| 1 | include "basics/types.ma". |
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| 2 | include "basics/bool.ma". |
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| 3 | |
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| 4 | inductive status_class : Type[0] ≝ |
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| 5 | | cl_return : status_class |
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| 6 | | cl_jump : status_class |
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| 7 | | cl_call : status_class |
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| 8 | | cl_other : status_class |
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| 9 | . |
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| 10 | |
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| 11 | record abstract_status : Type[1] ≝ { |
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| 12 | as_status :> Type[0]; |
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| 13 | as_execute : as_status → as_status → Prop; |
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| 14 | as_classifier : as_status → status_class → Prop; |
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| 15 | as_costed : as_status → Prop; |
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| 16 | as_after_return : (Σs:as_status. as_classifier s cl_call) → as_status → Prop |
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| 17 | }. |
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| 18 | |
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| 19 | inductive trace_ends_with_ret: Type[0] ≝ |
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| 20 | | ends_with_ret: trace_ends_with_ret |
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| 21 | | doesnt_end_with_ret: trace_ends_with_ret. |
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| 22 | |
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| 23 | inductive trace_label_return (S:abstract_status) : S → S → Type[0] ≝ |
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| 24 | | tlr_base: |
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| 25 | ∀status_before: S. |
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| 26 | ∀status_after: S. |
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| 27 | trace_label_label S ends_with_ret status_before status_after → |
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| 28 | trace_label_return S status_before status_after |
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| 29 | | tlr_step: |
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| 30 | ∀status_initial: S. |
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| 31 | ∀status_labelled: S. |
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| 32 | ∀status_final: S. |
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| 33 | trace_label_label S doesnt_end_with_ret status_initial status_labelled → |
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| 34 | trace_label_return S status_labelled status_final → |
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| 35 | trace_label_return S status_initial status_final |
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| 36 | with trace_label_label: trace_ends_with_ret → S → S → Type[0] ≝ |
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| 37 | | tll_base: |
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| 38 | ∀ends_flag: trace_ends_with_ret. |
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| 39 | ∀start_status: S. |
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| 40 | ∀end_status: S. |
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| 41 | trace_any_label S ends_flag start_status end_status → |
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| 42 | as_costed S start_status → |
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| 43 | trace_label_label S ends_flag start_status end_status |
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| 44 | with trace_any_label: trace_ends_with_ret → S → S → Type[0] ≝ |
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| 45 | | tal_base_not_return: |
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| 46 | ∀start_status: S. |
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| 47 | ∀final_status: S. |
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| 48 | as_execute S start_status final_status → |
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| 49 | ¬ as_classifier S start_status cl_return → |
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| 50 | as_costed S final_status → |
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| 51 | trace_any_label S doesnt_end_with_ret start_status final_status |
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| 52 | | tal_base_return: |
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| 53 | ∀start_status: S. |
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| 54 | ∀final_status: S. |
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| 55 | as_execute S start_status final_status → |
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| 56 | as_classifier S start_status cl_return → |
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| 57 | trace_any_label S ends_with_ret start_status final_status |
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| 58 | | tal_step_call: |
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| 59 | ∀end_flag: trace_ends_with_ret. |
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| 60 | ∀status_pre_fun_call: S. |
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| 61 | ∀status_start_fun_call: S. |
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| 62 | ∀status_after_fun_call: S. |
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| 63 | ∀status_final: S. |
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| 64 | as_execute S status_pre_fun_call status_start_fun_call → |
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| 65 | ∀H:as_classifier S status_pre_fun_call cl_call. |
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| 66 | as_after_return S (mk_Sig ?? status_pre_fun_call H) status_after_fun_call → |
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| 67 | trace_label_return S status_start_fun_call status_after_fun_call → |
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| 68 | trace_any_label S end_flag status_after_fun_call status_final → |
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| 69 | trace_any_label S end_flag status_pre_fun_call status_final |
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| 70 | | tal_step_default: |
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| 71 | ∀end_flag: trace_ends_with_ret. |
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| 72 | ∀status_pre: S. |
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| 73 | ∀status_init: S. |
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| 74 | ∀status_end: S. |
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| 75 | as_execute S status_pre status_init → |
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| 76 | trace_any_label S end_flag status_init status_end → |
