[1537] | 1 | |
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| 2 | include "RTLabs/semantics.ma". |
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[2218] | 3 | include "RTLabs/CostSpec.ma". |
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[1537] | 4 | include "common/StructuredTraces.ma". |
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| 5 | |
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[1552] | 6 | discriminator status_class. |
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[1537] | 7 | |
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[2044] | 8 | (* We augment states with a stack of function blocks (i.e. pointers) so that we |
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| 9 | can make sensible "program counters" for the abstract state definition, along |
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| 10 | with a proof that we will get the correct code when we do the lookup (which |
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| 11 | is done to find cost labels given a pc). |
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| 12 | |
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| 13 | Adding them to the semantics is an alternative, more direct approach. |
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| 14 | However, it makes animating the semantics extremely difficult, because it |
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| 15 | is hard to avoid normalising and displaying irrelevant parts of the global |
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| 16 | environment and proofs. |
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| 17 | |
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| 18 | We use blocks rather than identifiers because the global environments go |
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| 19 | |
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| 20 | identifier → block → definition |
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| 21 | |
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| 22 | and we'd have to introduce backwards lookups to find identifiers for |
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| 23 | function pointers. |
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| 24 | *) |
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| 25 | |
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| 26 | definition Ras_Fn_Match ≝ |
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| 27 | λge,state,fn_stack. |
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| 28 | match state with |
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| 29 | [ State f fs m ⇒ All2 … (λfr,b. find_funct_ptr ? ge b = Some ? (Internal ? (func fr))) (f::fs) fn_stack |
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| 30 | | Callstate fd _ _ fs _ ⇒ |
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| 31 | match fn_stack with |
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| 32 | [ nil ⇒ False |
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| 33 | | cons h t ⇒ find_funct_ptr ? ge h = Some ? fd ∧ |
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| 34 | All2 … (λfr,b. find_funct_ptr ? ge b = Some ? (Internal ? (func fr))) fs t |
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| 35 | ] |
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| 36 | | Returnstate _ _ fs _ ⇒ |
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| 37 | All2 … (λfr,b. find_funct_ptr ? ge b = Some ? (Internal ? (func fr))) fs fn_stack |
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| 38 | | Finalstate _ ⇒ fn_stack = [ ] |
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| 39 | ]. |
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| 40 | |
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| 41 | record RTLabs_state (ge:genv) : Type[0] ≝ { |
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| 42 | Ras_state :> state; |
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| 43 | Ras_fn_stack : list block; |
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| 44 | Ras_fn_match : Ras_Fn_Match ge Ras_state Ras_fn_stack |
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| 45 | }. |
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| 46 | |
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| 47 | (* Given a plain step of the RTLabs semantics, give the next state along with |
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| 48 | the shadow stack of function block numbers. Carefully defined so that the |
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| 49 | coercion back to the plain state always reduces. *) |
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| 50 | definition next_state : ∀ge. ∀s:RTLabs_state ge. ∀s':state. ∀t. |
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| 51 | eval_statement ge s = Value ??? 〈t,s'〉 → RTLabs_state ge ≝ |
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| 52 | λge,s,s',t,EX. mk_RTLabs_state ge s' |
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| 53 | (match s' return λs'. eval_statement ge s = Value ??? 〈t,s'〉 → ? with [ State _ _ _ ⇒ λ_. Ras_fn_stack … s | Callstate _ _ _ _ _ ⇒ λEX. func_block_of_exec … EX::Ras_fn_stack … s | Returnstate _ _ _ _ ⇒ λ_. tail … (Ras_fn_stack … s) | Finalstate _ ⇒ λ_. [ ] ] EX) |
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| 54 | ?. |
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| 55 | cases s' in EX ⊢ %; |
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| 56 | [ -s' #f #fs #m cases s -s #s #stk #mtc #EX whd in ⊢ (???%); inversion (eval_preserves … EX) |
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| 57 | [ #ge' #f1 #f2 #fs' #m1 #m2 #F #E1 #E2 #E3 #E4 destruct |
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| 58 | whd cases stk in mtc ⊢ %; [ * ] |
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| 59 | #hd #tl * #M1 #M2 % [ inversion F #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 destruct // |
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| 60 | | @M2 ] |
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| 61 | | #ge' #f1 #fs1 #m1 #fd #args #f' #dst #F #b #FFP #E1 #E2 #E3 #E4 destruct |
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| 62 | | #ge' #fn #locals #next #nok #sp #fs0 #m0 #args #dst #m' #E1 #E2 #E3 #E4 destruct |
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| 63 | whd cases stk in mtc ⊢ %; [ * ] |
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| 64 | #hd #tl #H @H |
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| 65 | | #H14 #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 destruct |
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| 66 | | #ge' #f0 #fs0 #rtv #dst #f' #m0 #F #E1 #E2 #E3 #E4 destruct |
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| 67 | cases stk in mtc ⊢ %; [ * ] #hd #tl * #M1 #M2 % |
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| 68 | [ inversion F #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 destruct // | @M2 ] |
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| 69 | | #ge' #r #dst #m0 #E1 #E2 #E3 #E4 destruct |
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| 70 | ] |
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| 71 | | -s' #fd #args #dst #fs #m #EX whd in ⊢ (???%); cases (func_block_of_exec … EX) #func_block #FFP |
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| 72 | whd % // -func_block cases s in EX ⊢ %; -s #s #stk #mtc #EX inversion (eval_preserves … EX) |
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| 73 | [ #ge' #f1 #f2 #fs' #m1 #m2 #F #E1 #E2 #E3 #E4 destruct |
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| 74 | | #ge' #f1 #fs1 #m1 #fd' #args' #f' #dst' #F #b #FFP #E1 #E2 #E3 #E4 destruct |
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| 75 | cases stk in mtc; [ * ] #b1 #bs * #M1 #M2 % |
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| 76 | [ inversion F #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 destruct // | @M2 ] |
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| 77 | | #ge' #fn #locals #next #nok #sp #fs0 #m0 #args #dst #m' #E1 #E2 #E3 #E4 destruct |
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| 78 | | #H14 #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 destruct |
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| 79 | | #ge' #f0 #fs0 #rtv #dst #f' #m0 #F #E1 #E2 #E3 #E4 destruct |
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| 80 | | #ge' #r #dst #m0 #E1 #E2 #E3 #E4 destruct |
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| 81 | ] |
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| 82 | | -s' #rtv #dst #fs #m #EX whd in ⊢ (???%); cases s in EX ⊢ %; -s #s #stk #mtc #EX inversion (eval_preserves … EX) |
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| 83 | [ #ge' #f1 #f2 #fs' #m1 #m2 #F #E1 #E2 #E3 #E4 destruct |
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| 84 | | #ge' #f1 #fs1 #m1 #fd' #args' #f' #dst' #F #b #FFP #E1 #E2 #E3 #E4 destruct |
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| 85 | | #ge' #fn #locals #next #nok #sp #fs0 #m0 #args #dst #m' #E1 #E2 #E3 #E4 destruct |
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| 86 | | #ge' #f #fs' #m' #rtv' #dst' #m' #E1 #E2 #E3 #E4 destruct |
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| 87 | cases stk in mtc ⊢ %; [ * ] #b #bs * #_ #H @H |
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| 88 | | #ge' #f0 #fs0 #rtv #dst #f' #m0 #F #E1 #E2 #E3 #E4 destruct |
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| 89 | | #ge' #r #dst #m0 #E1 #E2 #E3 #E4 destruct |
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| 90 | ] |
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| 91 | | #r #EX whd @refl |
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| 92 | ] qed. |
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| 93 | |
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| 94 | |
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[1565] | 95 | (* NB: For RTLabs we only classify branching behaviour as jumps. Other jumps |
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| 96 | will be added later (LTL → LIN). *) |
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[1552] | 97 | |
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[1563] | 98 | definition RTLabs_classify : state → status_class ≝ |
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| 99 | λs. match s with |
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[1565] | 100 | [ State f _ _ ⇒ |
