [ #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 #H13 destruct // |
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| 359 | | #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 @⊥ destruct |
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| 360 | >H21 in CL; * #E destruct |
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| 361 | | #H29 #H30 #H31 #H32 #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 @⊥ destruct |
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| 362 | >H35 in CL; * #E destruct |
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| 363 | | #H43 #H44 #H45 #H46 #H47 #H48 #H49 #H50 #H51 #H52 @⊥ destruct |
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| 364 | >H48 in CL; * #E destruct |
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| 365 | ] |
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| 366 | | @E |
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| 367 | ] |
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| 368 | | #s #tr #s' #d1 #EX #t #CL #T #IH #d2 #T2 #s2 #t2 #E @IH |
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| 369 | [ inversion T2 [ #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 #H13 @⊥ destruct |
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| 370 | >CL in H7; * #E destruct |
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| 371 | | #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 destruct // |
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| 372 | | #H29 #H30 #H31 #H32 #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 @⊥ destruct |
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| 373 | >H35 in CL; #E destruct |
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| 374 | | #H43 #H44 #H45 #H46 #H47 #H48 #H49 #H50 #H51 #H52 @⊥ destruct |
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| 375 | >H48 in CL; #E destruct |
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| 376 | ] |
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| 377 | | @E |
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| 378 | ] |
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| 379 | | #s #tr #s' #d1 #EX #t #CL #T #IH #d2 #T2 #s2 #t2 #E @IH |
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| 380 | [ inversion T2 [ #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 #H13 @⊥ destruct |
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| 381 | >CL in H7; * #E destruct |
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| 382 | | #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 @⊥ destruct |
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| 383 | >H21 in CL; #E destruct |
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| 384 | | #H29 #H30 #H31 #H32 #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 |
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| 385 | whd in H38:(??%??); destruct // |
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| 386 | | #H43 #H44 #H45 #H46 #H47 #H48 #H49 #H50 #H51 #H52 |
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| 387 | whd in H49:(??%??); @⊥ destruct |
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| 388 | ] |
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| 389 | | @E |
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| 390 | ] |
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| 391 | | #s #tr #s' #EX #t #CL #d2 #T2 #s2 #t2 #E whd in E:(??%?); destruct |
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| 392 | inversion T2 [ #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 #H13 @⊥ destruct |
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| 393 | >CL in H7; * #E destruct |
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| 394 | | #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 @⊥ destruct |
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| 395 | >H21 in CL; #E destruct |
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| 396 | | #H29 #H30 #H31 #H32 #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 |
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| 397 | whd in H38:(??%??); destruct // |
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| 398 | | #H43 #H44 #H45 #H46 #H47 #H48 #H49 #H50 #H51 #H52 |
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| 399 | whd in H49:(??%??); @⊥ destruct |
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| 400 | ] |
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| 401 | ] qed. |
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| 402 | |
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[1764] | 403 | lemma will_return_not_wrong : ∀ge,d,s,t,s',t'. |
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| 404 | ∀T:will_return ge d s t. |
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| 405 | not_wrong io_out io_in ge s t → |
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| 406 | will_return_end … T = ❬s', t'❭ → |
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| 407 | not_wrong io_out io_in ge s' t'. |
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| 408 | #ge #d #s #t #s' #t' #T elim T |
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| 409 | [ #s #tr #s' #d #EV #t1 #CL #T' #IH #NW #E @IH |
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| 410 | [ inversion NW |
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| 411 | [ #H1 #H2 #H3 #H4 #H5 destruct |
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| 412 | | #H7 #H8 #H9 #H10 #H11 #H12 #H13 #H14 #H15 destruct // |
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| 413 | ] |
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| 414 | | @E |
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| 415 | ] |
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| 416 | | #s #tr #s' #d #EV #t1 #CL #T' #IH #NW #E @IH |
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| 417 | [ inversion NW [ #H1 #H2 #H3 #H4 #H5 destruct | #H7 #H8 #H9 #H10 #H11 #H12 #H13 #H14 #H15 destruct // ] |
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| 418 | | @E |
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| 419 | ] |
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| 420 | | #s #tr #s' #d #EV #t1 #CL #T' #IH #NW #E @IH |
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| 421 | [ inversion NW [ #H1 #H2 #H3 #H4 #H5 destruct | #H7 #H8 #H9 #H10 #H11 #H12 #H13 #H14 #H15 destruct // ] |
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| 422 | | @E |
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| 423 | ] |
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| 424 | | #s #tr #s' #d #t1 #CL #NW #E normalize in E; destruct |
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| 425 | inversion NW [ #H1 #H2 #H3 #H4 #H5 destruct | #H7 #H8 #H9 #H10 #H11 #H12 #H13 #H14 #H15 destruct // ] |
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| 426 | ] qed. |
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| 427 | |
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[1806] | 428 | |
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| 429 | lemma will_return_lower : ∀ge,d,s,t. |
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| 430 | will_return ge d s t → |
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| 431 | ∀d'. d' ≤ d → |
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| 432 | will_return ge d' s t. |
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| 433 | #ge #d0 #s0 #t0 #TM elim TM |
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| 434 | [ #s #tr #s' #d #EX #tr #CL #TM1 #IH #d' #LE % /2/ |
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| 435 | | #s #tr #s' #d #EX #tr #CL #TM1 #IH #d' #LE %2 // @IH /2/ |
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| 436 | | #s #tr #s' #d #EX #tr #CL #TM1 #IH * |
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| 437 | [ #LE @wr_base // |
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| 438 | | #d' #LE %3 // @IH /2/ |
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| 439 | ] |
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| 440 | | #s #tr #s' #EX #tr #CL * |
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| 441 | [ #_ @wr_base // |
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| 442 | | #d' #LE @⊥ /2/ |
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| 443 | ] |
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| 444 | ] qed. |
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| 445 | |
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[1565] | 446 | (* We require that labels appear after branch instructions and at the start of |
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[1574] | 447 | functions. The first is required for preciseness, the latter for soundness. |
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| 448 | We will make a separate requirement for there to be a finite number of steps |
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| 449 | between labels to catch loops for soundness (is this sufficient?). *) |
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[1565] | 450 | |
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| 451 | definition well_cost_labelled_statement : ∀f:internal_function. ∀s. labels_present (f_graph f) s → Prop ≝ |
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| 452 | λf,s. match s return λs. labels_present ? s → Prop with |
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| 453 | [ St_cond _ l1 l2 ⇒ λH. |
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[1586] | 454 | is_cost_label (lookup_present … (f_graph f) l1 ?) = true ∧ |
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| 455 | is_cost_label (lookup_present … (f_graph f) l2 ?) = true |
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[1565] | 456 | | St_jumptable _ ls ⇒ λH. |
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[1586] | 457 | (* I did have a dependent version of All here, but it's a pain. *) |
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| 458 | All … (λl. ∃H. is_cost_label (lookup_present … (f_graph f) l H) = true) ls |
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[1565] | 459 | | _ ⇒ λ_. True |
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| 460 | ]. whd in H; |
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| 461 | [ @(proj1 … H) |
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| 462 | | @(proj2 … H) |
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| 463 | ] qed. |
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| 464 | |
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| 465 | definition well_cost_labelled_fn : internal_function → Prop ≝ |
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[1586] | 466 | λf. (∀l. ∀H:present … (f_graph f) l. |
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| 467 | well_cost_labelled_statement f (lookup_present … (f_graph f) l H) (f_closed f l …)) ∧ |
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| 468 | is_cost_label (lookup_present … (f_graph f) (f_entry f) ?) = true. |
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| 469 | [ @lookup_lookup_present | cases (f_entry f) // ] qed. |
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[1565] | 470 | |
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| 471 | (* We need to ensure that any code we come across is well-cost-labelled. We may |
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| 472 | get function code from either the global environment or the state. *) |
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| 473 | |
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| 474 | definition well_cost_labelled_ge : genv → Prop ≝ |
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[1583] | 475 | λge. ∀b,f. find_funct_ptr ?? ge b = Some ? (Internal ? f) → well_cost_labelled_fn f. |