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| 77 | as_classifier S status_pre cl_other → |
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| 78 | ¬ (as_costed S status_init) → |
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| 79 | trace_any_label S end_flag status_pre status_end. |
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| 80 | |
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| 81 | |
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| 82 | coinductive trace_label_diverges (S:abstract_status) : S → Type[0] ≝ |
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| 83 | | tld_step: |
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| 84 | ∀status_initial: S. |
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| 85 | ∀status_labelled: S. |
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| 86 | trace_label_label S doesnt_end_with_ret status_initial status_labelled → |
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| 87 | trace_label_diverges S status_labelled → |
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| 88 | trace_label_diverges S status_initial |
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| 89 | | tld_base: |
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| 90 | ∀status_initial: S. |
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| 91 | ∀status_pre_fun_call: S. |
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| 92 | ∀status_after_fun_call: S. |
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| 93 | ∀status_start_fun_call: S. |
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| 94 | trace_label_call S status_initial status_pre_fun_call → |
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| 95 | as_execute S status_pre_fun_call status_start_fun_call → |
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| 96 | ∀H:as_classifier S status_pre_fun_call cl_call. |
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| 97 | trace_label_diverges S status_start_fun_call → |
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| 98 | trace_label_diverges S status_initial |
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| 99 | |
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| 100 | with trace_label_call: S → S → Type[0] ≝ |
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| 101 | | tlc_base: |
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| 102 | ∀start_status: S. |
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| 103 | ∀end_status: S. |
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| 104 | trace_any_call S start_status end_status → |
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| 105 | as_costed S start_status → |
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| 106 | trace_label_call S start_status end_status |
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| 107 | |
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| 108 | with trace_any_call: S → S → Type[0] ≝ |
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| 109 | | tac_base: |
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| 110 | ∀status: S. |
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| 111 | as_classifier S status cl_call → |
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| 112 | trace_any_call S status status |
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| 113 | | tac_step_call: |
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| 114 | ∀status_pre_fun_call: S. |
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| 115 | ∀status_after_fun_call: S. |
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| 116 | ∀status_final: S. |
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| 117 | ∀status_start_fun_call: S. |
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| 118 | as_execute S status_pre_fun_call status_start_fun_call → |
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| 119 | ∀H:as_classifier S status_pre_fun_call cl_call. |
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| 120 | as_after_return S (mk_Sig ?? status_pre_fun_call H) status_after_fun_call → |
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| 121 | trace_label_return S status_start_fun_call status_after_fun_call → |
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| 122 | trace_any_call S status_after_fun_call status_final → |
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| 123 | trace_any_call S status_pre_fun_call status_final |
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| 124 | | tac_step_default: |
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| 125 | ∀status_pre: S. |
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| 126 | ∀status_end: S. |
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| 127 | ∀status_init: S. |
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| 128 | as_execute S status_pre status_init → |
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| 129 | trace_any_call S status_init status_end → |
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| 130 | as_classifier S status_pre cl_other → |
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| 131 | ¬ (as_costed S status_pre) → |
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| 132 | trace_any_call S status_pre status_end. |
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| 133 | |
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| 134 | let rec trace_any_label_label S s s' f |
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| 135 | (tr:trace_any_label S f s s') on tr : match f with [ doesnt_end_with_ret ⇒ as_costed S s' | ends_with_ret ⇒ True ] ≝ |
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| 136 | match tr return λf,s,s',tr. match f with [ doesnt_end_with_ret ⇒ as_costed S s' | ends_with_ret ⇒ True ] with |
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| 137 | [ tal_base_not_return start final _ _ C ⇒ C |
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| 138 | | tal_base_return _ _ _ _ ⇒ I |
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| 139 | | tal_step_call f pre start after final X C RET LR tr' ⇒ trace_any_label_label … tr' |
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| 140 | | tal_step_default f pre init end X tr' C C' ⇒ trace_any_label_label … tr' |
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| 141 | ]. |
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| 142 | |
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| 143 | lemma trace_label_label_label : ∀S,s,s',f. |
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| 144 | ∀tr:trace_label_label S f s s'. match f with [ doesnt_end_with_ret ⇒ as_costed S s' | ends_with_ret ⇒ True ]. |
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| 145 | #S #s #s' #f #tr |
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| 146 | cases tr |
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| 147 | #f #start #end #tr' #C @(trace_any_label_label … tr') |
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| 148 | qed. |
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| 149 | |
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