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| 101 | match lookup_present ?? (f_graph (func f)) (next f) (next_ok f) with |
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| 102 | [ St_cond _ _ _ ⇒ cl_jump |
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| 103 | | St_jumptable _ _ ⇒ cl_jump |
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| 104 | | _ ⇒ cl_other |
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| 105 | ] |
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[1563] | 106 | | Callstate _ _ _ _ _ ⇒ cl_call |
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| 107 | | Returnstate _ _ _ _ ⇒ cl_return |
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[1713] | 108 | | Finalstate _ ⇒ cl_other |
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[1563] | 109 | ]. |
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[1552] | 110 | |
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[2218] | 111 | (* As with is_cost_label/cost_label_of we define a boolean function as well |
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| 112 | as one which extracts the cost label so that we can use it in hypotheses |
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| 113 | without naming the cost label. *) |
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[1705] | 114 | |
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[1583] | 115 | definition RTLabs_cost : state → bool ≝ |
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| 116 | λs. match s with |
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| 117 | [ State f fs m ⇒ |
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[1586] | 118 | is_cost_label (lookup_present ?? (f_graph (func f)) (next f) (next_ok f)) |
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[1583] | 119 | | _ ⇒ false |
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| 120 | ]. |
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[1552] | 121 | |
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[1960] | 122 | definition RTLabs_cost_label : state → option costlabel ≝ |
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| 123 | λs. match s with |
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| 124 | [ State f fs m ⇒ |
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| 125 | cost_label_of (lookup_present ?? (f_graph (func f)) (next f) (next_ok f)) |
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| 126 | | _ ⇒ None ? |
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| 127 | ]. |
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| 128 | |
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[2218] | 129 | (* "Program counters" need to identify the current state, either as a pair of |
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| 130 | the function and current instruction, or the function being entered or |
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| 131 | left. Functions are identified by their function pointer block because |
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| 132 | this avoids introducing functions to map back pointers to function ids using |
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| 133 | the global environment. (See the comment at the start of this file, too.) *) |
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| 134 | |
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[2044] | 135 | inductive RTLabs_pc : Type[0] ≝ |
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| 136 | | rapc_state : block → label → RTLabs_pc |
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| 137 | | rapc_call : block → RTLabs_pc |
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| 138 | | rapc_ret : option block → RTLabs_pc |
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| 139 | | rapc_fin : RTLabs_pc |
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| 140 | . |
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| 141 | |
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| 142 | definition RTLabs_pc_eq : RTLabs_pc → RTLabs_pc → bool ≝ |
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| 143 | λx,y. match x with |
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| 144 | [ rapc_state b1 l1 ⇒ match y with [ rapc_state b2 l2 ⇒ eq_block b1 b2 ∧ eq_identifier … l1 l2 | _ ⇒ false ] |
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| 145 | | rapc_call b1 ⇒ match y with [ rapc_call b2 ⇒ eq_block b1 b2 | _ ⇒ false ] |
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| 146 | | rapc_ret b1 ⇒ match y with [ rapc_ret b2 ⇒ eq_option block_eq b1 b2 | _ ⇒ false ] |
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| 147 | | rapc_fin ⇒ match y with [ rapc_fin ⇒ true | _ ⇒ false ] |
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| 148 | ]. |
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| 149 | |
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| 150 | definition RTLabs_deqset : DeqSet ≝ mk_DeqSet RTLabs_pc RTLabs_pc_eq ?. |
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| 151 | whd in match RTLabs_pc_eq; whd in match eq_option; whd in match block_eq; |
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| 152 | * [ #b1 #l1 | #b1 | * [2: #b1 ] | ] |
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| 153 | * [ 1,5,9,13,17: #b2 #l2 | 2,6,10,14,18: #b2 | 3,7,11,15,19: * [2,4,6,8,10: #b2] | *: ] |
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| 154 | normalize nodelta |
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| 155 | [ 1,7,13: @eq_block_elim [ 1,3,5: #E destruct | *: * #NE ] ] |
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| 156 | [ 1,4: @eq_identifier_elim [ 1,3: #E destruct | *: * #NE ] ] |
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| 157 | normalize % #E destruct try (cases (NE (refl ??))) @refl |
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[1960] | 158 | qed. |
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| 159 | |
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[2044] | 160 | definition RTLabs_state_to_pc : ∀ge. RTLabs_state ge → RTLabs_deqset ≝ |
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| 161 | λge,s. match s with [ mk_RTLabs_state s' stk mtc0 ⇒ |
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| 162 | match s' return λs'. Ras_Fn_Match ? s' ? → ? with |
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| 163 | [ State f fs m ⇒ match stk return λstk. Ras_Fn_Match ?? stk → ? with [ nil ⇒ λmtc. ⊥ | cons b _ ⇒ λ_. rapc_state b (next … f) ] |
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| 164 | | Callstate _ _ _ _ _ ⇒ match stk return λstk. Ras_Fn_Match ?? stk → ? with [ nil ⇒ λmtc. ⊥ | cons b _ ⇒ λ_. rapc_call b ] |
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| 165 | | Returnstate _ _ _ _ ⇒ match stk with [ nil ⇒ λ_. rapc_ret (None ?) | cons b _ ⇒ λ_. rapc_ret (Some ? b) ] |
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| 166 | | Finalstate _ ⇒ λmtc. rapc_fin |
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| 167 | ] mtc0 ]. |
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| 168 | whd in mtc; cases mtc |
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| 169 | qed. |
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| 170 | |
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| 171 | definition RTLabs_pc_to_cost_label : ∀ge. RTLabs_pc → option costlabel ≝ |
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| 172 | λge,pc. |
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| 173 | match pc with |
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| 174 | [ rapc_state b l ⇒ |
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| 175 | match find_funct_ptr … ge b with [ None ⇒ None ? | Some fd ⇒ |
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| 176 | match fd with [ Internal f ⇒ match lookup ?? (f_graph f) l with [ Some s ⇒ cost_label_of s | _ ⇒ None ? ] | _ ⇒ None ? ] ] |
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| 177 | | _ ⇒ None ? |
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| 178 | ]. |
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| 179 | |
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[1960] | 180 | definition RTLabs_status : genv → abstract_status ≝ |
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[1537] | 181 | λge. |
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[1960] | 182 | mk_abstract_status |
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[2044] | 183 | (RTLabs_state ge) |
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| 184 | (λs,s'. ∃t,EX. next_state ge s s' t EX = s') |
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| 185 | RTLabs_deqset |
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| 186 | (RTLabs_state_to_pc ge) |
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[1563] | 187 | (λs,c. RTLabs_classify s = c) |
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[2044] | 188 | (RTLabs_pc_to_cost_label ge) |
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[1537] | 189 | (λs,s'. match s with |
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[1601] | 190 | [ mk_Sig s p ⇒ |
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[1563] | 191 | match s return λs. RTLabs_classify s = cl_call → ? with |
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[1537] | 192 | [ Callstate fd args dst stk m ⇒ |
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| 193 | λ_. match s' with |
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[1736] | 194 | [ 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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| 195 | | Finalstate r ⇒ stk = [ ] |
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[1537] | 196 | | _ ⇒ False |
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| 197 | ] |
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| 198 | | State f fs m ⇒ λH.⊥ |
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| 199 | | _ ⇒ λH.⊥ |
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| 200 | ] p |
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[1960] | 201 | ]) |
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[1880] | 202 | (λs. RTLabs_is_final s ≠ None ?). |
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[1960] | 203 | [ normalize in H; destruct |
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| 204 | | normalize in H; destruct |
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| 205 | | whd in H:(??%?); |
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[1565] | 206 | cases (lookup_present LabelTag statement (f_graph (func f)) (next f) (next_ok f)) in H; |