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[1565] | 476 | |
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| 477 | definition well_cost_labelled_state : state → Prop ≝ |
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| 478 | λs. match s with |
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| 479 | [ State f fs m ⇒ well_cost_labelled_fn (func f) ∧ All ? (λf. well_cost_labelled_fn (func f)) fs |
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| 480 | | Callstate fd _ _ fs _ ⇒ match fd with [ Internal fn ⇒ well_cost_labelled_fn fn | External _ ⇒ True ] ∧ |
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| 481 | All ? (λf. well_cost_labelled_fn (func f)) fs |
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| 482 | | Returnstate _ _ fs _ ⇒ All ? (λf. well_cost_labelled_fn (func f)) fs |
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[1713] | 483 | | Finalstate _ ⇒ True |
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[1565] | 484 | ]. |
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| 485 | |
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[1583] | 486 | lemma well_cost_labelled_state_step : ∀ge,s,tr,s'. |
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| 487 | eval_statement ge s = Value ??? 〈tr,s'〉 → |
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| 488 | well_cost_labelled_ge ge → |
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| 489 | well_cost_labelled_state s → |
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| 490 | well_cost_labelled_state s'. |
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| 491 | #ge #s #tr' #s' #EV cases (eval_perserves … EV) |
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| 492 | [ #ge #f #f' #fs #m #m' * #func #locals #next #next_ok #sp #retdst #locals' #next' #next_ok' #Hge * #H1 #H2 % // |
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| 493 | | #ge #f #fs #m * #fn #args #f' #dst * #func #locals #next #next_ok #sp #retdst #locals' #next' #next_ok' #b #Hfn #Hge * #H1 #H2 % /2/ |
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[1681] | 494 | (* |
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[1583] | 495 | | #ge #f #fs #m * #fn #args #f' #dst #m' #b #Hge * #H1 #H2 % /2/ |
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[1681] | 496 | *) |
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[1583] | 497 | | #ge #fn #locals #next #nok #sp #fs #m #args #dst #m' #Hge * #H1 #H2 % /2/ |
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| 498 | | #ge #f #fs #m #rtv #dst #m' #Hge * #H1 #H2 @H2 |
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| 499 | | #ge #f #fs #rtv #dst #f' #m * #func #locals #next #nok #sp #retdst #locals' #next' #nok' #Hge * #H1 #H2 % // |
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[1713] | 500 | | // |
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[1583] | 501 | ] qed. |
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| 502 | |
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[1586] | 503 | lemma rtlabs_jump_inv : ∀s. |
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| 504 | RTLabs_classify s = cl_jump → |
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| 505 | ∃f,fs,m. s = State f fs m ∧ |
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| 506 | let stmt ≝ lookup_present ?? (f_graph (func f)) (next f) (next_ok f) in |
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| 507 | (∃r,l1,l2. stmt = St_cond r l1 l2) ∨ (∃r,ls. stmt = St_jumptable r ls). |
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| 508 | * |
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| 509 | [ #f #fs #m #E |
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| 510 | %{f} %{fs} %{m} % |
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| 511 | [ @refl |
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| 512 | | whd in E:(??%?); cases (lookup_present ? statement ???) in E ⊢ %; |
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| 513 | try (normalize try #A try #B try #C try #D try #F try #G try #H destruct) |
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| 514 | [ %1 %{A} %{B} %{C} @refl |
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| 515 | | %2 %{A} %{B} @refl |
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| 516 | ] |
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| 517 | ] |
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| 518 | | normalize #H1 #H2 #H3 #H4 #H5 #H6 destruct |
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| 519 | | normalize #H8 #H9 #H10 #H11 #H12 destruct |
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[1713] | 520 | | #r #E normalize in E; destruct |
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[1586] | 521 | ] qed. |
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| 522 | |
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| 523 | lemma well_cost_labelled_jump : ∀ge,s,tr,s'. |
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| 524 | eval_statement ge s = Value ??? 〈tr,s'〉 → |
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| 525 | well_cost_labelled_state s → |
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| 526 | RTLabs_classify s = cl_jump → |
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| 527 | RTLabs_cost s' = true. |
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| 528 | #ge #s #tr #s' #EV #H #CL |
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| 529 | cases (rtlabs_jump_inv s CL) |
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| 530 | #fr * #fs * #m * #Es * |
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| 531 | [ * #r * #l1 * #l2 #Estmt |
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| 532 | >Es in H; whd in ⊢ (% → ?); * * #Hbody #_ #Hfs |
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| 533 | >Es in EV; whd in ⊢ (??%? → ?); generalize in ⊢ (??(?%)? → ?); |
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| 534 | >Estmt #LP whd in ⊢ (??%? → ?); |
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| 535 | (* replace with lemma on successors? *) |
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[1656] | 536 | @bind_value #v #Ev @bind_ok * #Eb whd in ⊢ (??%? → ?); #E destruct |
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[1586] | 537 | lapply (Hbody (next fr) (next_ok fr)) |
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| 538 | generalize in ⊢ (???% → ?); |
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| 539 | >Estmt #LP' |
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| 540 | whd in ⊢ (% → ?); |
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| 541 | * #H1 #H2 [ @H1 | @H2 ] |
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| 542 | | * #r * #ls #Estmt |
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| 543 | >Es in H; whd in ⊢ (% → ?); * * #Hbody #_ #Hfs |
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| 544 | >Es in EV; whd in ⊢ (??%? → ?); generalize in ⊢ (??(?%)? → ?); |
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| 545 | >Estmt #LP whd in ⊢ (??%? → ?); |
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| 546 | (* replace with lemma on successors? *) |
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[1656] | 547 | @bind_value #a cases a [ | #sz #i | #f | #r | #ptr ] #Ea whd in ⊢ (??%? → ?); |
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[1586] | 548 | [ 2: (* later *) |
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| 549 | | *: #E destruct |
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| 550 | ] |
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| 551 | lapply (Hbody (next fr) (next_ok fr)) |
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| 552 | generalize in ⊢ (???% → ?); >Estmt #LP' whd in ⊢ (% → ?); #CP |
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| 553 | generalize in ⊢ (??(?%)? → ?); |
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| 554 | cases (nth_opt label (nat_of_bitvector (bitsize_of_intsize sz) i) ls) in ⊢ (???% → ??(match % with [_⇒?|_⇒?]?)? → ?); |
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| 555 | [ #E1 #E2 whd in E2:(??%?); destruct |
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| 556 | | #l' #E1 #E2 whd in E2:(??%?); destruct |
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| 557 | cases (All_nth ???? CP ? E1) |
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| 558 | #H1 #H2 @H2 |
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| 559 | ] |
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| 560 | ] qed. |
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| 561 | |
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[1595] | 562 | lemma rtlabs_call_inv : ∀s. |
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| 563 | RTLabs_classify s = cl_call → |
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| 564 | ∃fd,args,dst,stk,m. s = Callstate fd args dst stk m. |
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| 565 | * [ #f #fs #m whd in ⊢ (??%? → ?); |
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| 566 | cases (lookup_present … (next f) (next_ok f)) normalize |
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| 567 | try #A try #B try #C try #D try #E try #F try #G destruct |
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| 568 | | #fd #args #dst #stk #m #E %{fd} %{args} %{dst} %{stk} %{m} @refl |
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| 569 | | normalize #H411 #H412 #H413 #H414 #H415 destruct |
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[1713] | 570 | | normalize #H1 #H2 destruct |
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[1595] | 571 | ] qed. |
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[1586] | 572 | |
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[1595] | 573 | lemma well_cost_labelled_call : ∀ge,s,tr,s'. |
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| 574 | eval_statement ge s = Value ??? 〈tr,s'〉 → |
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| 575 | well_cost_labelled_state s → |
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| 576 | RTLabs_classify s = cl_call → |
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| 577 | RTLabs_cost s' = true. |
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| 578 | #ge #s #tr #s' #EV #WCL #CL |
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| 579 | cases (rtlabs_call_inv s CL) |
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| 580 | #fd * #args * #dst * #stk * #m #E >E in EV WCL; |
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| 581 | whd in ⊢ (??%? → % → ?); |
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| 582 | cases fd |
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| 583 | [ #fn whd in ⊢ (??%? → % → ?); |
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[1656] | 584 | @bind_value #lcl #Elcl cases (alloc m O (f_stacksize fn) Any) |
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[1595] | 585 | #m' #b whd in ⊢ (??%? → ?); #E' destruct |
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| 586 | * whd in ⊢ (% → ?); * #WCL1 #WCL2 #WCL3 |
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| 587 | @WCL2 |
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| 588 | | #fn whd in ⊢ (??%? → % → ?); |
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| 589 | @bindIO_value #evargs #Eargs |
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[1656] | 590 | whd in ⊢ (??%? → ?); |
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| 591 | #E' destruct |
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[1595] | 592 | ] qed. |
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| 593 | |
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[1681] | 594 | |
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| 595 | (* The preservation of (most of) the stack is useful to show as_after_return. |
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[1682] | 596 | We do this by showing that during the execution of a function the lower stack |
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| 597 | frames never change, and that after returning from the function we preserve |
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| 598 | the identity of the next instruction to execute. |
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| 599 | *) |
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| 600 | |
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| 601 | inductive stack_of_state : list frame → state → Prop ≝ |
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| 602 | | sos_State : ∀f,fs,m. stack_of_state fs (State f fs m) |
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| 603 | | sos_Callstate : ∀fd,args,dst,f,fs,m. stack_of_state fs (Callstate fd args dst (f::fs) m) |
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| 604 | | sos_Returnstate : ∀rtv,dst,fs,m. stack_of_state fs (Returnstate rtv dst fs m) |
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| 605 | . |
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| 606 | |
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[1681] | 607 | inductive stack_preserved : trace_ends_with_ret → state → state → Prop ≝ |
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[1682] | 608 | | sp_normal : ∀fs,s1,s2. |
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| 609 | stack_of_state fs s1 → |
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| 610 | stack_of_state fs s2 → |
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| 611 | stack_preserved doesnt_end_with_ret s1 s2 |
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| 612 | | sp_finished : ∀s1,f,f',fs,m. |
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| 613 | next f = next f' → |
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[1736] | 614 | frame_rel f f' → |
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[1682] | 615 | stack_of_state (f::fs) s1 → |
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[1713] | 616 | stack_preserved ends_with_ret s1 (State f' fs m) |
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[1736] | 617 | | sp_stop : ∀s1,r. |
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| 618 | stack_of_state [ ] s1 → |
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| 619 | stack_preserved ends_with_ret s1 (Finalstate r) |
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| 620 | | sp_top : ∀fd,args,dst,m,r. |
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| 621 | stack_preserved doesnt_end_with_ret (Callstate fd args dst [ ] m) (Finalstate r) |
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[1713] | 622 | . |
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[1681] | 623 | |
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[1682] | 624 | discriminator list. |
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[1681] | 625 | |
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[1682] | 626 | lemma stack_of_state_eq : ∀fs,fs',s. |
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| 627 | stack_of_state fs s → |
---|
| 628 | stack_of_state fs' s → |
---|
| 629 | fs = fs'. |
---|
| 630 | #fs0 #fs0' #s0 * |
---|
| 631 | [ #f #fs #m #H inversion H |
---|
[1713] | 632 | #a #b #c #d try #e try #g try #h try #i try #j destruct @refl |
---|
[1682] | 633 | | #fd #args #dst #f #fs #m #H inversion H |
---|
[1713] | 634 | #a #b #c #d try #e try #g try #h try #i try #j destruct @refl |
---|
[1682] | 635 | | #rtv #dst #fs #m #H inversion H |
---|
[1713] | 636 | #a #b #c #d try #e try #g try #h try #i try #j destruct @refl |
---|
[1682] | 637 | ] qed. |
---|
| 638 | |
---|
[1713] | 639 | lemma stack_preserved_final : ∀e,r,s. |
---|
[1736] | 640 | ¬stack_preserved e (Finalstate r) s. |
---|
| 641 | #e #r #s % #H inversion H |
---|
[1713] | 642 | [ #H184 #H185 #H186 #SOS #H188 #H189 #H190 #H191 #H192 destruct |
---|
| 643 | inversion SOS #a #b #c #d #e #f try #g try #h destruct |
---|
[1736] | 644 | | #H194 #H195 #H196 #H197 #H198 #H199 #H200 #SOS #H201 #H202 #H203 #H204 destruct |
---|
[1713] | 645 | inversion SOS #a #b #c #d #e #f try #g try #h destruct |
---|
[1736] | 646 | | #s' #r' #SOS #E1 #E2 #E3 #E4 destruct |
---|
| 647 | inversion SOS #a #b #c #d #e #f try #g try #h destruct |
---|
| 648 | | #H24 #H25 #H26 #H27 #H28 #H29 #H30 #H31 #H32 destruct |
---|
[1713] | 649 | ] qed. |
---|
| 650 | |
---|
[1681] | 651 | lemma stack_preserved_join : ∀e,s1,s2,s3. |
---|
| 652 | stack_preserved doesnt_end_with_ret s1 s2 → |
---|
| 653 | stack_preserved e s2 s3 → |
---|
| 654 | stack_preserved e s1 s3. |
---|
| 655 | #e1 #s1 #s2 #s3 #H1 inversion H1 |
---|
[1682] | 656 | [ #fs #s1' #s2' #S1 #S2 #E1 #E2 #E3 #E4 destruct |
---|
| 657 | #H2 inversion H2 |
---|
| 658 | [ #fs' #s1'' #s2'' #S1' #S2' #E1 #E2 #E3 #E4 destruct |
---|
| 659 | @(sp_normal fs) // <(stack_of_state_eq … S1' S2) // |
---|
[1736] | 660 | | #s1'' #f #f' #fs' #m #N #F #S1' #E1 #E2 #E3 #E4 destruct |
---|
[1682] | 661 | @(sp_finished … N) >(stack_of_state_eq … S1' S2) // |
---|
[1736] | 662 | | #s1'' #r #S1'' #E1 #E2 #E3 #E4 destruct @sp_stop >(stack_of_state_eq … S1'' S2) // |
---|
| 663 | | #fd #args #dst #m #r #E1 #E2 #E3 #E4 destruct |
---|
| 664 | inversion S2 |
---|
| 665 | [ #H34 #H35 #H36 #H37 #H38 #H39 destruct |
---|
| 666 | | #fd' #args' #dst' #f #fs' #m' #E1 #E2 #E3 destruct |
---|
| 667 | | #H41 #H42 #H43 #H44 #H45 #H46 #H47 destruct |
---|
| 668 | ] |
---|
[1681] | 669 | ] |
---|
[1682] | 670 | | #H25 #H26 #H27 #H28 #H29 #H30 #H31 #H32 #H33 #H34 #H35 #H36 destruct |
---|
[1713] | 671 | | #H19 #H20 #H21 #H22 #H23 #H24 #H25 destruct #H |
---|
[1736] | 672 | cases (stack_preserved_final … H) #r #E destruct |
---|
| 673 | | #fd #args #dst #m #r #E1 #E2 #E3 #E4 destruct #F @⊥ |
---|
| 674 | @(absurd … F) // |
---|
[1681] | 675 | ] qed. |
---|
| 676 | |
---|
[1682] | 677 | lemma stack_preserved_return : ∀ge,s1,s2,tr. |
---|
[1681] | 678 | RTLabs_classify s1 = cl_return → |
---|
| 679 | eval_statement ge s1 = Value ??? 〈tr,s2〉 → |
---|
| 680 | stack_preserved ends_with_ret s1 s2. |
---|
| 681 | #ge * |
---|
| 682 | [ #f #fs #m #s2 #tr #E @⊥ whd in E:(??%?); |
---|
| 683 | cases (lookup_present ??? (next f) (next_ok f)) in E; |
---|
| 684 | normalize #a try #b try #c try #d try #e try #f try #g destruct |
---|
| 685 | | #fd #args #dst #fs #m #s2 #tr #E normalize in E; destruct |
---|
| 686 | | #res #dst * |
---|
[1736] | 687 | [ #m #s2 #tr #_ #EV whd in EV:(??%?); cases res in EV; |
---|
| 688 | [ normalize #EV destruct | * [ 2: * #r normalize #EV destruct /2/ | *: normalize #a try #b destruct ] ] |
---|
[1681] | 689 | | #f #fs #m #s2 #tr #_ whd in ⊢ (??%? → ?); @bind_value #locals #El #EV |
---|
[1682] | 690 | whd in EV:(??%?); destruct @(sp_finished ? f) // |
---|
[1736] | 691 | cases f // |
---|
[1681] | 692 | ] |
---|
[1713] | 693 | | #r #s2 #tr #E normalize in E; destruct |
---|
[1681] | 694 | ] qed. |
---|
| 695 | |
---|
| 696 | lemma stack_preserved_step : ∀ge,s1,s2,tr. |
---|
| 697 | RTLabs_classify s1 = cl_other ∨ RTLabs_classify s1 = cl_jump → |
---|
| 698 | eval_statement ge s1 = Value ??? 〈tr,s2〉 → |
---|
| 699 | stack_preserved doesnt_end_with_ret s1 s2. |
---|
| 700 | #ge0 #s1 #s2 #tr #CL #EV inversion (eval_perserves … EV) |
---|
| 701 | [ #ge #f #f' #fs #m #m' #F #E1 #E2 #E3 #E4 destruct /2/ |
---|
| 702 | | #ge #f #fs #m #fd #args #f' #dst #F #b #FFP #E1 #E2 #E3 #E4 /2/ |
---|
| 703 | | #ge #fn #locals #next #nok #sp #fs #m #args #dst #m' #E1 #E2 #E3 #E4 destruct |
---|
| 704 | normalize in CL; cases CL #E destruct |
---|
| 705 | | #ge #f #fs #m #rtv #dst #m' #E1 #E2 #E3 #E4 destruct /2/ |
---|
| 706 | | #ge #f #fs #rtv #dst #f' #m #F #E1 #E2 #E3 #E4 destruct cases CL |
---|
| 707 | #E normalize in E; destruct |
---|
[1713] | 708 | | #ge #r #dst #m #E1 #E2 destruct @⊥ cases CL normalize #E destruct |
---|
[1681] | 709 | ] qed. |
---|
| 710 | |
---|
| 711 | lemma stack_preserved_call : ∀ge,s1,s2,s3,tr. |
---|
| 712 | RTLabs_classify s1 = cl_call → |
---|
| 713 | eval_statement ge s1 = Value ??? 〈tr,s2〉 → |
---|
| 714 | stack_preserved ends_with_ret s2 s3 → |
---|
| 715 | stack_preserved doesnt_end_with_ret s1 s3. |
---|
| 716 | #ge #s1 #s2 #s3 #tr #CL #EV #SP |
---|
| 717 | cases (rtlabs_call_inv … CL) |
---|
| 718 | #fd * #args * #dst * #stk * #m #E destruct |
---|
[1682] | 719 | inversion SP |
---|
| 720 | [ #H38 #H39 #H40 #H41 #H42 #H43 #H44 #H45 #H46 destruct |
---|
[1736] | 721 | | #s2' #f #f' #fs #m' #N #F #S #E1 #E2 #E3 #E4 destruct |
---|
[1682] | 722 | inversion (eval_perserves … EV) |
---|
| 723 | [ #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 #H56 #H57 #H58 destruct |
---|
| 724 | | #H60 #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 #H70 #H71 #H72 #H73 #H74 destruct |
---|
| 725 | | #ge' #fn #locals #next #nok #sp #fs1 #m1 #args1 #dst1 #m2 #E1 #E2 #E3 #E4 destruct |
---|
| 726 | inversion S |
---|
| 727 | [ #fx #fsx #mx #E1 #E2 #E3 destruct /2/ |
---|
| 728 | | #H76 #H77 #H78 #H79 #H80 #H81 #H82 #H83 #H84 destruct |
---|
| 729 | | #H86 #H87 #H88 #H89 #H90 #H91 #H92 destruct |
---|
| 730 | ] |
---|
| 731 | | #H94 #H95 #H96 #H97 #H98 #H99 #H100 #H101 #H102 #H103 #H104 destruct |
---|
| 732 | | #H106 #H107 #H108 #H109 #H110 #H111 #H112 #H113 #H114 #H115 #H116 #H117 destruct |
---|
[1713] | 733 | | #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 destruct |
---|
[1682] | 734 | ] |
---|
[1736] | 735 | | #s1 #r #S1 #E1 #E2 #E3 #_ destruct |
---|
| 736 | inversion (eval_perserves … EV) |
---|
| 737 | [ #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 #H56 #H57 #H58 destruct |
---|
| 738 | | #H60 #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 #H70 #H71 #H72 #H73 #H74 destruct |
---|
| 739 | | #ge' #fn #locals #next #nok #sp #fs1 #m1 #args1 #dst1 #m2 #E1 #E2 #E3 #E4 destruct |
---|
| 740 | inversion S1 |
---|
| 741 | [ #fx #fsx #mx #E1 #E2 #E3 destruct /2/ |
---|
| 742 | | #H76 #H77 #H78 #H79 #H80 #H81 #H82 #H83 #H84 destruct |
---|
| 743 | | #H86 #H87 #H88 #H89 #H90 #H91 #H92 destruct |
---|
| 744 | ] |
---|
| 745 | | #H94 #H95 #H96 #H97 #H98 #H99 #H100 #H101 #H102 #H103 #H104 destruct |
---|
| 746 | | #H106 #H107 #H108 #H109 #H110 #H111 #H112 #H113 #H114 #H115 #H116 #H117 destruct |
---|
| 747 | | #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 destruct |
---|
| 748 | ] |
---|
| 749 | | #H49 #H50 #H51 #H52 #H53 #H54 #H55 #H56 #H57 destruct |
---|
[1682] | 750 | ] qed. |
---|
| 751 | |
---|
| 752 | lemma RTLabs_after_call : ∀ge,s1,s2,s3,tr. |
---|
| 753 | ∀CL : RTLabs_classify s1 = cl_call. |
---|
| 754 | eval_statement ge s1 = Value ??? 〈tr,s2〉 → |
---|
| 755 | stack_preserved ends_with_ret s2 s3 → |
---|
| 756 | as_after_return (RTLabs_status ge) «s1,CL» s3. |
---|
| 757 | #ge #s1 #s2 #s3 #tr #CL #EV #S23 |
---|
| 758 | cases (rtlabs_call_inv … CL) #fn * #args * #dst * #stk * #m #E destruct |
---|
| 759 | inversion S23 |
---|
| 760 | [ #H129 #H130 #H131 #H132 #H133 #H134 #H135 #H136 #H137 destruct |
---|
[1736] | 761 | | #s2' #f #f' #fs #m' #N #F #S #E1 #E2 #E3 #E4 destruct |
---|
[1682] | 762 | inversion (eval_perserves … EV) |
---|
| 763 | [ #H139 #H140 #H141 #H142 #H143 #H144 #H145 #H146 #H147 #H148 #H149 destruct |
---|
| 764 | | #H151 #H152 #H153 #H154 #H155 #H156 #H157 #H158 #H159 #H160 #H161 #H162 #H163 #H164 #H165 destruct |
---|
| 765 | | #gex #fnx #locals #next #nok #sp #fsx #mx #argsx #dstx #mx' #E1 #E2 #E3 #E4 destruct |
---|
| 766 | inversion S |
---|
[1736] | 767 | [ #fy #fsy #my #E1 #E2 #E3 destruct whd % [ @N | inversion F // ] |
---|
[1682] | 768 | | #H167 #H168 #H169 #H170 #H171 #H172 #H173 #H174 #H175 destruct |
---|
| 769 | | #H177 #H178 #H179 #H180 #H181 #H182 #H183 destruct |
---|
| 770 | ] |
---|
| 771 | | #H185 #H186 #H187 #H188 #H189 #H190 #H191 #H192 #H193 #H194 #H195 destruct |
---|