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[1877] | 207 | normalize try #a try #b try #c try #d try #e try #g try #h try #i try #j destruct |
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[1563] | 208 | ] qed. |
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[1559] | 209 | |
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[1960] | 210 | (* Allow us to move between the different notions of when a state is cost |
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| 211 | labelled. *) |
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| 212 | |
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[2044] | 213 | lemma RTLabs_costed : ∀ge. ∀s:RTLabs_state ge. |
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| 214 | RTLabs_cost s = true ↔ |
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[1960] | 215 | as_costed (RTLabs_status ge) s. |
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[2044] | 216 | cut (None (identifier CostTag) ≠ None ? → False) [ * /2/ ] #NONE |
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| 217 | #ge * * |
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| 218 | [ * #func #locals #next #nok #sp #r #fs #m #stk #mtc |
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| 219 | whd in ⊢ (??%); whd in ⊢ (??(?(??%?))); |
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| 220 | whd in match (as_pc_of ??); |
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| 221 | cases stk in mtc ⊢ %; [ * ] #func_block #stk' * #M1 #M2 |
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| 222 | whd in ⊢ (??(?(??%?))); >M1 whd in ⊢ (??(?(??%?))); |
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| 223 | >(lookup_lookup_present … nok) |
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| 224 | whd in ⊢ (?(??%?)(?(??%?))); |
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| 225 | % cases (lookup_present ?? (f_graph func) ??) normalize |
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[1960] | 226 | #A try #B try #C try #D try #E try #F try #G try #H try #G destruct |
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[2044] | 227 | try (% #E' destruct) |
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| 228 | cases (NONE ?) assumption |
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| 229 | | #fd #args #dst #fs #m #stk #mtc % |
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| 230 | [ #E normalize in E; destruct |
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| 231 | | whd in ⊢ (% → ?); whd in ⊢ (?(??%?) → ?); whd in match (as_pc_of ??); |
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| 232 | cases stk in mtc ⊢ %; [*] #fblk #fblks #mtc whd in ⊢ (?(??%?) → ?); |
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| 233 | #H cases (NONE H) |
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| 234 | ] |
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| 235 | | #v #dst #fs #m #stk #mtc % |
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| 236 | [ #E normalize in E; destruct |
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| 237 | | whd in ⊢ (% → ?); whd in ⊢ (?(??%?) → ?); whd in match (as_pc_of ??); |
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| 238 | cases stk in mtc ⊢ %; [2: #fblk #fblks ] #mtc whd in ⊢ (?(??%?) → ?); |
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| 239 | #H cases (NONE H) |
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| 240 | ] |
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| 241 | | #r #stk #mtc % |
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| 242 | [ #E normalize in E; destruct |
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| 243 | | #E normalize in E; cases (NONE E) |
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| 244 | ] |
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[1960] | 245 | ] qed. |
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| 246 | |
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[2044] | 247 | lemma RTLabs_not_cost : ∀ge. ∀s:RTLabs_state ge. |
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[1670] | 248 | RTLabs_cost s = false → |
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| 249 | ¬ as_costed (RTLabs_status ge) s. |
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[2044] | 250 | #ge #s #E % #C >(proj2 … (RTLabs_costed ??)) in E; // #E destruct |
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| 251 | qed. |
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[1670] | 252 | |
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[1559] | 253 | (* Before attempting to construct a structured trace, let's show that we can |
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| 254 | form flat traces with evidence that they were constructed from an execution. |
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| 255 | |
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| 256 | For now we don't consider I/O. *) |
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| 257 | |
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| 258 | |
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| 259 | coinductive exec_no_io (o:Type[0]) (i:o → Type[0]) : execution state o i → Prop ≝ |
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| 260 | | noio_stop : ∀a,b,c. exec_no_io o i (e_stop … a b c) |
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| 261 | | 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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| 262 | | noio_wrong : ∀m. exec_no_io o i (e_wrong … m). |
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| 263 | |
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| 264 | (* add I/O? *) |
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| 265 | coinductive flat_trace (o:Type[0]) (i:o → Type[0]) (ge:genv) : state → Type[0] ≝ |
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| 266 | | ft_stop : ∀s. RTLabs_is_final s ≠ None ? → flat_trace o i ge s |
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| 267 | | 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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[1880] | 268 | | ft_wrong : ∀s,m. RTLabs_is_final s = None ? → eval_statement ge s = Wrong ??? m → flat_trace o i ge s. |
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[1559] | 269 | |
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[1707] | 270 | 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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| 271 | | nw_stop : ∀s,H. not_wrong o i ge s (ft_stop o i ge s H) |
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| 272 | | 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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| 273 | |
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| 274 | lemma still_not_wrong : ∀o,i,ge,s,tr,s',H,tr'. |
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| 275 | not_wrong o i ge s (ft_step o i ge s tr s' H tr') → |
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| 276 | not_wrong o i ge s' tr'. |
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| 277 | #o #i #ge #s #tr #s' #H #tr' #NW inversion NW |
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| 278 | [ #H105 #H106 #H107 #H108 #H109 destruct |
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| 279 | | #H111 #H112 #H113 #H114 #H115 #H116 #H117 #H118 #H119 destruct // |
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| 280 | ] qed. |
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| 281 | |
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[1559] | 282 | let corec make_flat_trace ge s |
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[1880] | 283 | (NF:RTLabs_is_final s = None ?) |
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[1559] | 284 | (H:exec_no_io … (exec_inf_aux … RTLabs_fullexec ge (eval_statement ge s))) : |
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| 285 | flat_trace io_out io_in ge s ≝ |
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| 286 | let e ≝ exec_inf_aux … RTLabs_fullexec ge (eval_statement ge s) in |
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| 287 | match e return λx. e = x → ? with |
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| 288 | [ e_stop tr i s' ⇒ λE. ft_step … s tr s' ? (ft_stop … s' ?) |
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[1880] | 289 | | e_step tr s' e' ⇒ λE. ft_step … s tr s' ? (make_flat_trace ge s' ??) |
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| 290 | | e_wrong m ⇒ λE. ft_wrong … s m ?? |
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[1559] | 291 | | e_interact o f ⇒ λE. ⊥ |
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| 292 | ] (refl ? e). |
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| 293 | [ 1,2: whd in E:(??%?); >exec_inf_aux_unfold in E; |
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| 294 | cases (eval_statement ge s) |
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| 295 | [ 1,4: #O #K whd in ⊢ (??%? → ?); #E destruct |
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| 296 | | 2,5: * #tr #s1 whd in ⊢ (??%? → ?); |
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| 297 | >(?:is_final ????? = RTLabs_is_final s1) // |
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| 298 | lapply (refl ? (RTLabs_is_final s1)) |
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| 299 | cases (RTLabs_is_final s1) in ⊢ (???% → %); |
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| 300 | [ 1,3: #_ whd in ⊢ (??%? → ?); #E destruct |
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| 301 | | #i #_ whd in ⊢ (??%? → ?); #E destruct /2/ @refl |
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| 302 | | #i #E whd in ⊢ (??%? → ?); #E2 destruct >E % #E' destruct |
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| 303 | ] |
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| 304 | | *: #m whd in ⊢ (??%? → ?); #E destruct |
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| 305 | ] |
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| 306 | | whd in E:(??%?); >exec_inf_aux_unfold in E; |
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| 307 | cases (eval_statement ge s) |
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| 308 | [ #O #K whd in ⊢ (??%? → ?); #E destruct |
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| 309 | | * #tr #s1 whd in ⊢ (??%? → ?); |
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| 310 | cases (is_final ?????) |
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| 311 | [ whd in ⊢ (??%? → ?); #E destruct @refl |
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| 312 | | #i whd in ⊢ (??%? → ?); #E destruct |
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| 313 | ] |
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| 314 | | #m whd in ⊢ (??%? → ?); #E destruct |
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| 315 | ] |
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| 316 | | whd in E:(??%?); >E in H; #H >exec_inf_aux_unfold in E; |
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| 317 | cases (eval_statement ge s) |
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| 318 | [ #O #K whd in ⊢ (??%? → ?); #E destruct |
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| 319 | | * #tr #s1 whd in ⊢ (??%? → ?); |
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| 320 | cases (is_final ?????) |
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| 321 | [ whd in ⊢ (??%? → ?); #E |
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| 322 | change with (eval_statement ge s1) in E:(??(??????(?????%))?); |
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| 323 | destruct |
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| 324 | inversion H |
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| 325 | [ #a #b #c #E1 destruct |
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| 326 | | #trx #sx #ex #H1 #E2 #E3 destruct @H1 |
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| 327 | | #m #E1 destruct |
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| 328 | ] |
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| 329 | | #i whd in ⊢ (??%? → ?); #E destruct |
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| 330 | ] |
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| 331 | | #m whd in ⊢ (??%? → ?); #E destruct |
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| 332 | ] |
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| 333 | | whd in E:(??%?); >exec_inf_aux_unfold in E; |
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| 334 | cases (eval_statement ge s) |
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[1880] | 335 | [ #o #K whd in ⊢ (??%? → ?); #E destruct |
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| 336 | | * #tr #s1 whd in ⊢ (??%? → ?); |
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| 337 | lapply (refl ? (RTLabs_is_final s1)) |
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| 338 | change with (RTLabs_is_final s1) in ⊢ (? → ??(match % with [_⇒?|_⇒?])? → ?); |
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| 339 | cases (RTLabs_is_final s1) in ⊢ (???% → %); |
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| 340 | [ #F #E whd in E:(??%?); destruct @F |
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| 341 | | #r #F #E whd in E:(??%?); destruct |
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| 342 | ] |
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| 343 | | #m #E whd in E:(??%?); destruct |
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| 344 | ] |
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| 345 | | @NF |
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| 346 | | whd in E:(??%?); >exec_inf_aux_unfold in E; |
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| 347 | cases (eval_statement ge s) |
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[1559] | 348 | [ #O #K whd in ⊢ (??%? → ?); #E destruct |
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| 349 | | * #tr1 #s1 whd in ⊢ (??%? → ?); |
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| 350 | cases (is_final ?????) |
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| 351 | [ whd in ⊢ (??%? → ?); #E destruct |
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| 352 | | #i whd in ⊢ (??%? → ?); #E destruct |
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| 353 | ] |
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| 354 | | #m whd in ⊢ (??%? → ?); #E destruct @refl |
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| 355 | ] |
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| 356 | | whd in E:(??%?); >E in H; #H |
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| 357 | inversion H |
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| 358 | [ #a #b #c #E destruct |
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| 359 | | #a #b #c #d #E1 destruct |
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| 360 | | #m #E1 destruct |
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| 361 | ] |
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| 362 | ] qed. |
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| 363 | |
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| 364 | let corec make_whole_flat_trace p s |
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| 365 | (H:exec_no_io … (exec_inf … RTLabs_fullexec p)) |
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| 366 | (I:make_initial_state ??? p = OK ? s) : |
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| 367 | flat_trace io_out io_in (make_global … RTLabs_fullexec p) s ≝ |
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| 368 | let ge ≝ make_global … p in |
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| 369 | let e ≝ exec_inf_aux ?? RTLabs_fullexec ge (Value … 〈E0, s〉) in |
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| 370 | match e return λx. e = x → ? with |
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| 371 | [ e_stop tr i s' ⇒ λE. ft_stop ?? ge s ? |
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[1880] | 372 | | e_step _ _ e' ⇒ λE. make_flat_trace ge s ?? |
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[1559] | 373 | | e_wrong m ⇒ λE. ⊥ |
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| 374 | | e_interact o f ⇒ λE. ⊥ |
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| 375 | ] (refl ? e). |
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| 376 | [ whd in E:(??%?); >exec_inf_aux_unfold in E; |
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| 377 | whd in ⊢ (??%? → ?); |
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[1880] | 378 | change with (RTLabs_is_final s) in ⊢ (??(match % with[_⇒?|_⇒?])? → ?); |
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| 379 | cases (RTLabs_is_final s) |
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| 380 | [ #E whd in E:(??%?); destruct |
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| 381 | | #r #E % #E' destruct |
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[1559] | 382 | ] |
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[1880] | 383 | | @(initial_state_is_not_final … I) |
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[1559] | 384 | | whd in H:(???%); >I in H; whd in ⊢ (???% → ?); whd in E:(??%?); |
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| 385 | >exec_inf_aux_unfold in E ⊢ %; whd in ⊢ (??%? → ???% → ?); cases (is_final ?????) |
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| 386 | [ whd in ⊢ (??%? → ???% → ?); #E #H inversion H |
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| 387 | [ #a #b #c #E1 destruct |
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| 388 | | #tr1 #s1 #e1 #H1 #E1 #E2 -E2 -I destruct (E1) |
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| 389 | @H1 |
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| 390 | | #m #E1 destruct |
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| 391 | ] |
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| 392 | | #i whd in ⊢ (??%? → ???% → ?); #E destruct |
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| 393 | ] |
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| 394 | | whd in E:(??%?); >exec_inf_aux_unfold in E; whd in ⊢ (??%? → ?); |
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| 395 | cases (is_final ?????) [2:#i] whd in ⊢ (??%? → ?); #E destruct |
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| 396 | | whd in E:(??%?); >exec_inf_aux_unfold in E; whd in ⊢ (??%? → ?); |
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| 397 | cases (is_final ?????) [2:#i] whd in ⊢ (??%? → ?); #E destruct |
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| 398 | ] qed. |
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| 399 | |
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[1563] | 400 | (* Need a way to choose whether a called function terminates. Then, |
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| 401 | if the initial function terminates we generate a purely inductive structured trace, |
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| 402 | otherwise we start generating the coinductive one, and on every function call |
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| 403 | use the choice method again to decide whether to step over or keep going. |
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| 404 | |
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| 405 | Not quite what we need - have to decide on seeing each label whether we will see |
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| 406 | another or hit a non-terminating call? |
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| 407 | |
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| 408 | Also - need the notion of well-labelled in order to break loops. |
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| 409 | |
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| 410 | |
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| 411 | |
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| 412 | outline: |
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| 413 | |
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| 414 | does function terminate? |
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| 415 | - yes, get (bound on the number of steps until return), generate finite |
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| 416 | structure using bound as termination witness |
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| 417 | - no, get (¬ bound on steps to return), start building infinite trace out of |
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| 418 | finite steps. At calls, check for termination, generate appr. form. |
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| 419 | |
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| 420 | generating the finite parts: |
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| 421 | |
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| 422 | We start with the status after the call has been executed; well-labelling tells |