| 772 | | #H197 #H198 #H199 #H200 #H201 #H202 #H203 #H204 #H205 #H206 #H207 #H208 destruct |
---|
[1713] | 773 | | #H10 #H11 #H12 #H13 #H14 #H15 #H16 #H17 destruct |
---|
[1682] | 774 | ] |
---|
[1736] | 775 | | #s1 #r #S1 #E1 #E2 #E3 #E4 destruct whd |
---|
| 776 | inversion (eval_perserves … EV) |
---|
| 777 | [ #H59 #H60 #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 destruct |
---|
| 778 | | #H71 #H72 #H73 #H74 #H75 #H76 #H77 #H78 #H79 #H80 #H81 #H82 #H83 #H84 #H85 destruct |
---|
| 779 | | #ge' #fn' #locals #next #nok #sp #fs #m' #args' #dst' #m'' #E1 #E2 #E3 #E4 destruct |
---|
| 780 | inversion S1 |
---|
| 781 | [ #H103 #H104 #H105 #H106 #H107 #H108 destruct // |
---|
| 782 | | #H110 #H111 #H112 #H113 #H114 #H115 #H116 #H117 #H118 destruct |
---|
| 783 | | #H120 #H121 #H122 #H123 #H124 #H125 #H126 destruct |
---|
| 784 | ] |
---|
| 785 | | #H185 #H186 #H187 #H188 #H189 #H190 #H191 #H192 #H193 #H194 #H195 destruct |
---|
| 786 | | #H197 #H198 #H199 #H200 #H201 #H202 #H203 #H204 #H205 #H206 #H207 #H208 destruct |
---|
| 787 | | #H10 #H11 #H12 #H13 #H14 #H15 #H16 #H17 destruct |
---|
| 788 | ] |
---|
| 789 | | #H128 #H129 #H130 #H131 #H132 #H133 #H134 #H135 #H136 destruct |
---|
[1682] | 790 | ] qed. |
---|
[1681] | 791 | |
---|
[1574] | 792 | (* Don't need to know that labels break loops because we have termination. *) |
---|
| 793 | |
---|
[1596] | 794 | (* A bit of mucking around with the depth to avoid proving termination after |
---|
[1638] | 795 | termination. Note that we keep a proof that our upper bound on the length |
---|
| 796 | of the termination proof is respected. *) |
---|
[1719] | 797 | record trace_result (ge:genv) (depth:nat) (ends:trace_ends_with_ret) |
---|
| 798 | (start:state) (full:flat_trace io_out io_in ge start) |
---|
| 799 | (original_terminates: will_return ge depth start full) |
---|
| 800 | (T:state → Type[0]) (limit:nat) : Type[0] ≝ |
---|
| 801 | { |
---|
[1574] | 802 | new_state : state; |
---|
| 803 | remainder : flat_trace io_out io_in ge new_state; |
---|
| 804 | cost_labelled : well_cost_labelled_state new_state; |
---|
[1596] | 805 | new_trace : T new_state; |
---|
[1681] | 806 | stack_ok : stack_preserved ends start new_state; |
---|
[1719] | 807 | terminates : match (match ends with [ doesnt_end_with_ret ⇒ S depth | _ ⇒ depth ]) with |
---|
| 808 | [ O ⇒ will_return_end … original_terminates = ❬new_state, remainder❭ |
---|
| 809 | | S d ⇒ ΣTM:will_return ge d new_state remainder. |
---|
| 810 | limit > will_return_length … TM ∧ |
---|
| 811 | will_return_end … original_terminates = will_return_end … TM |
---|
[1596] | 812 | ] |
---|
[1574] | 813 | }. |
---|
| 814 | |
---|
[1638] | 815 | (* The same with a flag indicating whether the function returned, as opposed to |
---|
| 816 | encountering a label. *) |
---|
[1719] | 817 | record sub_trace_result (ge:genv) (depth:nat) |
---|
| 818 | (start:state) (full:flat_trace io_out io_in ge start) |
---|
| 819 | (original_terminates: will_return ge depth start full) |
---|
| 820 | (T:trace_ends_with_ret → state → Type[0]) (limit:nat) : Type[0] ≝ |
---|
| 821 | { |
---|
[1594] | 822 | ends : trace_ends_with_ret; |
---|
[1719] | 823 | trace_res :> trace_result ge depth ends start full original_terminates (T ends) limit |
---|
[1594] | 824 | }. |
---|
| 825 | |
---|
[1638] | 826 | (* We often return the result from a recursive call with an addition to the |
---|
| 827 | structured trace, so we define a couple of functions to help. The bound on |
---|
| 828 | the size of the termination proof might need to be relaxed, too. *) |
---|
| 829 | |
---|
[1719] | 830 | definition replace_trace : ∀ge,d,e,s1,s2,t1,t2,TM1,TM2,T1,T2,l1,l2. l2 ≥ l1 → |
---|
| 831 | ∀r:trace_result ge d e s1 t1 TM1 T1 l1. |
---|
| 832 | will_return_end … TM1 = will_return_end … TM2 → |
---|
[1712] | 833 | T2 (new_state … r) → |
---|
[1719] | 834 | stack_preserved e s2 (new_state … r) → |
---|
| 835 | trace_result ge d e s2 t2 TM2 T2 l2 ≝ |
---|
| 836 | λge,d,e,s1,s2,t1,t2,TM1,TM2,T1,T2,l1,l2,lGE,r,TME,trace,SP. |
---|
| 837 | mk_trace_result ge d e s2 t2 TM2 T2 l2 |
---|
[1574] | 838 | (new_state … r) |
---|
| 839 | (remainder … r) |
---|
| 840 | (cost_labelled … r) |
---|
[1594] | 841 | trace |
---|
[1681] | 842 | SP |
---|
[1719] | 843 | ? |
---|
| 844 | (*(match d return λd'.match d' with [ O ⇒ True | S d'' ⇒ ΣTM.l1 > will_return_length ge d'' (new_state … r) (remainder … r) TM] → |
---|
[1637] | 845 | match d' with [ O ⇒ True | S d'' ⇒ ΣTM.l2 > will_return_length ge d'' (new_state … r) (remainder … r) TM] with |
---|
[1719] | 846 | [O ⇒ λ_. I | _ ⇒ λTM. «pi1 … TM, ?» ] (terminates ???????? r))*) |
---|
| 847 | . |
---|
| 848 | cases e in r ⊢ %; |
---|
| 849 | [ <TME -TME * cases d in TM1 TM2 ⊢ %; |
---|
| 850 | [ #TM1 #TM2 #ns #rem #WCLS #T1NS #SP whd in ⊢ (% → %); #TMS @TMS |
---|
| 851 | | #d' #TM1 #TM2 #ns #rem #WCLS #T1NS #SP whd in ⊢ (% → %); * #TMa * #L1 #TME |
---|
| 852 | %{TMa} % // @(transitive_le … lGE) @L1 |
---|
| 853 | ] |
---|
| 854 | | <TME -TME * #ns #rem #WCLS #T1NS #SP whd in ⊢ (% → %); |
---|
| 855 | * #TMa * #L1 #TME |
---|
| 856 | %{TMa} % // @(transitive_le … lGE) @L1 |
---|
| 857 | ] qed. |
---|
[1574] | 858 | |
---|
[1719] | 859 | definition replace_sub_trace : ∀ge,d,s1,s2,t1,t2,TM1,TM2,T1,T2,l1,l2. l2 ≥ l1 → |
---|
| 860 | ∀r:sub_trace_result ge d s1 t1 TM1 T1 l1. |
---|
| 861 | will_return_end … TM1 = will_return_end … TM2 → |
---|
[1712] | 862 | T2 (ends … r) (new_state … r) → |
---|
| 863 | stack_preserved (ends … r) s2 (new_state … r) → |
---|
[1719] | 864 | sub_trace_result ge d s2 t2 TM2 T2 l2 ≝ |
---|
| 865 | λge,d,s1,s2,t1,t2,TM1,TM2,T1,T2,l1,l2,lGE,r,TME,trace,SP. |
---|
| 866 | mk_sub_trace_result ge d s2 t2 TM2 T2 l2 |
---|
[1637] | 867 | (ends … r) |
---|
[1719] | 868 | (replace_trace … lGE … r TME trace SP). |
---|
[1637] | 869 | |
---|
[1638] | 870 | (* Small syntax hack to avoid ambiguous input problems. *) |
---|
[1637] | 871 | definition myge : nat → nat → Prop ≝ ge. |
---|
| 872 | |
---|
[1596] | 873 | let rec make_label_return ge depth s |
---|
[1565] | 874 | (trace: flat_trace io_out io_in ge s) |
---|
| 875 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
[1574] | 876 | (STATE_COSTLABELLED: well_cost_labelled_state s) (* functions in the state *) |
---|
[1583] | 877 | (STATEMENT_COSTLABEL: RTLabs_cost s = true) (* current statement is a cost label *) |
---|
[1596] | 878 | (TERMINATES: will_return ge depth s trace) |
---|
[1637] | 879 | (TIME: nat) |
---|
| 880 | (TERMINATES_IN_TIME: myge TIME (plus 2 (times 3 (will_return_length … TERMINATES)))) |
---|
[1719] | 881 | on TIME : trace_result ge depth ends_with_ret s trace TERMINATES |
---|
[1638] | 882 | (trace_label_return (RTLabs_status ge) s) |
---|
| 883 | (will_return_length … TERMINATES) ≝ |
---|
| 884 | |
---|
[1637] | 885 | match TIME return λTIME. TIME ≥ ? → ? with |
---|
| 886 | [ O ⇒ λTERMINATES_IN_TIME. ⊥ |
---|
| 887 | | S TIME ⇒ λTERMINATES_IN_TIME. |
---|
[1638] | 888 | |
---|
| 889 | let r ≝ make_label_label ge depth s |
---|
| 890 | trace |
---|
| 891 | ENV_COSTLABELLED |
---|
| 892 | STATE_COSTLABELLED |
---|
| 893 | STATEMENT_COSTLABEL |
---|
| 894 | TERMINATES |
---|
| 895 | TIME ? in |
---|
[1719] | 896 | match ends … r return λx. trace_result ge depth x s trace TERMINATES (trace_label_label (RTLabs_status ge) x s) ? → |
---|
| 897 | trace_result ge depth ends_with_ret s trace TERMINATES (trace_label_return (RTLabs_status ge) s) (will_return_length … TERMINATES) with |
---|
[1596] | 898 | [ ends_with_ret ⇒ λr. |
---|
[1712] | 899 | replace_trace … r ? (tlr_base (RTLabs_status ge) s (new_state … r) (new_trace … r)) (stack_ok … r) |
---|
[1596] | 900 | | doesnt_end_with_ret ⇒ λr. |
---|
| 901 | let r' ≝ make_label_return ge depth (new_state … r) |
---|
[1638] | 902 | (remainder … r) |
---|
| 903 | ENV_COSTLABELLED |
---|
| 904 | (cost_labelled … r) ? |
---|
| 905 | (pi1 … (terminates … r)) TIME ? in |
---|
[1712] | 906 | replace_trace … r' ? |
---|
[1638] | 907 | (tlr_step (RTLabs_status ge) s (new_state … r) |
---|
[1681] | 908 | (new_state … r') (new_trace … r) (new_trace … r')) ? |
---|
[1596] | 909 | ] (trace_res … r) |
---|
[1638] | 910 | |
---|
[1637] | 911 | ] TERMINATES_IN_TIME |
---|
[1574] | 912 | |
---|
[1638] | 913 | |
---|
[1596] | 914 | and make_label_label ge depth s |
---|
[1574] | 915 | (trace: flat_trace io_out io_in ge s) |
---|
| 916 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
| 917 | (STATE_COSTLABELLED: well_cost_labelled_state s) (* functions in the state *) |
---|
[1583] | 918 | (STATEMENT_COSTLABEL: RTLabs_cost s = true) (* current statement is a cost label *) |
---|
[1596] | 919 | (TERMINATES: will_return ge depth s trace) |
---|
[1637] | 920 | (TIME: nat) |
---|
| 921 | (TERMINATES_IN_TIME: myge TIME (plus 1 (times 3 (will_return_length … TERMINATES)))) |
---|
[1719] | 922 | on TIME : sub_trace_result ge depth s trace TERMINATES |
---|
[1638] | 923 | (λends. trace_label_label (RTLabs_status ge) ends s) |
---|
| 924 | (will_return_length … TERMINATES) ≝ |
---|
| 925 | |
---|
[1637] | 926 | match TIME return λTIME. TIME ≥ ? → ? with |
---|
| 927 | [ O ⇒ λTERMINATES_IN_TIME. ⊥ |
---|
| 928 | | S TIME ⇒ λTERMINATES_IN_TIME. |
---|
[1638] | 929 | |
---|
[1637] | 930 | let r ≝ make_any_label ge depth s trace ENV_COSTLABELLED STATE_COSTLABELLED TERMINATES TIME ? in |
---|
[1712] | 931 | replace_sub_trace … r ? |
---|
[1681] | 932 | (tll_base (RTLabs_status ge) (ends … r) s (new_state … r) (new_trace … r) STATEMENT_COSTLABEL) (stack_ok … r) |
---|
[1638] | 933 | |
---|
[1637] | 934 | ] TERMINATES_IN_TIME |
---|
[1574] | 935 | |
---|
[1638] | 936 | |
---|
[1596] | 937 | and make_any_label ge depth s |
---|
[1574] | 938 | (trace: flat_trace io_out io_in ge s) |
---|
| 939 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
| 940 | (STATE_COSTLABELLED: well_cost_labelled_state s) (* functions in the state *) |
---|
[1596] | 941 | (TERMINATES: will_return ge depth s trace) |
---|
[1637] | 942 | (TIME: nat) |
---|
| 943 | (TERMINATES_IN_TIME: myge TIME (times 3 (will_return_length … TERMINATES))) |
---|
[1719] | 944 | on TIME : sub_trace_result ge depth s trace TERMINATES |
---|
[1638] | 945 | (λends. trace_any_label (RTLabs_status ge) ends s) |
---|
| 946 | (will_return_length … TERMINATES) ≝ |
---|
[1637] | 947 | |
---|
| 948 | match TIME return λTIME. TIME ≥ ? → ? with |
---|
| 949 | [ O ⇒ λTERMINATES_IN_TIME. ⊥ |
---|
| 950 | | S TIME ⇒ λTERMINATES_IN_TIME. |
---|
[1638] | 951 | |
---|
[1719] | 952 | match trace return λs,trace. well_cost_labelled_state s → |
---|
| 953 | ∀TM:will_return ??? trace. |
---|
| 954 | myge ? (times 3 (will_return_length ??? trace TM)) → |
---|
| 955 | sub_trace_result ge depth s trace TM (λends. trace_any_label (RTLabs_status ge) ends s) (will_return_length … TM) with |
---|
[1638] | 956 | [ ft_stop st FINAL ⇒ |
---|
[1713] | 957 | λSTATE_COSTLABELLED,TERMINATES,TERMINATES_IN_TIME. ⊥ |
---|
[1638] | 958 | |
---|
[1637] | 959 | | ft_step start tr next EV trace' ⇒ λSTATE_COSTLABELLED,TERMINATES,TERMINATES_IN_TIME. |
---|
[1719] | 960 | match RTLabs_classify start return λx. RTLabs_classify start = x → sub_trace_result ge depth start ?? (λends. trace_any_label (RTLabs_status ge) ends ?) (will_return_length … TERMINATES) with |
---|
[1583] | 961 | [ cl_other ⇒ λCL. |
---|
[1719] | 962 | match RTLabs_cost next return λx. RTLabs_cost next = x → sub_trace_result ge depth start ?? (λends. trace_any_label (RTLabs_status ge) ends ?) (will_return_length … TERMINATES) with |
---|
[1638] | 963 | (* We're about to run into a label. *) |
---|
[1583] | 964 | [ true ⇒ λCS. |
---|
[1719] | 965 | mk_sub_trace_result ge depth start ? TERMINATES (λends. trace_any_label (RTLabs_status ge) ends start) ? |
---|
[1596] | 966 | doesnt_end_with_ret |
---|
[1719] | 967 | (mk_trace_result ge … next trace' ? |
---|
[1681] | 968 | (tal_base_not_return (RTLabs_status ge) start next ?? CS) ??) |
---|
[1638] | 969 | (* An ordinary step, keep going. *) |
---|
[1583] | 970 | | false ⇒ λCS. |
---|
[1638] | 971 | let r ≝ make_any_label ge depth next trace' ENV_COSTLABELLED ? (will_return_notfn … TERMINATES) TIME ? in |
---|
[1712] | 972 | replace_sub_trace … r ? |
---|
[1638] | 973 | (tal_step_default (RTLabs_status ge) (ends … r) |
---|
[1681] | 974 | start next (new_state … r) ? (new_trace … r) ? (RTLabs_not_cost … CS)) ? |
---|
[1583] | 975 | ] (refl ??) |
---|
[1638] | 976 | |
---|
[1586] | 977 | | cl_jump ⇒ λCL. |
---|
[1719] | 978 | mk_sub_trace_result ge depth start ? TERMINATES (λends. trace_any_label (RTLabs_status ge) ends start) ? |
---|
[1596] | 979 | doesnt_end_with_ret |
---|
[1719] | 980 | (mk_trace_result ge … next trace' ? |
---|
[1681] | 981 | (tal_base_not_return (RTLabs_status ge) start next ???) ??) |
---|
[1638] | 982 | |
---|
[1595] | 983 | | cl_call ⇒ λCL. |
---|
[1719] | 984 | let r ≝ make_label_return ge (S depth) next trace' ENV_COSTLABELLED ?? (will_return_call … CL TERMINATES) TIME ? in |
---|
| 985 | match RTLabs_cost (new_state … r) return λx. RTLabs_cost (new_state … r) = x → sub_trace_result ge depth start ?? (λends. trace_any_label (RTLabs_status ge) ends ?) (will_return_length … TERMINATES) with |
---|
[1654] | 986 | (* We're about to run into a label, use base case for call *) |
---|
| 987 | [ true ⇒ λCS. |
---|
[1719] | 988 | mk_sub_trace_result ge depth start ? TERMINATES (λends. trace_any_label (RTLabs_status ge) ends start) ? |
---|
[1654] | 989 | doesnt_end_with_ret |
---|
[1719] | 990 | (mk_trace_result ge … |
---|
[1654] | 991 | (tal_base_call (RTLabs_status ge) start next (new_state … r) |
---|
[1719] | 992 | ? CL ? (new_trace … r) CS) ??) |
---|
[1654] | 993 | (* otherwise use step case *) |
---|
| 994 | | false ⇒ λCS. |
---|
| 995 | let r' ≝ make_any_label ge depth |
---|
| 996 | (new_state … r) (remainder … r) ENV_COSTLABELLED ? |
---|
| 997 | (pi1 … (terminates … r)) TIME ? in |
---|
[1712] | 998 | replace_sub_trace … r' ? |
---|
[1654] | 999 | (tal_step_call (RTLabs_status ge) (ends … r') |
---|
| 1000 | start next (new_state … r) (new_state … r') ? CL ? |
---|
[1681] | 1001 | (new_trace … r) (RTLabs_not_cost … CS) (new_trace … r')) ? |
---|
[1654] | 1002 | ] (refl ??) |
---|
[1638] | 1003 | |
---|
[1594] | 1004 | | cl_return ⇒ λCL. |
---|
[1719] | 1005 | mk_sub_trace_result ge depth start ? TERMINATES (λends. trace_any_label (RTLabs_status ge) ends start) ? |
---|
[1596] | 1006 | ends_with_ret |
---|
[1719] | 1007 | (mk_trace_result ge … |
---|
[1596] | 1008 | next |
---|
| 1009 | trace' |
---|
| 1010 | ? |
---|
| 1011 | (tal_base_return (RTLabs_status ge) start next ? CL) |
---|
[1681] | 1012 | ? |
---|
[1596] | 1013 | ?) |
---|
[1583] | 1014 | ] (refl ? (RTLabs_classify start)) |
---|
[1638] | 1015 | |
---|
[1637] | 1016 | | ft_wrong start m EV ⇒ λSTATE_COSTLABELLED,TERMINATES. ⊥ |
---|
[1638] | 1017 | |
---|
[1637] | 1018 | ] STATE_COSTLABELLED TERMINATES TERMINATES_IN_TIME |
---|
| 1019 | ] TERMINATES_IN_TIME. |
---|
[1574] | 1020 | |
---|
[1637] | 1021 | [ cases (not_le_Sn_O ?) [ #H @H @TERMINATES_IN_TIME ] |
---|
| 1022 | | // |
---|
[1712] | 1023 | | // |
---|
[1719] | 1024 | | cases r #H1 #H2 #H3 #H4 #H5 * #H7 * #GT #_ @(le_S_to_le … GT) |
---|
| 1025 | | cases r #H1 #H2 #H3 #H4 #H5 * #H7 * #_ #EEQ // |
---|
[1681] | 1026 | | @(stack_preserved_join … (stack_ok … r)) // |
---|
[1637] | 1027 | | @(trace_label_label_label … (new_trace … r)) |
---|
[1719] | 1028 | | cases r #H1 #H2 #H3 #H4 #H5 * #H7 * #LT #_ |
---|
[1637] | 1029 | @(le_plus_to_le … 1) @(transitive_le … TERMINATES_IN_TIME) |
---|
| 1030 | @(transitive_le … (3*(will_return_length … TERMINATES))) |
---|
| 1031 | [ >commutative_times change with ((S ?) * 3 ≤ ?) >commutative_times |
---|
[1681] | 1032 | @(monotonic_le_times_r 3 … LT) |
---|
[1637] | 1033 | | @le_S @le_S @le_n |
---|
| 1034 | ] |
---|
| 1035 | | @le_S_S_to_le @TERMINATES_IN_TIME |
---|
| 1036 | | cases (not_le_Sn_O ?) [ #H @H @TERMINATES_IN_TIME ] |
---|
| 1037 | | @le_n |
---|
[1712] | 1038 | | // |
---|
[1637] | 1039 | | @le_S_S_to_le @TERMINATES_IN_TIME |
---|
| 1040 | | @(wrl_nonzero … TERMINATES_IN_TIME) |
---|
[1713] | 1041 | | (* We can't reach the final state because the function terminates with a |
---|
| 1042 | return *) |
---|
| 1043 | inversion TERMINATES |
---|
| 1044 | [ #H214 #H215 #H216 #H217 #H218 #H219 #H220 #H221 #H222 #H223 #H224 #H225 #_ -TERMINATES -TERMINATES destruct |
---|
| 1045 | | #H228 #H229 #H230 #H231 #H232 #H233 #H234 #H235 #H236 #H237 #H238 #H239 #H240 -TERMINATES -TERMINATES destruct |
---|
| 1046 | | #H242 #H243 #H244 #H245 #H246 #H247 #H248 #H249 #H250 #H251 #H252 #H253 #H254 -TERMINATES -TERMINATES destruct |
---|
| 1047 | | #H256 #H257 #H258 #H259 #H260 #H261 #H262 #H263 #H264 #H265 -TERMINATES -TERMINATES destruct |
---|
| 1048 | ] |
---|
[1637] | 1049 | | @(will_return_return … CL TERMINATES) |
---|
[1682] | 1050 | | /2 by stack_preserved_return/ |
---|
[1594] | 1051 | | %{tr} @EV |
---|
[1586] | 1052 | | @(well_cost_labelled_state_step … EV) // |
---|
[1596] | 1053 | | whd @(will_return_notfn … TERMINATES) %2 @CL |
---|
[1681] | 1054 | | @stack_preserved_step /2/ |
---|
[1586] | 1055 | | %{tr} @EV |
---|
[1654] | 1056 | | %1 whd @CL |
---|
[1586] | 1057 | | @(well_cost_labelled_jump … EV) // |
---|
[1594] | 1058 | | @(well_cost_labelled_state_step … EV) // |
---|
[1719] | 1059 | | whd cases (terminates ???????? r) #TMr * #LTr #EQr %{TMr} % |
---|
| 1060 | [ @(transitive_lt … LTr) cases (will_return_call … CL TERMINATES) |
---|
| 1061 | #TMx * #LT' #_ @LT' |
---|
| 1062 | | <EQr cases (will_return_call … CL TERMINATES) |
---|
| 1063 | #TM' * #_ #EQ' @EQ' |
---|
| 1064 | ] |
---|
[1682] | 1065 | | @(stack_preserved_call … EV (stack_ok … r)) // |
---|
[1654] | 1066 | | %{tr} @EV |
---|
[1682] | 1067 | | @RTLabs_after_call // |
---|
[1719] | 1068 | | @(cost_labelled … r) |
---|
| 1069 | | skip |
---|
| 1070 | | cases r #ns #rm #WS #TLR #SP * #TM * #LT #_ @le_S_to_le |
---|
| 1071 | @(transitive_lt … LT) |
---|
| 1072 | cases (will_return_call … CL TERMINATES) #TM' * #LT' #_ @LT' |
---|
| 1073 | | cases r #ns #rm #WS #TLR #SP * #TM * #_ #EQ <EQ |
---|
| 1074 | cases (will_return_call … CL TERMINATES) #TM' * #_ #EQ' // |
---|
[1682] | 1075 | | @RTLabs_after_call // |
---|
[1595] | 1076 | | %{tr} @EV |
---|
[1682] | 1077 | | @(stack_preserved_join … (stack_ok … r')) @(stack_preserved_call … EV (stack_ok … r)) // |
---|
[1595] | 1078 | | @(cost_labelled … r) |
---|
[1719] | 1079 | | cases r #H72 #H73 #H74 #H75 #HX * #HY * #GT #H78 |
---|
[1637] | 1080 | @(le_plus_to_le … 1) @(transitive_le … TERMINATES_IN_TIME) |
---|
[1719] | 1081 | cases (will_return_call … TERMINATES) in GT; |
---|
| 1082 | #X * #Y #_ #Z |
---|
[1637] | 1083 | @(transitive_le … (monotonic_lt_times_r 3 … Y)) |
---|
| 1084 | [ @(transitive_le … (monotonic_lt_times_r 3 … Z)) // |
---|
| 1085 | | // |
---|
| 1086 | ] |
---|
[1596] | 1087 | | @(well_cost_labelled_state_step … EV) // |
---|
| 1088 | | @(well_cost_labelled_call … EV) // |
---|
[1638] | 1089 | | cases (will_return_call … TERMINATES) |
---|
[1719] | 1090 | #TM * #GT #_ @le_S_S_to_le |
---|
[1637] | 1091 | >commutative_times change with ((S ?) * 3 ≤ ?) >commutative_times |
---|
| 1092 | @(transitive_le … TERMINATES_IN_TIME) |
---|
| 1093 | @(monotonic_le_times_r 3 … GT) |
---|
[1596] | 1094 | | whd @(will_return_notfn … TERMINATES) %1 @CL |
---|
[1682] | 1095 | | @(stack_preserved_step … EV) /2/ |
---|
[1596] | 1096 | | %{tr} @EV |
---|
[1654] | 1097 | | %2 whd @CL |
---|
[1596] | 1098 | | @(well_cost_labelled_state_step … EV) // |
---|
[1719] | 1099 | | cases (will_return_notfn … TERMINATES) #TM * #GT #_ @(le_S_to_le … GT) |
---|
| 1100 | | cases (will_return_notfn … TERMINATES) #TM * #_ #EQ // |
---|
[1594] | 1101 | | @CL |
---|
[1583] | 1102 | | %{tr} @EV |
---|
[1682] | 1103 | | @(stack_preserved_join … (stack_ok … r)) @(stack_preserved_step … EV) /2/ |
---|
[1594] | 1104 | | @(well_cost_labelled_state_step … EV) // |
---|
[1638] | 1105 | | %1 @CL |
---|
[1719] | 1106 | | cases (will_return_notfn … TERMINATES) #TM * #GT #_ |
---|
[1637] | 1107 | @le_S_S_to_le |
---|
| 1108 | @(transitive_le … (monotonic_lt_times_r … GT) TERMINATES_IN_TIME) |
---|
| 1109 | // |
---|
[1574] | 1110 | | inversion TERMINATES |
---|
[1637] | 1111 | [ #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 #H13 -TERMINATES -TERMINATES destruct |
---|
| 1112 | | #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 #H29 -TERMINATES -TERMINATES destruct |
---|
| 1113 | | #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 #H42 #H43 #H44 #H45 -TERMINATES -TERMINATES destruct |
---|
| 1114 | | #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 -TERMINATES -TERMINATES destruct |
---|
[1583] | 1115 | ] |
---|
[1713] | 1116 | ] qed. |
---|
[1583] | 1117 | |
---|
[1638] | 1118 | (* We can initialise TIME with a suitably large value based on the length of the |
---|
| 1119 | termination proof. *) |
---|
[1637] | 1120 | let rec make_label_return' ge depth s |
---|
| 1121 | (trace: flat_trace io_out io_in ge s) |
---|
| 1122 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
| 1123 | (STATE_COSTLABELLED: well_cost_labelled_state s) (* functions in the state *) |
---|
| 1124 | (STATEMENT_COSTLABEL: RTLabs_cost s = true) (* current statement is a cost label *) |
---|
| 1125 | (TERMINATES: will_return ge depth s trace) |
---|
[1719] | 1126 | : trace_result ge depth ends_with_ret s trace TERMINATES (trace_label_return (RTLabs_status ge) s) (will_return_length … TERMINATES) ≝ |
---|
[1637] | 1127 | make_label_return ge depth s trace ENV_COSTLABELLED STATE_COSTLABELLED STATEMENT_COSTLABEL TERMINATES |
---|
| 1128 | (2 + 3 * will_return_length … TERMINATES) ?. |
---|
| 1129 | @le_n |
---|
| 1130 | qed. |
---|
[1574] | 1131 | |
---|
[1713] | 1132 | (* Tail-calls would not be handled properly (which means that if we try to show the |
---|
[1617] | 1133 | full version with non-termination we'll fail because calls and returns aren't |
---|
| 1134 | balanced. |
---|
[1651] | 1135 | *) |
---|
| 1136 | |
---|
| 1137 | inductive inhabited (T:Type[0]) : Prop ≝ |
---|
| 1138 | | witness : T → inhabited T. |
---|
| 1139 | |
---|
| 1140 | (* We also require that program's traces are soundly labelled: for any state |
---|
| 1141 | in the execution, we can give a distance to a labelled state or termination. |
---|
[1617] | 1142 | |
---|
[1651] | 1143 | Note that this differs from the syntactic notions in earlier languages |
---|
| 1144 | because it is a global property. In principle, we would have a loop broken |
---|
| 1145 | only by a call to a function (which necessarily has a label) and no local |
---|
| 1146 | cost label. |
---|
| 1147 | *) |
---|
| 1148 | |
---|
| 1149 | let rec nth_state ge s |
---|
| 1150 | (trace: flat_trace io_out io_in ge s) |
---|
| 1151 | n |
---|
| 1152 | on n : option state ≝ |
---|
| 1153 | match n with |
---|
| 1154 | [ O ⇒ Some ? s |
---|
| 1155 | | S n' ⇒ |
---|
| 1156 | match trace with |
---|
| 1157 | [ ft_step _ _ s' _ trace' ⇒ nth_state ge s' trace' n' |
---|
| 1158 | | _ ⇒ None ? |
---|
| 1159 | ] |
---|
| 1160 | ]. |
---|
| 1161 | |
---|
| 1162 | definition soundly_labelled_trace : ∀ge,s. flat_trace io_out io_in ge s → Prop ≝ |
---|
| 1163 | λge,s,trace. ∀n.∃m. ∀s'. nth_state ge s trace (n+m) = Some ? s' → RTLabs_cost s' = true. |
---|
| 1164 | |
---|
| 1165 | lemma soundly_labelled_step : ∀ge,s,tr,s',EV,trace'. |
---|
| 1166 | soundly_labelled_trace ge s (ft_step … ge s tr s' EV trace') → |
---|
| 1167 | soundly_labelled_trace ge s' trace'. |
---|
| 1168 | #ge #s #tr #s' #EV #trace' #H |
---|
| 1169 | #n cases (H (S n)) #m #H' %{m} @H' |
---|
| 1170 | qed. |
---|
| 1171 | |
---|
[1705] | 1172 | (* Define a notion of sound labellings of RTLabs programs. *) |
---|
[1675] | 1173 | |
---|