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| 423 | us that this is a labelled state. Now we want to generate a trace_label_return |
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| 424 | and also return the remainder of the flat trace. |
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| 425 | |
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| 426 | *) |
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| 427 | |
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[1595] | 428 | (* [will_return ge depth s trace] says that after a finite number of steps of |
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| 429 | [trace] from [s] we reach the return state for the current function. [depth] |
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| 430 | performs the call/return counting necessary for handling deeper function |
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| 431 | calls. It should be zero at the top level. *) |
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[1637] | 432 | inductive will_return (ge:genv) : nat → ∀s. flat_trace io_out io_in ge s → Type[0] ≝ |
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[1595] | 433 | | wr_step : ∀s,tr,s',depth,EX,trace. |
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[1565] | 434 | RTLabs_classify s = cl_other ∨ RTLabs_classify s = cl_jump → |
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[1595] | 435 | will_return ge depth s' trace → |
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| 436 | will_return ge depth s (ft_step ?? ge s tr s' EX trace) |
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| 437 | | wr_call : ∀s,tr,s',depth,EX,trace. |
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[1563] | 438 | RTLabs_classify s = cl_call → |
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[1595] | 439 | will_return ge (S depth) s' trace → |
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| 440 | will_return ge depth s (ft_step ?? ge s tr s' EX trace) |
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| 441 | | wr_ret : ∀s,tr,s',depth,EX,trace. |
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[1563] | 442 | RTLabs_classify s = cl_return → |
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[1595] | 443 | will_return ge depth s' trace → |
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| 444 | will_return ge (S depth) s (ft_step ?? ge s tr s' EX trace) |
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[1583] | 445 | (* Note that we require the ability to make a step after the return (this |
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| 446 | corresponds to somewhere that will be guaranteed to be a label at the |
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| 447 | end of the compilation chain). *) |
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[1595] | 448 | | wr_base : ∀s,tr,s',EX,trace. |
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[1563] | 449 | RTLabs_classify s = cl_return → |
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[1595] | 450 | will_return ge O s (ft_step ?? ge s tr s' EX trace) |
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[1563] | 451 | . |
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| 452 | |
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[1638] | 453 | (* The way we will use [will_return] won't satisfy Matita's guardedness check, |
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| 454 | so we will measure the length of these termination proofs and use an upper |
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| 455 | bound to show termination of the finite structured trace construction |
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| 456 | functions. *) |
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| 457 | |
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[1637] | 458 | let rec will_return_length ge d s tr (T:will_return ge d s tr) on T : nat ≝ |
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| 459 | match T with |
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| 460 | [ wr_step _ _ _ _ _ _ _ T' ⇒ S (will_return_length … T') |
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| 461 | | wr_call _ _ _ _ _ _ _ T' ⇒ S (will_return_length … T') |
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| 462 | | wr_ret _ _ _ _ _ _ _ T' ⇒ S (will_return_length … T') |
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| 463 | | wr_base _ _ _ _ _ _ ⇒ S O |
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| 464 | ]. |
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[1638] | 465 | |
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[1637] | 466 | include alias "arithmetics/nat.ma". |
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| 467 | |
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[1638] | 468 | (* Specialised to the particular situation it is used in. *) |
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[1637] | 469 | lemma wrl_nonzero : ∀ge,d,s,tr,T. O ≥ 3 * (will_return_length ge d s tr T) → False. |
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| 470 | #ge #d #s #tr * #s1 #tr1 #s2 [ 1,2,3: #d ] #EX #tr' #CL [1,2,3:#IH] |
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| 471 | whd in ⊢ (??(??%) → ?); |
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| 472 | >commutative_times |
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| 473 | #H lapply (le_plus_b … H) |
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| 474 | #H lapply (le_to_leb_true … H) |
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| 475 | normalize #E destruct |
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| 476 | qed. |
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[1719] | 477 | |
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| 478 | 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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| 479 | match T with |
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| 480 | [ wr_step _ _ _ _ _ _ _ T' ⇒ will_return_end … T' |
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| 481 | | wr_call _ _ _ _ _ _ _ T' ⇒ will_return_end … T' |
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| 482 | | wr_ret _ _ _ _ _ _ _ T' ⇒ will_return_end … T' |
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| 483 | | wr_base _ _ _ _ tr' _ ⇒ mk_DPair ? (λs.flat_trace io_out io_in ge s) ? tr' |
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| 484 | ]. |
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[1563] | 485 | |
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[1638] | 486 | (* Inversion lemmas on [will_return] that also note the effect on the length |
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| 487 | of the proof. *) |
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| 488 | lemma will_return_call : ∀ge,d,s,tr,s',EX,trace. |
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[1637] | 489 | RTLabs_classify s = cl_call → |
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| 490 | ∀TM:will_return ge d s (ft_step ?? ge s tr s' EX trace). |
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[1719] | 491 | 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] | 492 | #ge #d #s #tr #s' #EX #trace #CL #TERM inversion TERM |
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| 493 | [ #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 #H29 #H30 #H31 @⊥ destruct >CL in H25; * #E destruct |
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[1719] | 494 | | #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 #H42 #H43 #H44 #H45 destruct % /2/ |
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[1637] | 495 | | #H47 #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 #H56 #H57 #H58 #H59 @⊥ destruct >CL in H53; #E destruct |
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| 496 | | #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 #H70 @⊥ destruct >CL in H66; #E destruct |
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| 497 | ] qed. |
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[1595] | 498 | |
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[1637] | 499 | lemma will_return_return : ∀ge,d,s,tr,s',EX,trace. |
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| 500 | RTLabs_classify s = cl_return → |
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| 501 | ∀TM:will_return ge d s (ft_step ?? ge s tr s' EX trace). |
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| 502 | match d with |
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[1719] | 503 | [ O ⇒ will_return_end … TM = ❬s', trace❭ |
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[1637] | 504 | | S d' ⇒ |
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[1719] | 505 | 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] | 506 | ]. |
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| 507 | #ge #d #s #tr #s' #EX #trace #CL #TERM inversion TERM |
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| 508 | [ #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 #H29 #H30 #H31 @⊥ destruct >CL in H25; * #E destruct |
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| 509 | | #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 #H42 #H43 #H44 #H45 @⊥ destruct >CL in H39; #E destruct |
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[1719] | 510 | | #H47 #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 #H56 #H57 #H58 #H59 destruct % /2/ |
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| 511 | | #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 #H70 destruct @refl |
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[1637] | 512 | ] qed. |
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| 513 | |
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[1596] | 514 | lemma will_return_notfn : ∀ge,d,s,tr,s',EX,trace. |
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[1637] | 515 | (RTLabs_classify s = cl_other) ⊎ (RTLabs_classify s = cl_jump) → |
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| 516 | ∀TM:will_return ge d s (ft_step ?? ge s tr s' EX trace). |
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[1719] | 517 | 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] | 518 | #ge #d #s #tr #s' #EX #trace * #CL #TERM inversion TERM |
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[1719] | 519 | [ #H290 #H291 #H292 #H293 #H294 #H295 #H296 #H297 #H298 #H299 #H300 #H301 #H302 destruct % /2/ |
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[1637] | 520 | | #H304 #H305 #H306 #H307 #H308 #H309 #H310 #H311 #H312 #H313 #H314 #H315 #H316 @⊥ destruct >CL in H310; #E destruct |
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| 521 | | #H318 #H319 #H320 #H321 #H322 #H323 #H324 #H325 #H326 #H327 #H328 #H329 #H330 @⊥ destruct >CL in H324; #E destruct |
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| 522 | | #H332 #H333 #H334 #H335 #H336 #H337 #H338 #H339 #H340 #H341 @⊥ destruct >CL in H337; #E destruct |
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[1719] | 523 | | #H343 #H344 #H345 #H346 #H347 #H348 #H349 #H350 #H351 #H352 #H353 #H354 #H355 destruct % /2/ |
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[1637] | 524 | | #H357 #H358 #H359 #H360 #H361 #H362 #H363 #H364 #H365 #H366 #H367 #H368 #H369 @⊥ destruct >CL in H363; #E destruct |
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| 525 | | #H371 #H372 #H373 #H374 #H375 #H376 #H377 #H378 #H379 #H380 #H381 #H382 #H383 @⊥ destruct >CL in H377; #E destruct |
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| 526 | | #H385 #H386 #H387 #H388 #H389 #H390 #H391 #H392 #H393 #H394 @⊥ destruct >CL in H390; #E destruct |
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[1595] | 527 | ] qed. |
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| 528 | |
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[1719] | 529 | (* When it comes to building bits of nonterminating executions we'll need to be |
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| 530 | able to glue termination proofs together. *) |
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| 531 | |
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| 532 | lemma will_return_prepend : ∀ge,d1,s1,t1. |
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| 533 | ∀T1:will_return ge d1 s1 t1. |
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| 534 | ∀d2,s2,t2. |
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| 535 | will_return ge d2 s2 t2 → |
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| 536 | will_return_end … T1 = ❬s2, t2❭ → |
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| 537 | will_return ge (d1 + S d2) s1 t1. |
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| 538 | #ge #d1 #s1 #tr1 #T1 elim T1 |
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| 539 | [ #s #tr #s' #depth #EX #t #CL #T #IH #d2 #s2 #t2 #T2 #E |
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| 540 | %1 // @(IH … T2) @E |
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| 541 | | #s #tr #s' #depth #EX #t #CL #T #IH #d2 #s2 #t2 #T2 #E %2 // @(IH … T2) @E |
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| 542 | | #s #tr #s' #depth #EX #t #CL #T #IH #s2 #s2 #t2 #T2 #E %3 // @(IH … T2) @E |
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| 543 | | #s #tr #s' #EX #t #CL #d2 #s2 #t2 #T2 #E normalize in E; destruct %3 // |
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| 544 | ] qed. |
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| 545 | |
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| 546 | discriminator nat. |
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| 547 | |
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| 548 | lemma will_return_remove_call : ∀ge,d1,s1,t1. |
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| 549 | ∀T1:will_return ge d1 s1 t1. |
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| 550 | ∀d2. |
---|
| 551 | will_return ge (d1 + S d2) s1 t1 → |
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| 552 | ∀s2,t2. |
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| 553 | will_return_end … T1 = ❬s2, t2❭ → |
---|
| 554 | will_return ge d2 s2 t2. |
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| 555 | (* The key part of the proof is to show that the two termination proofs follow |
---|
| 556 | the same pattern. *) |
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| 557 | #ge #d1x #s1x #t1x #T1 elim T1 |
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| 558 | [ #s #tr #s' #d1 #EX #t #CL #T #IH #d2 #T2 #s2 #t2 #E @IH |
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| 559 | [ inversion T2 [ #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 #H13 destruct // |
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| 560 | | #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 @⊥ destruct |
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| 561 | >H21 in CL; * #E destruct |
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| 562 | | #H29 #H30 #H31 #H32 #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 @⊥ destruct |
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| 563 | >H35 in CL; * #E destruct |
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| 564 | | #H43 #H44 #H45 #H46 #H47 #H48 #H49 #H50 #H51 #H52 @⊥ destruct |
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| 565 | >H48 in CL; * #E destruct |
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| 566 | ] |
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| 567 | | @E |
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| 568 | ] |
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| 569 | | #s #tr #s' #d1 #EX #t #CL #T #IH #d2 #T2 #s2 #t2 #E @IH |
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| 570 | [ inversion T2 [ #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 #H13 @⊥ destruct |
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| 571 | >CL in H7; * #E destruct |
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| 572 | | #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 destruct // |
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| 573 | | #H29 #H30 #H31 #H32 #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 @⊥ destruct |
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| 574 | >H35 in CL; #E destruct |
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| 575 | | #H43 #H44 #H45 #H46 #H47 #H48 #H49 #H50 #H51 #H52 @⊥ destruct |
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| 576 | >H48 in CL; #E destruct |
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| 577 | ] |
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| 578 | | @E |
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| 579 | ] |
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| 580 | | #s #tr #s' #d1 #EX #t #CL #T #IH #d2 #T2 #s2 #t2 #E @IH |
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| 581 | [ inversion T2 [ #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 #H13 @⊥ destruct |
---|
| 582 | >CL in H7; * #E destruct |
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| 583 | | #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 @⊥ destruct |
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| 584 | >H21 in CL; #E destruct |
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| 585 | | #H29 #H30 #H31 #H32 #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 |
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| 586 | whd in H38:(??%??); destruct // |
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| 587 | | #H43 #H44 #H45 #H46 #H47 #H48 #H49 #H50 #H51 #H52 |
---|
| 588 | whd in H49:(??%??); @⊥ destruct |
---|
| 589 | ] |
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| 590 | | @E |
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| 591 | ] |
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| 592 | | #s #tr #s' #EX #t #CL #d2 #T2 #s2 #t2 #E whd in E:(??%?); destruct |
---|
| 593 | inversion T2 [ #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 #H13 @⊥ destruct |
---|
| 594 | >CL in H7; * #E destruct |
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| 595 | | #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 @⊥ destruct |
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| 596 | >H21 in CL; #E destruct |
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| 597 | | #H29 #H30 #H31 #H32 #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 |
---|
| 598 | whd in H38:(??%??); destruct // |
---|
| 599 | | #H43 #H44 #H45 #H46 #H47 #H48 #H49 #H50 #H51 #H52 |
---|
| 600 | whd in H49:(??%??); @⊥ destruct |
---|
| 601 | ] |
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| 602 | ] qed. |
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| 603 | |
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[1764] | 604 | lemma will_return_not_wrong : ∀ge,d,s,t,s',t'. |
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| 605 | ∀T:will_return ge d s t. |
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| 606 | not_wrong io_out io_in ge s t → |
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| 607 | will_return_end … T = ❬s', t'❭ → |
---|
| 608 | not_wrong io_out io_in ge s' t'. |
---|
| 609 | #ge #d #s #t #s' #t' #T elim T |
---|
| 610 | [ #s #tr #s' #d #EV #t1 #CL #T' #IH #NW #E @IH |
---|
| 611 | [ inversion NW |
---|
| 612 | [ #H1 #H2 #H3 #H4 #H5 destruct |
---|
| 613 | | #H7 #H8 #H9 #H10 #H11 #H12 #H13 #H14 #H15 destruct // |
---|
| 614 | ] |
---|
| 615 | | @E |
---|
| 616 | ] |
---|
| 617 | | #s #tr #s' #d #EV #t1 #CL #T' #IH #NW #E @IH |
---|
| 618 | [ inversion NW [ #H1 #H2 #H3 #H4 #H5 destruct | #H7 #H8 #H9 #H10 #H11 #H12 #H13 #H14 #H15 destruct // ] |
---|
| 619 | | @E |
---|
| 620 | ] |
---|
| 621 | | #s #tr #s' #d #EV #t1 #CL #T' #IH #NW #E @IH |
---|
| 622 | [ inversion NW [ #H1 #H2 #H3 #H4 #H5 destruct | #H7 #H8 #H9 #H10 #H11 #H12 #H13 #H14 #H15 destruct // ] |
---|
| 623 | | @E |
---|
| 624 | ] |
---|
| 625 | | #s #tr #s' #d #t1 #CL #NW #E normalize in E; destruct |
---|
| 626 | inversion NW [ #H1 #H2 #H3 #H4 #H5 destruct | #H7 #H8 #H9 #H10 #H11 #H12 #H13 #H14 #H15 destruct // ] |
---|
| 627 | ] qed. |
---|
| 628 | |
---|
[1806] | 629 | |
---|
| 630 | lemma will_return_lower : ∀ge,d,s,t. |
---|
| 631 | will_return ge d s t → |
---|
| 632 | ∀d'. d' ≤ d → |
---|
| 633 | will_return ge d' s t. |
---|
| 634 | #ge #d0 #s0 #t0 #TM elim TM |
---|
| 635 | [ #s #tr #s' #d #EX #tr #CL #TM1 #IH #d' #LE % /2/ |
---|
| 636 | | #s #tr #s' #d #EX #tr #CL #TM1 #IH #d' #LE %2 // @IH /2/ |
---|
| 637 | | #s #tr #s' #d #EX #tr #CL #TM1 #IH * |
---|
| 638 | [ #LE @wr_base // |
---|
| 639 | | #d' #LE %3 // @IH /2/ |
---|
| 640 | ] |
---|
| 641 | | #s #tr #s' #EX #tr #CL * |
---|
| 642 | [ #_ @wr_base // |
---|
| 643 | | #d' #LE @⊥ /2/ |
---|
| 644 | ] |
---|
| 645 | ] qed. |
---|
| 646 | |
---|
[1565] | 647 | (* We need to ensure that any code we come across is well-cost-labelled. We may |
---|
| 648 | get function code from either the global environment or the state. *) |
---|
| 649 | |
---|
| 650 | definition well_cost_labelled_ge : genv → Prop ≝ |
---|