[1705] | 1174 | let rec successors (s : statement) : list label ≝ |
---|
| 1175 | match s with |
---|
| 1176 | [ St_skip l ⇒ [l] |
---|
| 1177 | | St_cost _ l ⇒ [l] |
---|
| 1178 | | St_const _ _ l ⇒ [l] |
---|
| 1179 | | St_op1 _ _ _ _ _ l ⇒ [l] |
---|
| 1180 | | St_op2 _ _ _ _ l ⇒ [l] |
---|
| 1181 | | St_load _ _ _ l ⇒ [l] |
---|
| 1182 | | St_store _ _ _ l ⇒ [l] |
---|
| 1183 | | St_call_id _ _ _ l ⇒ [l] |
---|
| 1184 | | St_call_ptr _ _ _ l ⇒ [l] |
---|
| 1185 | (* |
---|
| 1186 | | St_tailcall_id _ _ ⇒ [ ] |
---|
| 1187 | | St_tailcall_ptr _ _ ⇒ [ ] |
---|
| 1188 | *) |
---|
| 1189 | | St_cond _ l1 l2 ⇒ [l1; l2] |
---|
| 1190 | | St_jumptable _ ls ⇒ ls |
---|
| 1191 | | St_return ⇒ [ ] |
---|
| 1192 | ]. |
---|
[1675] | 1193 | |
---|
[1705] | 1194 | definition actual_successor : state → option label ≝ |
---|
| 1195 | λs. match s with |
---|
| 1196 | [ State f fs m ⇒ Some ? (next f) |
---|
| 1197 | | Callstate _ _ _ fs _ ⇒ match fs with [ cons f _ ⇒ Some ? (next f) | _ ⇒ None ? ] |
---|
| 1198 | | Returnstate _ _ _ _ ⇒ None ? |
---|
[1713] | 1199 | | Finalstate _ ⇒ None ? |
---|
[1705] | 1200 | ]. |
---|
| 1201 | |
---|
| 1202 | lemma nth_opt_Exists : ∀A,n,l,a. |
---|
| 1203 | nth_opt A n l = Some A a → |
---|
| 1204 | Exists A (λa'. a' = a) l. |
---|
| 1205 | #A #n elim n |
---|
| 1206 | [ * [ #a #E normalize in E; destruct | #a #l #a' #E normalize in E; destruct % // ] |
---|
| 1207 | | #m #IH * |
---|
| 1208 | [ #a #E normalize in E; destruct |
---|
| 1209 | | #a #l #a' #E %2 @IH @E |
---|
| 1210 | ] |
---|
| 1211 | ] qed. |
---|
| 1212 | |
---|
| 1213 | lemma eval_successor : ∀ge,f,fs,m,tr,s'. |
---|
| 1214 | eval_statement ge (State f fs m) = Value ??? 〈tr,s'〉 → |
---|
| 1215 | RTLabs_classify s' = cl_return ∨ |
---|
| 1216 | ∃l. actual_successor s' = Some ? l ∧ Exists ? (λl0. l0 = l) (successors (lookup_present … (f_graph (func f)) (next f) (next_ok f))). |
---|
| 1217 | #ge * #func #locals #next #next_ok #sp #dst #fs #m #tr #s' |
---|
| 1218 | whd in ⊢ (??%? → ?); |
---|
| 1219 | generalize in ⊢ (??(?%)? → ?); cases (lookup_present ??? next next_ok) |
---|
| 1220 | [ #l #LP whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1221 | | #cl #l #LP whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1222 | | #r #c #l #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1223 | | #ty #ty' #op #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev @bind_ok #v' #Ev' @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1224 | | #op #r1 #r2 #r3 #l #LP whd in ⊢ (??%? → ?); @bind_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #v' #Ev' @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1225 | | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1226 | | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #m' #Em whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1227 | | #id #rs #r #l #LP whd in ⊢ (??%? → ?); @bind_value #b #Eb @bind_ok #fd #Efd @bind_ok #vs #Evs whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1228 | | #r #rs #r' #l #LP whd in ⊢ (??%? → ?); @bind_value #fv #Efv @bind_ok #fd #Efd @bind_ok #vs #Evs whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1229 | | #r #l1 #l2 #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev @bind_ok #b #Eb whd in ⊢ (??%? → ?); #E destruct %2 cases b [ %{l1} | %{l2} ] % // [ % | %2 %] // |
---|
| 1230 | | #r #ls #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev |
---|
| 1231 | cases v [ #E normalize in E; destruct | #sz #i | #f #E normalize in E; destruct | #r #E normalize in E; destruct | #p #E normalize in E; destruct ] |
---|
| 1232 | whd in ⊢ (??%? → ?); |
---|
| 1233 | generalize in ⊢ (??(?%)? → ?); |
---|
| 1234 | cases (nth_opt label (nat_of_bitvector (bitsize_of_intsize sz) i) ls) in ⊢ (???% → ??(match % with [ _ ⇒ ? | _ ⇒ ? ] ?)? → ?); |
---|
| 1235 | [ #e #E normalize in E; destruct |
---|
| 1236 | | #l #El whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // @(nth_opt_Exists … El) |
---|
| 1237 | ] |
---|
| 1238 | | #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev whd in ⊢ (??%? → ?); #E destruct %1 @refl |
---|
| 1239 | ] qed. |
---|
| 1240 | |
---|
| 1241 | definition steps_for_statement : statement → nat ≝ |
---|
| 1242 | λs. S (match s with [ St_call_id _ _ _ _ ⇒ 1 | St_call_ptr _ _ _ _ ⇒ 1 | St_return ⇒ 1 | _ ⇒ 0 ]). |
---|
| 1243 | |
---|
[1707] | 1244 | inductive bound_on_steps_to_cost (g:graph statement) : label → nat → Prop ≝ |
---|
| 1245 | | bostc_here : ∀l,n,H. is_cost_label (lookup_present … g l H) → bound_on_steps_to_cost g l n |
---|
| 1246 | | bostc_later : ∀l,n. bound_on_steps_to_cost1 g l n → bound_on_steps_to_cost g l n |
---|
| 1247 | with bound_on_steps_to_cost1 : label → nat → Prop ≝ |
---|
| 1248 | | bostc_step : ∀l,n,H. |
---|
[1705] | 1249 | let stmt ≝ lookup_present … g l H in |
---|
[1706] | 1250 | (∀l'. Exists label (λl0. l0 = l') (successors stmt) → |
---|
[1707] | 1251 | bound_on_steps_to_cost g l' n) → |
---|
| 1252 | bound_on_steps_to_cost1 g l (steps_for_statement stmt + n). |
---|
[1705] | 1253 | |
---|
[1707] | 1254 | (* |
---|
[1706] | 1255 | lemma steps_to_label_bound_inv : ∀g,l,n. |
---|
[1705] | 1256 | steps_to_label_bound g l n → |
---|
[1706] | 1257 | ∀H. let stmt ≝ lookup_present … g l H in |
---|
| 1258 | ∃n'. n = steps_for_statement stmt + n' ∧ |
---|
[1705] | 1259 | (∀l'. Exists label (λl0. l0 = l') (successors stmt) → |
---|
[1706] | 1260 | (∃H'. bool_to_Prop (is_cost_label (lookup_present … g l' H'))) ∨ |
---|
| 1261 | steps_to_label_bound g l' n'). |
---|
| 1262 | #g #l0 #n0 #S inversion S #l #n #H #IH #E1 #E2 #_ destruct #H' |
---|
| 1263 | % [2: % [ @refl | #l' #EX cases (IH l' EX) /2/ ] | skip ] |
---|
| 1264 | qed. |
---|
[1707] | 1265 | *) |
---|
[1719] | 1266 | |
---|
[1707] | 1267 | (* |
---|
[1705] | 1268 | definition soundly_labelled_pc ≝ λg,l. ∃n. steps_to_label_bound g l n. |
---|
| 1269 | |
---|
| 1270 | let rec soundly_labelled_fn (fn : internal_function) : Prop ≝ |
---|
| 1271 | soundly_labelled_pc (f_graph fn) (f_entry fn). |
---|
| 1272 | |
---|
| 1273 | |
---|
[1675] | 1274 | definition soundly_labelled_frame : frame → Prop ≝ |
---|
| 1275 | λf. soundly_labelled_pc (f_graph (func f)) (next f). |
---|
| 1276 | |
---|
| 1277 | definition soundly_labelled_state : state → Prop ≝ |
---|
| 1278 | λs. match s with |
---|
| 1279 | [ State f _ _ ⇒ soundly_labelled_frame f |
---|
| 1280 | | Callstate _ _ _ stk _ ⇒ match stk with [ nil ⇒ False | cons f _ ⇒ soundly_labelled_frame f ] |
---|
| 1281 | | Returnstate _ _ stk _ ⇒ match stk with [ nil ⇒ False | cons f _ ⇒ soundly_labelled_frame f ] |
---|
| 1282 | ]. |
---|
[1707] | 1283 | *) |
---|
| 1284 | definition frame_bound_on_steps_to_cost : frame → nat → Prop ≝ |
---|
| 1285 | λf. bound_on_steps_to_cost (f_graph (func f)) (next f). |
---|
| 1286 | definition frame_bound_on_steps_to_cost1 : frame → nat → Prop ≝ |
---|
| 1287 | λf. bound_on_steps_to_cost1 (f_graph (func f)) (next f). |
---|
[1705] | 1288 | |
---|
[1707] | 1289 | inductive state_bound_on_steps_to_cost : state → nat → Prop ≝ |
---|
| 1290 | | sbostc_state : ∀f,fs,m,n. frame_bound_on_steps_to_cost1 f n → state_bound_on_steps_to_cost (State f fs m) n |
---|
| 1291 | | sbostc_call : ∀fd,args,dst,f,fs,m,n. frame_bound_on_steps_to_cost f n → state_bound_on_steps_to_cost (Callstate fd args dst (f::fs) m) (S n) |
---|
| 1292 | | sbostc_ret : ∀rtv,dst,f,fs,m,n. frame_bound_on_steps_to_cost f n → state_bound_on_steps_to_cost (Returnstate rtv dst (f::fs) m) (S n) |
---|
[1675] | 1293 | . |
---|
| 1294 | |
---|
[1707] | 1295 | lemma state_bound_on_steps_to_cost_zero : ∀s. |
---|
| 1296 | ¬ state_bound_on_steps_to_cost s O. |
---|
[1705] | 1297 | #s % #H inversion H |
---|
[1707] | 1298 | [ #H46 #H47 #H48 #H49 #H50 #H51 #H52 #H53 destruct |
---|
| 1299 | whd in H50; @(bound_on_steps_to_cost1_inv_ind … H50) (* XXX inversion H50*) |
---|
| 1300 | #H55 #H56 #H57 #H58 #H59 #H60 #H61 normalize in H60; destruct |
---|
[1705] | 1301 | | #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 destruct |
---|
| 1302 | | #H13 #H14 #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 destruct |
---|
| 1303 | ] qed. |
---|
| 1304 | |
---|
| 1305 | lemma eval_steps : ∀ge,f,fs,m,tr,s'. |
---|
| 1306 | eval_statement ge (State f fs m) = Value ??? 〈tr,s'〉 → |
---|
| 1307 | steps_for_statement (lookup_present ?? (f_graph (func f)) (next f) (next_ok f)) = |
---|
[1713] | 1308 | match s' with [ State _ _ _ ⇒ 1 | Callstate _ _ _ _ _ ⇒ 2 | Returnstate _ _ _ _ ⇒ 2 | Finalstate _ ⇒ 1 ]. |
---|
[1705] | 1309 | #ge * #func #locals #next #next_ok #sp #dst #fs #m #tr #s' |
---|
| 1310 | whd in ⊢ (??%? → ?); |
---|
| 1311 | generalize in ⊢ (??(?%)? → ?); cases (lookup_present ??? next next_ok) |
---|
| 1312 | [ #l #LP whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1313 | | #cl #l #LP whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1314 | | #r #c #l #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1315 | | #ty #ty' #op #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev @bind_ok #v' #Ev' @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1316 | | #op #r1 #r2 #r3 #l #LP whd in ⊢ (??%? → ?); @bind_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #v' #Ev' @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1317 | | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1318 | | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #m' #Em whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1319 | | #id #rs #r #l #LP whd in ⊢ (??%? → ?); @bind_value #b #Eb @bind_ok #fd #Efd @bind_ok #vs #Evs whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1320 | | #r #rs #r' #l #LP whd in ⊢ (??%? → ?); @bind_value #fv #Efv @bind_ok #fd #Efd @bind_ok #vs #Evs whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1321 | | #r #l1 #l2 #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev @bind_ok #b #Eb whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1322 | | #r #ls #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev |
---|
| 1323 | cases v [ #E normalize in E; destruct | #sz #i | #f #E normalize in E; destruct | #r #E normalize in E; destruct | #p #E normalize in E; destruct ] |
---|
| 1324 | whd in ⊢ (??%? → ?); |
---|
| 1325 | generalize in ⊢ (??(?%)? → ?); |
---|
| 1326 | cases (nth_opt label (nat_of_bitvector (bitsize_of_intsize sz) i) ls) in ⊢ (???% → ??(match % with [ _ ⇒ ? | _ ⇒ ? ] ?)? → ?); |
---|
| 1327 | [ #e #E normalize in E; destruct |
---|
| 1328 | | #l #El whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1329 | ] |
---|
| 1330 | | #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1331 | ] qed. |
---|
| 1332 | |
---|
[1736] | 1333 | lemma bound_after_call : ∀ge,s,s',n. |
---|
| 1334 | state_bound_on_steps_to_cost s (S n) → |
---|
| 1335 | ∀CL:RTLabs_classify s = cl_call. |
---|
| 1336 | as_after_return (RTLabs_status ge) «s, CL» s' → |
---|
| 1337 | RTLabs_cost s' = false → |
---|
| 1338 | state_bound_on_steps_to_cost s' n. |
---|
| 1339 | #ge #s #s' #n #H inversion H |
---|
| 1340 | [ #f #fs #m #n' #S #E1 #E2 #_ #CL @⊥ cases (rtlabs_call_inv … CL) |
---|
| 1341 | #fn * #args * #dst * #stk * #m' #E destruct |
---|
| 1342 | | #fd #args #dst #f #fs #m #n' #S #E1 #E2 #_ destruct |
---|
| 1343 | whd in S; #CL cases s' |
---|
| 1344 | [ #f' #fs' #m' * #N #F #CS |
---|
| 1345 | %1 whd |
---|
| 1346 | inversion S |
---|
| 1347 | [ #l #n #P #CS' #E1 #E2 #_ destruct @⊥ |
---|
| 1348 | change with (is_cost_label ?) in CS:(??%?); >N in P CS'; >F >CS #P * |
---|
| 1349 | | #l #n #B #E1 #E2 #_ destruct <N <F @B |
---|
| 1350 | ] |
---|
| 1351 | | #fd' #args' #dst' #fs' #m' * |
---|
| 1352 | | #rv #dst' #fs' #m' * |
---|
| 1353 | | #r #E normalize in E; destruct |
---|
| 1354 | ] |
---|
| 1355 | | #rtv #dst #f #fs #m #n' #S #E1 #E2 #E3 destruct #CL normalize in CL; destruct |
---|
| 1356 | ] qed. |
---|
| 1357 | |
---|
[1707] | 1358 | lemma bound_after_step : ∀ge,s,tr,s',n. |
---|
| 1359 | state_bound_on_steps_to_cost s (S n) → |
---|