[2044] | 651 | λge. ∀b,f. find_funct_ptr … ge b = Some ? (Internal ? f) → well_cost_labelled_fn f. |
---|
[1565] | 652 | |
---|
| 653 | definition well_cost_labelled_state : state → Prop ≝ |
---|
| 654 | λs. match s with |
---|
| 655 | [ State f fs m ⇒ well_cost_labelled_fn (func f) ∧ All ? (λf. well_cost_labelled_fn (func f)) fs |
---|
| 656 | | Callstate fd _ _ fs _ ⇒ match fd with [ Internal fn ⇒ well_cost_labelled_fn fn | External _ ⇒ True ] ∧ |
---|
| 657 | All ? (λf. well_cost_labelled_fn (func f)) fs |
---|
| 658 | | Returnstate _ _ fs _ ⇒ All ? (λf. well_cost_labelled_fn (func f)) fs |
---|
[1713] | 659 | | Finalstate _ ⇒ True |
---|
[1565] | 660 | ]. |
---|
| 661 | |
---|
[1583] | 662 | lemma well_cost_labelled_state_step : ∀ge,s,tr,s'. |
---|
| 663 | eval_statement ge s = Value ??? 〈tr,s'〉 → |
---|
| 664 | well_cost_labelled_ge ge → |
---|
| 665 | well_cost_labelled_state s → |
---|
| 666 | well_cost_labelled_state s'. |
---|
[2025] | 667 | #ge #s #tr' #s' #EV cases (eval_preserves … EV) |
---|
[1583] | 668 | [ #ge #f #f' #fs #m #m' * #func #locals #next #next_ok #sp #retdst #locals' #next' #next_ok' #Hge * #H1 #H2 % // |
---|
| 669 | | #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/ |
---|
[1681] | 670 | (* |
---|
[1583] | 671 | | #ge #f #fs #m * #fn #args #f' #dst #m' #b #Hge * #H1 #H2 % /2/ |
---|
[1681] | 672 | *) |
---|
[1583] | 673 | | #ge #fn #locals #next #nok #sp #fs #m #args #dst #m' #Hge * #H1 #H2 % /2/ |
---|
| 674 | | #ge #f #fs #m #rtv #dst #m' #Hge * #H1 #H2 @H2 |
---|
| 675 | | #ge #f #fs #rtv #dst #f' #m * #func #locals #next #nok #sp #retdst #locals' #next' #nok' #Hge * #H1 #H2 % // |
---|
[1713] | 676 | | // |
---|
[1583] | 677 | ] qed. |
---|
| 678 | |
---|
[1586] | 679 | lemma rtlabs_jump_inv : ∀s. |
---|
| 680 | RTLabs_classify s = cl_jump → |
---|
| 681 | ∃f,fs,m. s = State f fs m ∧ |
---|
| 682 | let stmt ≝ lookup_present ?? (f_graph (func f)) (next f) (next_ok f) in |
---|
| 683 | (∃r,l1,l2. stmt = St_cond r l1 l2) ∨ (∃r,ls. stmt = St_jumptable r ls). |
---|
| 684 | * |
---|
| 685 | [ #f #fs #m #E |
---|
| 686 | %{f} %{fs} %{m} % |
---|
| 687 | [ @refl |
---|
| 688 | | whd in E:(??%?); cases (lookup_present ? statement ???) in E ⊢ %; |
---|
[1877] | 689 | try (normalize try #A try #B try #C try #D try #F try #G try #H try #I try #J destruct) |
---|
[1586] | 690 | [ %1 %{A} %{B} %{C} @refl |
---|
| 691 | | %2 %{A} %{B} @refl |
---|
| 692 | ] |
---|
| 693 | ] |
---|
| 694 | | normalize #H1 #H2 #H3 #H4 #H5 #H6 destruct |
---|
| 695 | | normalize #H8 #H9 #H10 #H11 #H12 destruct |
---|
[1713] | 696 | | #r #E normalize in E; destruct |
---|
[1586] | 697 | ] qed. |
---|
| 698 | |
---|
| 699 | lemma well_cost_labelled_jump : ∀ge,s,tr,s'. |
---|
| 700 | eval_statement ge s = Value ??? 〈tr,s'〉 → |
---|
| 701 | well_cost_labelled_state s → |
---|
| 702 | RTLabs_classify s = cl_jump → |
---|
| 703 | RTLabs_cost s' = true. |
---|
| 704 | #ge #s #tr #s' #EV #H #CL |
---|
| 705 | cases (rtlabs_jump_inv s CL) |
---|
| 706 | #fr * #fs * #m * #Es * |
---|
| 707 | [ * #r * #l1 * #l2 #Estmt |
---|
| 708 | >Es in H; whd in ⊢ (% → ?); * * #Hbody #_ #Hfs |
---|
| 709 | >Es in EV; whd in ⊢ (??%? → ?); generalize in ⊢ (??(?%)? → ?); |
---|
| 710 | >Estmt #LP whd in ⊢ (??%? → ?); |
---|
| 711 | (* replace with lemma on successors? *) |
---|
[1960] | 712 | @bind_res_value #v #Ev @bind_ok * #Eb whd in ⊢ (??%? → ?); #E destruct |
---|
[1586] | 713 | lapply (Hbody (next fr) (next_ok fr)) |
---|
| 714 | generalize in ⊢ (???% → ?); |
---|
| 715 | >Estmt #LP' |
---|
| 716 | whd in ⊢ (% → ?); |
---|
| 717 | * #H1 #H2 [ @H1 | @H2 ] |
---|
| 718 | | * #r * #ls #Estmt |
---|
| 719 | >Es in H; whd in ⊢ (% → ?); * * #Hbody #_ #Hfs |
---|
| 720 | >Es in EV; whd in ⊢ (??%? → ?); generalize in ⊢ (??(?%)? → ?); |
---|
| 721 | >Estmt #LP whd in ⊢ (??%? → ?); |
---|
| 722 | (* replace with lemma on successors? *) |
---|
[2184] | 723 | @bind_res_value #a cases a [ | #sz #i | #f | | #ptr ] #Ea whd in ⊢ (??%? → ?); |
---|
[1586] | 724 | [ 2: (* later *) |
---|
| 725 | | *: #E destruct |
---|
| 726 | ] |
---|
| 727 | lapply (Hbody (next fr) (next_ok fr)) |
---|
| 728 | generalize in ⊢ (???% → ?); >Estmt #LP' whd in ⊢ (% → ?); #CP |
---|
| 729 | generalize in ⊢ (??(?%)? → ?); |
---|
| 730 | cases (nth_opt label (nat_of_bitvector (bitsize_of_intsize sz) i) ls) in ⊢ (???% → ??(match % with [_⇒?|_⇒?]?)? → ?); |
---|
| 731 | [ #E1 #E2 whd in E2:(??%?); destruct |
---|
| 732 | | #l' #E1 #E2 whd in E2:(??%?); destruct |
---|
| 733 | cases (All_nth ???? CP ? E1) |
---|
| 734 | #H1 #H2 @H2 |
---|
| 735 | ] |
---|
| 736 | ] qed. |
---|
| 737 | |
---|
[1595] | 738 | lemma rtlabs_call_inv : ∀s. |
---|
| 739 | RTLabs_classify s = cl_call → |
---|
| 740 | ∃fd,args,dst,stk,m. s = Callstate fd args dst stk m. |
---|
| 741 | * [ #f #fs #m whd in ⊢ (??%? → ?); |
---|
| 742 | cases (lookup_present … (next f) (next_ok f)) normalize |
---|
[1877] | 743 | try #A try #B try #C try #D try #E try #F try #G try #I try #J destruct |
---|
[1595] | 744 | | #fd #args #dst #stk #m #E %{fd} %{args} %{dst} %{stk} %{m} @refl |
---|
| 745 | | normalize #H411 #H412 #H413 #H414 #H415 destruct |
---|
[1713] | 746 | | normalize #H1 #H2 destruct |
---|
[1595] | 747 | ] qed. |
---|
[1586] | 748 | |
---|
[1595] | 749 | lemma well_cost_labelled_call : ∀ge,s,tr,s'. |
---|
| 750 | eval_statement ge s = Value ??? 〈tr,s'〉 → |
---|
| 751 | well_cost_labelled_state s → |
---|
| 752 | RTLabs_classify s = cl_call → |
---|
| 753 | RTLabs_cost s' = true. |
---|
| 754 | #ge #s #tr #s' #EV #WCL #CL |
---|
| 755 | cases (rtlabs_call_inv s CL) |
---|
| 756 | #fd * #args * #dst * #stk * #m #E >E in EV WCL; |
---|
| 757 | whd in ⊢ (??%? → % → ?); |
---|
| 758 | cases fd |
---|
| 759 | [ #fn whd in ⊢ (??%? → % → ?); |
---|
[2184] | 760 | @bind_res_value #lcl #Elcl cases (alloc m O (f_stacksize fn) XData) |
---|
[1595] | 761 | #m' #b whd in ⊢ (??%? → ?); #E' destruct |
---|
| 762 | * whd in ⊢ (% → ?); * #WCL1 #WCL2 #WCL3 |
---|
| 763 | @WCL2 |
---|
| 764 | | #fn whd in ⊢ (??%? → % → ?); |
---|
| 765 | @bindIO_value #evargs #Eargs |
---|
[1656] | 766 | whd in ⊢ (??%? → ?); |
---|
| 767 | #E' destruct |
---|
[1595] | 768 | ] qed. |
---|
| 769 | |
---|
[1681] | 770 | |
---|
| 771 | (* The preservation of (most of) the stack is useful to show as_after_return. |
---|
[1682] | 772 | We do this by showing that during the execution of a function the lower stack |
---|
| 773 | frames never change, and that after returning from the function we preserve |
---|
| 774 | the identity of the next instruction to execute. |
---|
[2044] | 775 | |
---|
| 776 | Note: since this was first written I've introduced the shadow stack of |
---|
| 777 | function blocks. It might be possible to replace some or all of the stack |
---|
| 778 | preservation with that. |
---|
[1682] | 779 | *) |
---|
| 780 | |
---|
| 781 | inductive stack_of_state : list frame → state → Prop ≝ |
---|
| 782 | | sos_State : ∀f,fs,m. stack_of_state fs (State f fs m) |
---|
| 783 | | sos_Callstate : ∀fd,args,dst,f,fs,m. stack_of_state fs (Callstate fd args dst (f::fs) m) |
---|
| 784 | | sos_Returnstate : ∀rtv,dst,fs,m. stack_of_state fs (Returnstate rtv dst fs m) |
---|
| 785 | . |
---|
| 786 | |
---|
[1681] | 787 | inductive stack_preserved : trace_ends_with_ret → state → state → Prop ≝ |
---|
[1682] | 788 | | sp_normal : ∀fs,s1,s2. |
---|
| 789 | stack_of_state fs s1 → |
---|
| 790 | stack_of_state fs s2 → |
---|
| 791 | stack_preserved doesnt_end_with_ret s1 s2 |
---|
| 792 | | sp_finished : ∀s1,f,f',fs,m. |
---|
| 793 | next f = next f' → |
---|
[1736] | 794 | frame_rel f f' → |
---|
[1682] | 795 | stack_of_state (f::fs) s1 → |
---|
[1713] | 796 | stack_preserved ends_with_ret s1 (State f' fs m) |
---|
[1736] | 797 | | sp_stop : ∀s1,r. |
---|
| 798 | stack_of_state [ ] s1 → |
---|
| 799 | stack_preserved ends_with_ret s1 (Finalstate r) |
---|
| 800 | | sp_top : ∀fd,args,dst,m,r. |
---|
| 801 | stack_preserved doesnt_end_with_ret (Callstate fd args dst [ ] m) (Finalstate r) |
---|
[1713] | 802 | . |
---|
[1681] | 803 | |
---|
[1682] | 804 | discriminator list. |
---|
[1681] | 805 | |
---|
[1682] | 806 | lemma stack_of_state_eq : ∀fs,fs',s. |
---|
| 807 | stack_of_state fs s → |
---|
| 808 | stack_of_state fs' s → |
---|
| 809 | fs = fs'. |
---|
| 810 | #fs0 #fs0' #s0 * |
---|
| 811 | [ #f #fs #m #H inversion H |
---|
[1713] | 812 | #a #b #c #d try #e try #g try #h try #i try #j destruct @refl |
---|
[1682] | 813 | | #fd #args #dst #f #fs #m #H inversion H |
---|
[1713] | 814 | #a #b #c #d try #e try #g try #h try #i try #j destruct @refl |
---|
[1682] | 815 | | #rtv #dst #fs #m #H inversion H |
---|
[1713] | 816 | #a #b #c #d try #e try #g try #h try #i try #j destruct @refl |
---|
[1682] | 817 | ] qed. |
---|
| 818 | |
---|
[1713] | 819 | lemma stack_preserved_final : ∀e,r,s. |
---|
[1736] | 820 | ¬stack_preserved e (Finalstate r) s. |
---|
| 821 | #e #r #s % #H inversion H |
---|
[1713] | 822 | [ #H184 #H185 #H186 #SOS #H188 #H189 #H190 #H191 #H192 destruct |
---|
| 823 | inversion SOS #a #b #c #d #e #f try #g try #h destruct |
---|
[1736] | 824 | | #H194 #H195 #H196 #H197 #H198 #H199 #H200 #SOS #H201 #H202 #H203 #H204 destruct |
---|
[1713] | 825 | inversion SOS #a #b #c #d #e #f try #g try #h destruct |
---|
[1736] | 826 | | #s' #r' #SOS #E1 #E2 #E3 #E4 destruct |
---|
| 827 | inversion SOS #a #b #c #d #e #f try #g try #h destruct |
---|
| 828 | | #H24 #H25 #H26 #H27 #H28 #H29 #H30 #H31 #H32 destruct |
---|
[1713] | 829 | ] qed. |
---|
| 830 | |
---|
[1681] | 831 | lemma stack_preserved_join : ∀e,s1,s2,s3. |
---|
| 832 | stack_preserved doesnt_end_with_ret s1 s2 → |
---|
| 833 | stack_preserved e s2 s3 → |
---|
| 834 | stack_preserved e s1 s3. |
---|
| 835 | #e1 #s1 #s2 #s3 #H1 inversion H1 |
---|
[1682] | 836 | [ #fs #s1' #s2' #S1 #S2 #E1 #E2 #E3 #E4 destruct |
---|
| 837 | #H2 inversion H2 |
---|
| 838 | [ #fs' #s1'' #s2'' #S1' #S2' #E1 #E2 #E3 #E4 destruct |
---|
| 839 | @(sp_normal fs) // <(stack_of_state_eq … S1' S2) // |
---|
[1736] | 840 | | #s1'' #f #f' #fs' #m #N #F #S1' #E1 #E2 #E3 #E4 destruct |
---|
[1682] | 841 | @(sp_finished … N) >(stack_of_state_eq … S1' S2) // |
---|
[1736] | 842 | | #s1'' #r #S1'' #E1 #E2 #E3 #E4 destruct @sp_stop >(stack_of_state_eq … S1'' S2) // |
---|
| 843 | | #fd #args #dst #m #r #E1 #E2 #E3 #E4 destruct |
---|
| 844 | inversion S2 |
---|
| 845 | [ #H34 #H35 #H36 #H37 #H38 #H39 destruct |
---|
| 846 | | #fd' #args' #dst' #f #fs' #m' #E1 #E2 #E3 destruct |
---|
| 847 | | #H41 #H42 #H43 #H44 #H45 #H46 #H47 destruct |
---|
| 848 | ] |
---|
[1681] | 849 | ] |
---|
[1682] | 850 | | #H25 #H26 #H27 #H28 #H29 #H30 #H31 #H32 #H33 #H34 #H35 #H36 destruct |
---|