[1705] | 1360 | eval_statement ge s = Value ??? 〈tr, s'〉 → |
---|
[1706] | 1361 | RTLabs_cost s' = false → |
---|
[1705] | 1362 | (RTLabs_classify s' = cl_return ∨ RTLabs_classify s = cl_call) ∨ |
---|
[1707] | 1363 | state_bound_on_steps_to_cost s' n. |
---|
| 1364 | #ge #s #tr #s' #n #BOUND1 inversion BOUND1 |
---|
[1705] | 1365 | [ #f #fs #m #m #FS #E1 #E2 #_ destruct |
---|
| 1366 | #EVAL cases (eval_successor … EVAL) |
---|
| 1367 | [ /3/ |
---|
| 1368 | | * #l * #S1 #S2 #NC %2 |
---|
[1707] | 1369 | (* |
---|
| 1370 | cases (bound_on_steps_to_cost1_inv … FS ?) [2: @(next_ok f) ] |
---|
| 1371 | *) |
---|
| 1372 | @(bound_on_steps_to_cost1_inv_ind … FS) #next #n' #next_ok #IH #E1 #E2 #E3 destruct |
---|
[1705] | 1373 | inversion (eval_perserves … EVAL) |
---|
[1707] | 1374 | [ #ge0 #f0 #f' #fs' #m0 #m' #F #E4 #E5 #E6 #_ destruct |
---|
| 1375 | >(eval_steps … EVAL) in E2; #En normalize in En; |
---|
| 1376 | inversion F #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 destruct |
---|
| 1377 | %1 inversion (IH … S2) |
---|
| 1378 | [ #lx #nx #LPx #CSx #E1x #E2x @⊥ destruct |
---|
| 1379 | change with (RTLabs_cost (State (mk_frame H1 H7 lx LPx H5 H6) fs' m')) in CSx:(?%); |
---|
| 1380 | whd in S1:(??%?); destruct >NC in CSx; * |
---|
| 1381 | | whd in S1:(??%?); destruct #H71 #H72 #H73 #H74 #H75 #H76 destruct @H73 |
---|
[1706] | 1382 | ] |
---|
[1707] | 1383 | | #ge0 #f0 #fs' #m0 #fd #args #f' #dst #F #b #FFP #E4 #E5 #E6 #_ destruct |
---|
| 1384 | >(eval_steps … EVAL) in E2; #En normalize in En; |
---|
| 1385 | inversion F #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 destruct |
---|
| 1386 | %2 @IH normalize in S1; destruct @S2 |
---|
| 1387 | | #H14 #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 |
---|
[1705] | 1388 | destruct |
---|
[1707] | 1389 | | #H31 #H32 #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 destruct |
---|
[1705] | 1390 | normalize in S1; destruct |
---|
[1707] | 1391 | | #H44 #H45 #H46 #H47 #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 destruct |
---|
[1713] | 1392 | | #H267 #H268 #H269 #H270 #H271 #H272 #H273 #H274 destruct |
---|
[1705] | 1393 | ] |
---|
| 1394 | ] |
---|
| 1395 | | #H58 #H59 #H60 #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 #H70 destruct |
---|
| 1396 | /3/ |
---|
| 1397 | | #rtv #dst #f #fs #m #n' #FS #E1 #E2 #_ destruct |
---|
| 1398 | #EVAL #NC %2 inversion (eval_perserves … EVAL) |
---|
| 1399 | [ #H72 #H73 #H74 #H75 #H76 #H77 #H78 #H79 #H80 #H81 #H82 destruct |
---|
| 1400 | | #H84 #H85 #H86 #H87 #H88 #H89 #H90 #H91 #H92 #H93 #H94 #H95 #H96 #H97 #H98 destruct |
---|
| 1401 | | #H100 #H101 #H102 #H103 #H104 #H105 #H106 #H107 #H108 #H109 #H110 #H111 #H112 #H113 #H114 destruct |
---|
| 1402 | | #H116 #H117 #H118 #H119 #H120 #H121 #H122 #H123 #H124 #H125 #H126 destruct |
---|
| 1403 | | #ge' #f' #fs' #rtv' #dst' #f'' #m' #F #E1 #E2 #E3 #_ destruct |
---|
| 1404 | %1 whd in FS ⊢ %; |
---|
[1736] | 1405 | inversion (stack_preserved_return … EVAL) [ @refl | 2,4,5: #H141 #H142 #H143 #H144 #H145 #H146 #H147 try #H148 try #H149 destruct ] |
---|
| 1406 | #s1 #f1 #f2 #fs #m #FE #FR #SS1 #_ #E1 #E2 #_ destruct <FE |
---|
[1705] | 1407 | inversion SS1 [ #H163 #H164 #H165 #H166 #H167 #H168 destruct | #H170 #H171 #H172 #H173 #H174 #H175 #H176 #H177 #H178 destruct | #rtv #dst #fs0 #m0 #E1 #E2 #_ destruct ] |
---|
| 1408 | inversion F #func #locals #next #next_ok #sp #retdst #locals' #next' #next_ok' #E1 #E2 #_ destruct |
---|
[1707] | 1409 | inversion FS |
---|
| 1410 | [ #lx #nx #LPx #CSx #E1x #E2x @⊥ destruct |
---|
| 1411 | change with (RTLabs_cost (State (mk_frame func locals' lx ? sp retdst) fs m0)) in CSx:(?%); |
---|
| 1412 | >NC in CSx; * |
---|
| 1413 | | #lx #nx #H #E1x #E2x #_ destruct @H |
---|
| 1414 | ] |
---|
[1713] | 1415 | | #H284 #H285 #H286 #H287 #H288 #H289 #H290 #H291 destruct |
---|
[1705] | 1416 | ] |
---|
| 1417 | ] qed. |
---|
[1806] | 1418 | |
---|
| 1419 | |
---|
| 1420 | |
---|
| 1421 | |
---|
| 1422 | definition soundly_labelled_fn : internal_function → Prop ≝ |
---|
| 1423 | λf. ∀l. present … (f_graph f) l → ∃n. bound_on_steps_to_cost1 (f_graph f) l n. |
---|
| 1424 | |
---|
| 1425 | definition soundly_labelled_ge : genv → Prop ≝ |
---|
| 1426 | λge. ∀b,f. find_funct_ptr ?? ge b = Some ? (Internal ? f) → soundly_labelled_fn f. |
---|
| 1427 | |
---|
| 1428 | definition soundly_labelled_state : state → Prop ≝ |
---|
| 1429 | λs. match s with |
---|
| 1430 | [ State f fs m ⇒ soundly_labelled_fn (func f) ∧ All ? (λf. soundly_labelled_fn (func f)) fs |
---|
| 1431 | | Callstate fd _ _ fs _ ⇒ match fd with [ Internal fn ⇒ soundly_labelled_fn fn | External _ ⇒ True ] ∧ |
---|
| 1432 | All ? (λf. soundly_labelled_fn (func f)) fs |
---|
| 1433 | | Returnstate _ _ fs _ ⇒ All ? (λf. soundly_labelled_fn (func f)) fs |
---|
| 1434 | | Finalstate _ ⇒ True |
---|
| 1435 | ]. |
---|
| 1436 | |
---|
| 1437 | lemma steps_from_sound : ∀s. |
---|
| 1438 | RTLabs_cost s = true → |
---|
| 1439 | soundly_labelled_state s → |
---|
| 1440 | ∃n. state_bound_on_steps_to_cost s n. |
---|
| 1441 | * [ #f #fs #m #CS | #a #b #c #d #e #E normalize in E; destruct | #a #b #c #d #E normalize in E; destruct | #a #E normalize in E; destruct ] |
---|
| 1442 | whd in ⊢ (% → ?); * #SLF #_ |
---|
| 1443 | cases (SLF (next f) (next_ok f)) #n #B1 |
---|
| 1444 | %{n} % @B1 |
---|
| 1445 | qed. |
---|
| 1446 | |
---|
| 1447 | lemma soundly_labelled_state_step : ∀ge,s,tr,s'. |
---|
| 1448 | soundly_labelled_ge ge → |
---|
| 1449 | eval_statement ge s = Value ??? 〈tr,s'〉 → |
---|
| 1450 | soundly_labelled_state s → |
---|
| 1451 | soundly_labelled_state s'. |
---|
| 1452 | #ge #s #tr #s' #ENV #EV #S |
---|
| 1453 | inversion (eval_perserves … EV) |
---|
| 1454 | [ #ge' #f #f' #fs #m #m' #F #E1 #E2 #E3 #_ destruct |
---|
| 1455 | whd in S ⊢ %; inversion F #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 destruct @S |
---|
| 1456 | | #ge' #f #fs #m #fd #args #f' #dst #F #b #FFP #E1 #E2 #E3 #_ destruct |
---|
| 1457 | whd in S ⊢ %; % |
---|
| 1458 | [ cases fd in FFP ⊢ %; // #fn #FFP @ENV // |
---|
| 1459 | | inversion F #H30 #H31 #H32 #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 destruct @S |
---|
| 1460 | ] |
---|
| 1461 | | #ge' #fn #locals #next #nok #sp #fs #m #args #dst #m' #E1 #E2 #E3 #E4 destruct |
---|
| 1462 | whd in S ⊢ %; @S |
---|
| 1463 | | #ge' #f #fs #m #rtv #dst #m' #E1 #E2 #E3 #E4 destruct |
---|
| 1464 | whd in S ⊢ %; cases S // |
---|
| 1465 | | #ge' #f #fs #rtv #dst #f' #m #F #E1 #E2 #E3 #E4 destruct |
---|
| 1466 | whd in S ⊢ %; inversion F #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 destruct @S |
---|
| 1467 | | #ge' #r #dst #m #E1 #E2 #E3 #E4 destruct @I |
---|
| 1468 | ] qed. |
---|
| 1469 | |
---|
| 1470 | lemma soundly_labelled_state_preserved : ∀s,s'. |
---|
| 1471 | stack_preserved ends_with_ret s s' → |
---|
| 1472 | soundly_labelled_state s → |
---|
| 1473 | soundly_labelled_state s'. |
---|
| 1474 | #s0 #s0' #SP inversion SP |
---|
| 1475 | [ #H73 #H74 #H75 #H76 #H77 #H78 #H79 #H80 #H81 #H82 destruct |
---|
| 1476 | | #s1 #f #f' #fs #m #N #F #S1 #E1 #E2 #E3 #E4 destruct |
---|
| 1477 | inversion S1 |
---|
| 1478 | [ #f1 #fs1 #m1 #E1 #E2 #E3 destruct |
---|
| 1479 | * #_ #S whd in S; |
---|
| 1480 | inversion F #H96 #H97 #H98 #H99 #H100 #H101 #H102 #H103 #H104 #H105 #H106 #H107 |
---|
| 1481 | destruct @S |
---|
| 1482 | | #fd #args #dst #f1 #fs1 #m1 #E1 #E2 #E3 destruct * #_ * #_ #S |
---|
| 1483 | inversion F #H96 #H97 #H98 #H99 #H100 #H101 #H102 #H103 #H104 #H105 #H106 #H107 |
---|
| 1484 | destruct @S |
---|
| 1485 | | #rtv #dst #fs1 #m1 #E1 #E2 #E3 destruct #S |
---|
| 1486 | inversion F #H96 #H97 #H98 #H99 #H100 #H101 #H102 #H103 #H104 #H105 #H106 #H107 |
---|
| 1487 | destruct @S |
---|
| 1488 | ] |
---|
| 1489 | | // |
---|
| 1490 | | // |
---|
| 1491 | ] qed. |
---|
| 1492 | |
---|
[1653] | 1493 | (* When constructing an infinite trace, we need to be able to grab the finite |
---|
| 1494 | portion of the trace for the next [trace_label_diverges] constructor. We |
---|
| 1495 | use the fact that the trace is soundly labelled to achieve this. *) |
---|
| 1496 | |
---|
[1805] | 1497 | record remainder_ok (ge:genv) (s:state) (t:flat_trace io_out io_in ge s) : Type[0] ≝ { |
---|
| 1498 | ro_well_cost_labelled: well_cost_labelled_state s; |
---|
[1806] | 1499 | ro_soundly_labelled: soundly_labelled_state s; |
---|
[1805] | 1500 | ro_no_termination: Not (∃depth. inhabited (will_return ge depth s t)); |
---|
| 1501 | ro_not_undefined: not_wrong … t; |
---|
| 1502 | ro_not_final: RTLabs_is_final s = None ? |
---|
| 1503 | }. |
---|
| 1504 | |
---|
[1670] | 1505 | inductive finite_prefix (ge:genv) : state → Prop ≝ |
---|
[1653] | 1506 | | fp_tal : ∀s,s'. |
---|
| 1507 | trace_any_label (RTLabs_status ge) doesnt_end_with_ret s s' → |
---|
[1805] | 1508 | ∀t:flat_trace io_out io_in ge s'. |
---|
| 1509 | remainder_ok ge s' t → |
---|
[1653] | 1510 | finite_prefix ge s |
---|
[1806] | 1511 | | fp_tac : ∀s1,s2,s3,tr. |
---|
| 1512 | trace_any_call (RTLabs_status ge) s1 s2 → |
---|
| 1513 | well_cost_labelled_state s2 → |
---|
| 1514 | eval_statement ge s2 = Value ??? 〈tr,s3〉 → |
---|
| 1515 | ∀t:flat_trace io_out io_in ge s3. |
---|
| 1516 | remainder_ok ge s3 t → |
---|
| 1517 | finite_prefix ge s1 |
---|
[1653] | 1518 | . |
---|
| 1519 | |
---|
| 1520 | definition fp_add_default : ∀ge,s,s'. |
---|
| 1521 | RTLabs_classify s = cl_other → |
---|
| 1522 | finite_prefix ge s' → |
---|
| 1523 | (∃t. eval_statement ge s = Value ??? 〈t,s'〉) → |
---|
| 1524 | RTLabs_cost s' = false → |
---|
| 1525 | finite_prefix ge s ≝ |
---|
| 1526 | λge,s,s',OTHER,fp. |
---|
| 1527 | match fp return λs'.λ_. (∃t. eval_statement ge ? = Value ??? 〈t,s'〉) → RTLabs_cost s' = false → finite_prefix ge s with |
---|
[1805] | 1528 | [ fp_tal s' sf TAL rem rok ⇒ λEVAL, NOT_COST. fp_tal ge s sf |
---|
[1670] | 1529 | (tal_step_default (RTLabs_status ge) doesnt_end_with_ret s s' sf EVAL TAL OTHER (RTLabs_not_cost … NOT_COST)) |
---|
[1805] | 1530 | rem rok |
---|
[1806] | 1531 | | fp_tac s1 s2 s3 tr TAC WCL2 EV rem rok ⇒ λEVAL, NOT_COST. fp_tac ge s s2 s3 tr |
---|
| 1532 | (tac_step_default (RTLabs_status ge) ??? EVAL TAC OTHER (RTLabs_not_cost … NOT_COST)) |
---|
| 1533 | WCL2 EV rem rok |
---|
[1653] | 1534 | ]. |
---|
[1670] | 1535 | |
---|
[1806] | 1536 | definition fp_add_terminating_call : ∀ge,s,s1,s''. |
---|
[1653] | 1537 | (∃t. eval_statement ge s = Value ??? 〈t,s1〉) → |
---|
| 1538 | ∀CALL:RTLabs_classify s = cl_call. |
---|
[1806] | 1539 | finite_prefix ge s'' → |
---|
| 1540 | trace_label_return (RTLabs_status ge) s1 s'' → |
---|
| 1541 | as_after_return (RTLabs_status ge) (mk_Sig ?? s CALL) s'' → |
---|
| 1542 | RTLabs_cost s'' = false → |
---|
[1653] | 1543 | finite_prefix ge s ≝ |
---|
[1806] | 1544 | λge,s,s1,s'',EVAL,CALL,fp. |
---|
| 1545 | match fp return λs''.λ_. trace_label_return (RTLabs_status ge) ? s'' → as_after_return (RTLabs_status ge) ? s'' → RTLabs_cost s'' = false → finite_prefix ge s with |
---|
| 1546 | [ fp_tal s'' sf TAL rem rok ⇒ λTLR,RET,NOT_COST. fp_tal ge s sf |
---|
| 1547 | (tal_step_call (RTLabs_status ge) doesnt_end_with_ret s s1 s'' sf EVAL CALL RET TLR (RTLabs_not_cost … NOT_COST) TAL) |
---|
[1805] | 1548 | rem rok |
---|
[1806] | 1549 | | fp_tac s'' s2 s3 tr TAC WCL2 EV rem rok ⇒ λTLR,RET,NOT_COST. fp_tac ge s s2 s3 tr |
---|
| 1550 | (tac_step_call (RTLabs_status ge) s s'' s2 s1 EVAL CALL RET TLR (RTLabs_not_cost … NOT_COST) TAC) |
---|
| 1551 | WCL2 EV rem rok |
---|
[1653] | 1552 | ]. |
---|
[1670] | 1553 | |
---|
[1765] | 1554 | lemma not_return_to_not_final : ∀ge,s,tr,s'. |
---|
| 1555 | eval_statement ge s = Value ??? 〈tr, s'〉 → |
---|
| 1556 | RTLabs_classify s ≠ cl_return → |
---|
| 1557 | RTLabs_is_final s' = None ?. |
---|
| 1558 | #ge #s #tr #s' #EV |
---|
| 1559 | inversion (eval_perserves … EV) // |
---|
| 1560 | #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 #CL |
---|
| 1561 | @⊥ @(absurd ?? CL) @refl |
---|
| 1562 | qed. |
---|
| 1563 | |
---|
[1670] | 1564 | definition termination_oracle ≝ ∀ge,depth,s,trace. |
---|
[1671] | 1565 | inhabited (will_return ge depth s trace) ∨ ¬ inhabited (will_return ge depth s trace). |
---|
[1670] | 1566 | |
---|
| 1567 | let rec finite_segment ge s n trace |
---|
| 1568 | (ORACLE: termination_oracle) |
---|
[1805] | 1569 | (TRACE_OK: remainder_ok ge s trace) |
---|
[1670] | 1570 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
[1806] | 1571 | (ENV_SOUNDLY_LABELLED: soundly_labelled_ge ge) |
---|
| 1572 | (STATE_SOUNDLY_LABELLED: soundly_labelled_state s) |
---|
[1707] | 1573 | (LABEL_LIMIT: state_bound_on_steps_to_cost s n) |
---|
[1671] | 1574 | on n : finite_prefix ge s ≝ |
---|
[1707] | 1575 | match n return λn. state_bound_on_steps_to_cost s n → finite_prefix ge s with |
---|
[1705] | 1576 | [ O ⇒ λLABEL_LIMIT. ⊥ |
---|
| 1577 | | S n' ⇒ |
---|
[1805] | 1578 | match trace return λs,trace. remainder_ok ge s trace → state_bound_on_steps_to_cost s (S n') → finite_prefix ge s with |
---|
| 1579 | [ ft_stop st FINAL ⇒ λTRACE_OK,LABEL_LIMIT. ⊥ |
---|
| 1580 | | ft_step start tr next EV trace' ⇒ λTRACE_OK,LABEL_LIMIT. |
---|
[1670] | 1581 | match RTLabs_classify start return λx. RTLabs_classify start = x → ? with |
---|
| 1582 | [ cl_other ⇒ λCL. |
---|
[1805] | 1583 | let TRACE_OK' ≝ ? in |
---|
[1670] | 1584 | match RTLabs_cost next return λx. RTLabs_cost next = x → ? with |
---|
| 1585 | [ true ⇒ λCS. |
---|
[1805] | 1586 | fp_tal ge start next (tal_base_not_return (RTLabs_status ge) start next ?? CS) trace' TRACE_OK' |
---|
[1670] | 1587 | | false ⇒ λCS. |
---|
[1806] | 1588 | let fs ≝ finite_segment ge next n' trace' ORACLE TRACE_OK' ENV_COSTLABELLED ENV_SOUNDLY_LABELLED ?? in |
---|
[1670] | 1589 | fp_add_default ge ?? CL fs ? CS |
---|
| 1590 | ] (refl ??) |
---|
| 1591 | | cl_jump ⇒ λCL. |
---|
[1805] | 1592 | fp_tal ge start next (tal_base_not_return (RTLabs_status ge) start next ?? ?) trace' ? |
---|
[1707] | 1593 | | cl_call ⇒ λCL. |
---|
[1671] | 1594 | match ORACLE ge O next trace' return λ_. finite_prefix ge start with |
---|
| 1595 | [ or_introl TERMINATES ⇒ |
---|
| 1596 | match TERMINATES with [ witness TERMINATES ⇒ |
---|
| 1597 | let tlr ≝ make_label_return' ge O next trace' ENV_COSTLABELLED ?? TERMINATES in |
---|
[1805] | 1598 | let TRACE_OK' ≝ ? in |
---|
[1671] | 1599 | match RTLabs_cost (new_state … tlr) return λx. RTLabs_cost (new_state … tlr) = x → finite_prefix ge start with |
---|
[1805] | 1600 | [ true ⇒ λCS. fp_tal ge start (new_state … tlr) (tal_base_call (RTLabs_status ge) start next (new_state … tlr) ? CL ? (new_trace … tlr) CS) (remainder … tlr) TRACE_OK' |
---|
[1707] | 1601 | | false ⇒ λCS. |
---|
[1806] | 1602 | let fs ≝ finite_segment ge (new_state … tlr) n' (remainder … tlr) ORACLE TRACE_OK' ENV_COSTLABELLED ENV_SOUNDLY_LABELLED ?? in |
---|
[1707] | 1603 | fp_add_terminating_call … fs (new_trace … tlr) ? CS |
---|
[1671] | 1604 | ] (refl ??) |
---|
| 1605 | ] |
---|
| 1606 | | or_intror NO_TERMINATION ⇒ |
---|
[1806] | 1607 | fp_tac ????? (tac_base (RTLabs_status ge) start CL) ? EV trace' ? |
---|
[1707] | 1608 | ] |
---|
[1670] | 1609 | | cl_return ⇒ λCL. ⊥ |
---|
| 1610 | ] (refl ??) |
---|
[1805] | 1611 | | ft_wrong start m EV ⇒ λTRACE_OK,LABEL_LIMIT. ⊥ |
---|
| 1612 | ] TRACE_OK |
---|
[1705] | 1613 | ] LABEL_LIMIT. |
---|
[1707] | 1614 | [ cases (state_bound_on_steps_to_cost_zero s) /2/ |
---|
[1805] | 1615 | | @(absurd … (ro_not_final … TRACE_OK) FINAL) |
---|
| 1616 | | @(absurd ?? (ro_no_termination … TRACE_OK)) |
---|
[1670] | 1617 | %{0} % @wr_base // |
---|
[1805] | 1618 | | @(well_cost_labelled_jump … EV) /2/ |
---|
| 1619 | | 5,6,8,9,10,11: /3/ |
---|
| 1620 | | % [ @(well_cost_labelled_state_step … EV) /2/ |
---|
[1806] | 1621 | | @(soundly_labelled_state_step … EV) /2/ |
---|
[1805] | 1622 | | @(not_to_not … (ro_no_termination … TRACE_OK)) * #depth * #TM1 %{depth} % @wr_step /2/ |
---|
| 1623 | | @(still_not_wrong … EV) /2/ |
---|
| 1624 | | @(not_return_to_not_final … EV) >CL % #E destruct |
---|
| 1625 | ] |
---|
[1707] | 1626 | | /2/ |
---|
[1806] | 1627 | | @(soundly_labelled_state_preserved … (stack_ok … tlr)) |
---|
| 1628 | @(soundly_labelled_state_step … EV) /2/ |
---|
[1736] | 1629 | | @(bound_after_call ge … LABEL_LIMIT CL ? CS) |
---|
| 1630 | @(RTLabs_after_call … CL EV) @(stack_ok … tlr) |
---|
[1805] | 1631 | | % [ /2/ |
---|
[1806] | 1632 | | @(soundly_labelled_state_preserved … (stack_ok … tlr)) |
---|
| 1633 | @(soundly_labelled_state_step … EV) /2/ |
---|
[1805] | 1634 | | @(not_to_not … (ro_no_termination … TRACE_OK)) * #depth * #TM1 %{depth} % |
---|
| 1635 | @wr_call // |
---|
| 1636 | @(will_return_prepend … TERMINATES … TM1) |
---|
| 1637 | cases (terminates … tlr) // |
---|
| 1638 | | @(will_return_not_wrong … TERMINATES) |
---|
| 1639 | [ @(still_not_wrong … EV) /2/ |
---|
| 1640 | | cases (terminates … tlr) // |
---|
| 1641 | ] |
---|
| 1642 | | (* By stack preservation we cannot be in the final state *) |
---|
| 1643 | inversion (stack_ok … tlr) |
---|
| 1644 | [ #H101 #H102 #H103 #H104 #H105 #H106 #H107 #H108 #H109 destruct |
---|
| 1645 | | #s1 #f #f' #fs #m #N #F #S #E1 #E2 #E3 #E4 -TERMINATES destruct @refl |
---|
| 1646 | | #s1 #r #S #E1 #E2 #E3 #E4 -TERMINATES destruct |
---|
| 1647 | cases (rtlabs_call_inv … CL) #fd * #args * #dst * #stk * #m #E destruct |
---|
| 1648 | inversion (eval_perserves … EV) |
---|
| 1649 | [ 1,2,4,5,6: #H111 #H112 #H113 #H114 #H115 #H116 #H117 #H118 try #H119 try #H120 try #H121 try #H122 try #H123 -TRACE_OK destruct ] |
---|
| 1650 | #ge' #fn #locals #next #nok #sp #fs #m' #args' #dst' #m'' #E1 #E2 #E3 #E4 -TRACE_OK destruct |
---|
| 1651 | inversion S |
---|
| 1652 | [ #f #fs0 #m #E1 #E2 #E3 destruct | *: #H123 #H124 #H125 #H126 #H127 #H128 #H129 try #H130 destruct ] |
---|
| 1653 | (* state_bound_on_steps_to_cost needs to know about the current stack frame, |
---|
| 1654 | so we can use it as a witness that at least one frame exists *) |
---|
| 1655 | inversion LABEL_LIMIT |
---|
| 1656 | #H141 #H142 #H143 #H144 #H145 #H146 #H147 #H148 try #H150 destruct |
---|
| 1657 | | #H173 #H174 #H175 #H176 #H177 #H178 #H179 #H180 #H181 destruct |
---|
| 1658 | ] |
---|
| 1659 | ] |
---|
| 1660 | | @(well_cost_labelled_state_step … EV) /2/ |
---|
| 1661 | | @(well_cost_labelled_call … EV) /2/ |
---|
[1806] | 1662 | | /2/ |
---|
| 1663 | | % [ @(well_cost_labelled_state_step … EV) /2/ |
---|
| 1664 | | @(soundly_labelled_state_step … EV) /2/ |
---|
| 1665 | | @(not_to_not … NO_TERMINATION) * #depth * #TM % |
---|
| 1666 | @(will_return_lower … TM) // |
---|
| 1667 | | @(still_not_wrong … EV) /2/ |
---|
| 1668 | | @(not_return_to_not_final … EV) >CL % #E destruct |
---|
| 1669 | ] |
---|
| 1670 | | 20,21,22: /2/ |
---|
| 1671 | | @(soundly_labelled_state_step … EV) /2/ |
---|
[1707] | 1672 | | cases (bound_after_step … LABEL_LIMIT EV ?) |
---|
[1805] | 1673 | [ * [ #TERMINATES @⊥ @(absurd ?? (ro_no_termination … TRACE_OK)) %{0} % @wr_step [ %1 // | |
---|
[1707] | 1674 | inversion trace' |
---|
[1805] | 1675 | [ #s0 #FINAL #E1 #E2 -TRACE_OK' destruct @⊥ |
---|
[1765] | 1676 | @(absurd ?? FINAL) @(not_return_to_not_final … EV) |
---|
| 1677 | % >CL #E destruct |
---|
[1805] | 1678 | | #s1 #tr1 #s2 #EVAL' #trace'' #E1 #E2 -TRACE_OK' destruct |
---|
[1765] | 1679 | @wr_base // |
---|
[1805] | 1680 | | #H99 #H100 #H101 #H102 #H103 -TRACE_OK' destruct |
---|
| 1681 | inversion (ro_not_undefined … TRACE_OK) |
---|
[1707] | 1682 | [ #H137 #H138 #H139 #H140 #H141 destruct |
---|
| 1683 | | #H143 #H144 #H145 #H146 #H147 #H148 #H149 #H150 #H151 destruct |
---|
| 1684 | inversion H148 |
---|
| 1685 | [ #H153 #H154 #H155 #H156 #H157 destruct |
---|
| 1686 | | #H159 #H160 #H161 #H162 #H163 #H164 #H165 #H166 #H167 destruct |
---|
| 1687 | ] |
---|
| 1688 | ] |
---|
| 1689 | ] |
---|
| 1690 | ] |
---|
| 1691 | | >CL #E destruct |
---|
| 1692 | ] |
---|
| 1693 | | // |
---|
| 1694 | | // |
---|
| 1695 | ] |
---|
[1805] | 1696 | | % [ @(well_cost_labelled_state_step … EV) /2/ |
---|
[1806] | 1697 | | @(soundly_labelled_state_step … EV) /2/ |
---|
[1805] | 1698 | | @(not_to_not … (ro_no_termination … TRACE_OK)) |
---|
| 1699 | * #depth * #TERM %{depth} % @wr_step /2/ |
---|
| 1700 | | @(still_not_wrong … (ro_not_undefined … TRACE_OK)) |
---|
| 1701 | | @(not_return_to_not_final … EV) >CL % #E destruct |
---|
| 1702 | ] |
---|
| 1703 | | inversion (ro_not_undefined … TRACE_OK) |
---|
[1707] | 1704 | [ #H169 #H170 #H171 #H172 #H173 destruct |
---|
| 1705 | | #H175 #H176 #H177 #H178 #H179 #H180 #H181 #H182 #H183 destruct |
---|
| 1706 | ] |
---|
[1765] | 1707 | ] qed. |
---|
[1670] | 1708 | |
---|
[1806] | 1709 | (* FIXME: |
---|
[1784] | 1710 | |
---|
[1806] | 1711 | This isn't going to work in this form: the existential isn't the coinductive |
---|
| 1712 | type, so this isn't technically a cofixpoint. Try to return just the |
---|
| 1713 | structured trace won't either, because the termination oracle is in Prop. |
---|
| 1714 | |
---|
| 1715 | Not sure how to get out of this situation... |
---|
| 1716 | *) |
---|
[1784] | 1717 | |
---|
[1651] | 1718 | let corec make_label_diverges ge s |
---|
| 1719 | (trace: flat_trace io_out io_in ge s) |
---|
[1784] | 1720 | (ORACLE: termination_oracle) |
---|
[1805] | 1721 | (TRACE_OK: remainder_ok ge s trace) |
---|
[1651] | 1722 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
[1784] | 1723 | (ENV_SOUNDLY_LABELLED: soundly_labelled_ge ge) |
---|
| 1724 | (STATE_SOUNDLY_LABELLED: soundly_labelled_state s) |
---|
[1651] | 1725 | (STATEMENT_COSTLABEL: RTLabs_cost s = true) (* current statement is a cost label *) |
---|
[1784] | 1726 | : ∃T:trace_label_diverges (RTLabs_status ge) s. True ≝ |
---|
| 1727 | match steps_from_sound s STATEMENT_COSTLABEL STATE_SOUNDLY_LABELLED with |
---|
| 1728 | [ ex_intro n B ⇒ |
---|
[1806] | 1729 | match finite_segment ge s n trace ORACLE TRACE_OK ENV_COSTLABELLED ENV_SOUNDLY_LABELLED STATE_SOUNDLY_LABELLED B |
---|
[1784] | 1730 | return λs.λ_. RTLabs_cost s = true → ∃T:trace_label_diverges (RTLabs_status ge) s. True |
---|
| 1731 | with |
---|
[1805] | 1732 | [ fp_tal s s' T t tOK ⇒ λSTATEMENT_COSTLABEL. |
---|
| 1733 | match make_label_diverges ge s' t ORACLE tOK ENV_COSTLABELLED ENV_SOUNDLY_LABELLED ?? with |
---|
[1784] | 1734 | [ ex_intro T' _ ⇒ |
---|
| 1735 | ex_intro ?? (tld_step (RTLabs_status ge) s s' (tll_base … T STATEMENT_COSTLABEL) T') I |
---|
| 1736 | ] |
---|
[1806] | 1737 | | fp_tac s s2 s3 tr T WCL2 EV t tOK ⇒ λSTATEMENT_COSTLABEL. |
---|
| 1738 | match make_label_diverges ge s3 t ORACLE tOK ENV_COSTLABELLED ENV_SOUNDLY_LABELLED ?? with |
---|
| 1739 | [ ex_intro T' _ ⇒ |
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| 1740 | ex_intro ?? (tld_base (RTLabs_status ge) s s2 s3 (tlc_base … T STATEMENT_COSTLABEL) ?? T') ? |
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| 1741 | ] |
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[1784] | 1742 | ] STATEMENT_COSTLABEL |
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| 1743 | ]. |
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[1806] | 1744 | [ /2/ |
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| 1745 | | @(trace_any_label_label … T) |
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| 1746 | | @(trace_any_call_call … T) |
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| 1747 | | %{tr} @EV |
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| 1748 | | @I |
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| 1749 | | /2/ |
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| 1750 | | @(well_cost_labelled_call … EV) // @(trace_any_call_call … T) |
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| 1751 | ] (* XXX fails, see above. *) qed. |
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