[1713] | 851 | | #H19 #H20 #H21 #H22 #H23 #H24 #H25 destruct #H |
---|
[1736] | 852 | cases (stack_preserved_final … H) #r #E destruct |
---|
| 853 | | #fd #args #dst #m #r #E1 #E2 #E3 #E4 destruct #F @⊥ |
---|
| 854 | @(absurd … F) // |
---|
[1681] | 855 | ] qed. |
---|
| 856 | |
---|
[1682] | 857 | lemma stack_preserved_return : ∀ge,s1,s2,tr. |
---|
[1681] | 858 | RTLabs_classify s1 = cl_return → |
---|
| 859 | eval_statement ge s1 = Value ??? 〈tr,s2〉 → |
---|
| 860 | stack_preserved ends_with_ret s1 s2. |
---|
| 861 | #ge * |
---|
| 862 | [ #f #fs #m #s2 #tr #E @⊥ whd in E:(??%?); |
---|
| 863 | cases (lookup_present ??? (next f) (next_ok f)) in E; |
---|
[1877] | 864 | normalize #a try #b try #c try #d try #e try #f try #g try #i try #j destruct |
---|
[1681] | 865 | | #fd #args #dst #fs #m #s2 #tr #E normalize in E; destruct |
---|
| 866 | | #res #dst * |
---|
[1736] | 867 | [ #m #s2 #tr #_ #EV whd in EV:(??%?); cases res in EV; |
---|
| 868 | [ normalize #EV destruct | * [ 2: * #r normalize #EV destruct /2/ | *: normalize #a try #b destruct ] ] |
---|
[1960] | 869 | | #f #fs #m #s2 #tr #_ whd in ⊢ (??%? → ?); @bind_res_value #locals #El #EV |
---|
[1682] | 870 | whd in EV:(??%?); destruct @(sp_finished ? f) // |
---|
[1736] | 871 | cases f // |
---|
[1681] | 872 | ] |
---|
[1713] | 873 | | #r #s2 #tr #E normalize in E; destruct |
---|
[1681] | 874 | ] qed. |
---|
| 875 | |
---|
| 876 | lemma stack_preserved_step : ∀ge,s1,s2,tr. |
---|
| 877 | RTLabs_classify s1 = cl_other ∨ RTLabs_classify s1 = cl_jump → |
---|
| 878 | eval_statement ge s1 = Value ??? 〈tr,s2〉 → |
---|
| 879 | stack_preserved doesnt_end_with_ret s1 s2. |
---|
[2025] | 880 | #ge0 #s1 #s2 #tr #CL #EV inversion (eval_preserves … EV) |
---|
[1681] | 881 | [ #ge #f #f' #fs #m #m' #F #E1 #E2 #E3 #E4 destruct /2/ |
---|
| 882 | | #ge #f #fs #m #fd #args #f' #dst #F #b #FFP #E1 #E2 #E3 #E4 /2/ |
---|
| 883 | | #ge #fn #locals #next #nok #sp #fs #m #args #dst #m' #E1 #E2 #E3 #E4 destruct |
---|
| 884 | normalize in CL; cases CL #E destruct |
---|
| 885 | | #ge #f #fs #m #rtv #dst #m' #E1 #E2 #E3 #E4 destruct /2/ |
---|
| 886 | | #ge #f #fs #rtv #dst #f' #m #F #E1 #E2 #E3 #E4 destruct cases CL |
---|
| 887 | #E normalize in E; destruct |
---|
[1713] | 888 | | #ge #r #dst #m #E1 #E2 destruct @⊥ cases CL normalize #E destruct |
---|
[1681] | 889 | ] qed. |
---|
| 890 | |
---|
| 891 | lemma stack_preserved_call : ∀ge,s1,s2,s3,tr. |
---|
| 892 | RTLabs_classify s1 = cl_call → |
---|
| 893 | eval_statement ge s1 = Value ??? 〈tr,s2〉 → |
---|
| 894 | stack_preserved ends_with_ret s2 s3 → |
---|
| 895 | stack_preserved doesnt_end_with_ret s1 s3. |
---|
| 896 | #ge #s1 #s2 #s3 #tr #CL #EV #SP |
---|
| 897 | cases (rtlabs_call_inv … CL) |
---|
| 898 | #fd * #args * #dst * #stk * #m #E destruct |
---|
[1682] | 899 | inversion SP |
---|
| 900 | [ #H38 #H39 #H40 #H41 #H42 #H43 #H44 #H45 #H46 destruct |
---|
[1736] | 901 | | #s2' #f #f' #fs #m' #N #F #S #E1 #E2 #E3 #E4 destruct |
---|
[2025] | 902 | inversion (eval_preserves … EV) |
---|
[1682] | 903 | [ #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 #H56 #H57 #H58 destruct |
---|
| 904 | | #H60 #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 #H70 #H71 #H72 #H73 #H74 destruct |
---|
| 905 | | #ge' #fn #locals #next #nok #sp #fs1 #m1 #args1 #dst1 #m2 #E1 #E2 #E3 #E4 destruct |
---|
| 906 | inversion S |
---|
| 907 | [ #fx #fsx #mx #E1 #E2 #E3 destruct /2/ |
---|
| 908 | | #H76 #H77 #H78 #H79 #H80 #H81 #H82 #H83 #H84 destruct |
---|
| 909 | | #H86 #H87 #H88 #H89 #H90 #H91 #H92 destruct |
---|
| 910 | ] |
---|
| 911 | | #H94 #H95 #H96 #H97 #H98 #H99 #H100 #H101 #H102 #H103 #H104 destruct |
---|
| 912 | | #H106 #H107 #H108 #H109 #H110 #H111 #H112 #H113 #H114 #H115 #H116 #H117 destruct |
---|
[1713] | 913 | | #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 destruct |
---|
[1682] | 914 | ] |
---|
[1736] | 915 | | #s1 #r #S1 #E1 #E2 #E3 #_ destruct |
---|
[2025] | 916 | inversion (eval_preserves … EV) |
---|
[1736] | 917 | [ #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 #H56 #H57 #H58 destruct |
---|
| 918 | | #H60 #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 #H70 #H71 #H72 #H73 #H74 destruct |
---|
| 919 | | #ge' #fn #locals #next #nok #sp #fs1 #m1 #args1 #dst1 #m2 #E1 #E2 #E3 #E4 destruct |
---|
| 920 | inversion S1 |
---|
| 921 | [ #fx #fsx #mx #E1 #E2 #E3 destruct /2/ |
---|
| 922 | | #H76 #H77 #H78 #H79 #H80 #H81 #H82 #H83 #H84 destruct |
---|
| 923 | | #H86 #H87 #H88 #H89 #H90 #H91 #H92 destruct |
---|
| 924 | ] |
---|
| 925 | | #H94 #H95 #H96 #H97 #H98 #H99 #H100 #H101 #H102 #H103 #H104 destruct |
---|
| 926 | | #H106 #H107 #H108 #H109 #H110 #H111 #H112 #H113 #H114 #H115 #H116 #H117 destruct |
---|
| 927 | | #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 destruct |
---|
| 928 | ] |
---|
| 929 | | #H49 #H50 #H51 #H52 #H53 #H54 #H55 #H56 #H57 destruct |
---|
[1682] | 930 | ] qed. |
---|
| 931 | |
---|
[2044] | 932 | lemma RTLabs_after_call : ∀ge.∀s1,s2,s3:RTLabs_state ge.∀tr. |
---|
[1682] | 933 | ∀CL : RTLabs_classify s1 = cl_call. |
---|
| 934 | eval_statement ge s1 = Value ??? 〈tr,s2〉 → |
---|
| 935 | stack_preserved ends_with_ret s2 s3 → |
---|
| 936 | as_after_return (RTLabs_status ge) «s1,CL» s3. |
---|
[2044] | 937 | #ge * #s1 #stk1 #M1 * #s2 #stk2 #M2 * #s3 #stk3 #M3 #tr #CL #EV #S23 |
---|
[1682] | 938 | cases (rtlabs_call_inv … CL) #fn * #args * #dst * #stk * #m #E destruct |
---|
[2044] | 939 | whd |
---|
[1682] | 940 | inversion S23 |
---|
| 941 | [ #H129 #H130 #H131 #H132 #H133 #H134 #H135 #H136 #H137 destruct |
---|
[2044] | 942 | | #s2' #f #f' #fs #m' #N #F #S #E1 #E2 #E3 #E4 destruct whd |
---|
[2025] | 943 | inversion (eval_preserves … EV) |
---|
[1682] | 944 | [ #H139 #H140 #H141 #H142 #H143 #H144 #H145 #H146 #H147 #H148 #H149 destruct |
---|
| 945 | | #H151 #H152 #H153 #H154 #H155 #H156 #H157 #H158 #H159 #H160 #H161 #H162 #H163 #H164 #H165 destruct |
---|
| 946 | | #gex #fnx #locals #next #nok #sp #fsx #mx #argsx #dstx #mx' #E1 #E2 #E3 #E4 destruct |
---|
| 947 | inversion S |
---|
[1736] | 948 | [ #fy #fsy #my #E1 #E2 #E3 destruct whd % [ @N | inversion F // ] |
---|
[1682] | 949 | | #H167 #H168 #H169 #H170 #H171 #H172 #H173 #H174 #H175 destruct |
---|
| 950 | | #H177 #H178 #H179 #H180 #H181 #H182 #H183 destruct |
---|
| 951 | ] |
---|
| 952 | | #H185 #H186 #H187 #H188 #H189 #H190 #H191 #H192 #H193 #H194 #H195 destruct |
---|
| 953 | | #H197 #H198 #H199 #H200 #H201 #H202 #H203 #H204 #H205 #H206 #H207 #H208 destruct |
---|
[1713] | 954 | | #H10 #H11 #H12 #H13 #H14 #H15 #H16 #H17 destruct |
---|
[1682] | 955 | ] |
---|
[1736] | 956 | | #s1 #r #S1 #E1 #E2 #E3 #E4 destruct whd |
---|
[2025] | 957 | inversion (eval_preserves … EV) |
---|
[1736] | 958 | [ #H59 #H60 #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 destruct |
---|
| 959 | | #H71 #H72 #H73 #H74 #H75 #H76 #H77 #H78 #H79 #H80 #H81 #H82 #H83 #H84 #H85 destruct |
---|
| 960 | | #ge' #fn' #locals #next #nok #sp #fs #m' #args' #dst' #m'' #E1 #E2 #E3 #E4 destruct |
---|
| 961 | inversion S1 |
---|
| 962 | [ #H103 #H104 #H105 #H106 #H107 #H108 destruct // |
---|
| 963 | | #H110 #H111 #H112 #H113 #H114 #H115 #H116 #H117 #H118 destruct |
---|
| 964 | | #H120 #H121 #H122 #H123 #H124 #H125 #H126 destruct |
---|
| 965 | ] |
---|
| 966 | | #H185 #H186 #H187 #H188 #H189 #H190 #H191 #H192 #H193 #H194 #H195 destruct |
---|
| 967 | | #H197 #H198 #H199 #H200 #H201 #H202 #H203 #H204 #H205 #H206 #H207 #H208 destruct |
---|
| 968 | | #H10 #H11 #H12 #H13 #H14 #H15 #H16 #H17 destruct |
---|
| 969 | ] |
---|
| 970 | | #H128 #H129 #H130 #H131 #H132 #H133 #H134 #H135 #H136 destruct |
---|
[1682] | 971 | ] qed. |
---|
[1681] | 972 | |
---|
[1574] | 973 | (* Don't need to know that labels break loops because we have termination. *) |
---|
| 974 | |
---|
[1596] | 975 | (* A bit of mucking around with the depth to avoid proving termination after |
---|
[1638] | 976 | termination. Note that we keep a proof that our upper bound on the length |
---|
| 977 | of the termination proof is respected. *) |
---|
[1719] | 978 | record trace_result (ge:genv) (depth:nat) (ends:trace_ends_with_ret) |
---|
[2044] | 979 | (start:RTLabs_state ge) (full:flat_trace io_out io_in ge start) |
---|
[1719] | 980 | (original_terminates: will_return ge depth start full) |
---|
[2044] | 981 | (T:RTLabs_state ge → Type[0]) (limit:nat) : Type[0] ≝ |
---|
[1719] | 982 | { |
---|
[2044] | 983 | new_state : RTLabs_state ge; |
---|
[1574] | 984 | remainder : flat_trace io_out io_in ge new_state; |
---|
| 985 | cost_labelled : well_cost_labelled_state new_state; |
---|
[1596] | 986 | new_trace : T new_state; |
---|
[1681] | 987 | stack_ok : stack_preserved ends start new_state; |
---|
[1719] | 988 | terminates : match (match ends with [ doesnt_end_with_ret ⇒ S depth | _ ⇒ depth ]) with |
---|
| 989 | [ O ⇒ will_return_end … original_terminates = ❬new_state, remainder❭ |
---|
| 990 | | S d ⇒ ΣTM:will_return ge d new_state remainder. |
---|
[2044] | 991 | gt limit (will_return_length … TM) ∧ |
---|
[1719] | 992 | will_return_end … original_terminates = will_return_end … TM |
---|
[1596] | 993 | ] |
---|
[1574] | 994 | }. |
---|
| 995 | |
---|
[1638] | 996 | (* The same with a flag indicating whether the function returned, as opposed to |
---|
| 997 | encountering a label. *) |
---|
[1719] | 998 | record sub_trace_result (ge:genv) (depth:nat) |
---|
[2044] | 999 | (start:RTLabs_state ge) (full:flat_trace io_out io_in ge start) |
---|
[1719] | 1000 | (original_terminates: will_return ge depth start full) |
---|
[2044] | 1001 | (T:trace_ends_with_ret → RTLabs_state ge → Type[0]) (limit:nat) : Type[0] ≝ |
---|
[1719] | 1002 | { |
---|
[1594] | 1003 | ends : trace_ends_with_ret; |
---|
[1719] | 1004 | trace_res :> trace_result ge depth ends start full original_terminates (T ends) limit |
---|
[1594] | 1005 | }. |
---|
| 1006 | |
---|
[1638] | 1007 | (* We often return the result from a recursive call with an addition to the |
---|
| 1008 | structured trace, so we define a couple of functions to help. The bound on |
---|
| 1009 | the size of the termination proof might need to be relaxed, too. *) |
---|
| 1010 | |
---|
[2044] | 1011 | definition replace_trace : ∀ge,d,e.∀s1,s2:RTLabs_state ge.∀t1,t2,TM1,TM2,T1,T2,l1,l2. l2 ≥ l1 → |
---|
[1719] | 1012 | ∀r:trace_result ge d e s1 t1 TM1 T1 l1. |
---|
| 1013 | will_return_end … TM1 = will_return_end … TM2 → |
---|
[1712] | 1014 | T2 (new_state … r) → |
---|
[1719] | 1015 | stack_preserved e s2 (new_state … r) → |
---|
| 1016 | trace_result ge d e s2 t2 TM2 T2 l2 ≝ |
---|
| 1017 | λge,d,e,s1,s2,t1,t2,TM1,TM2,T1,T2,l1,l2,lGE,r,TME,trace,SP. |
---|
| 1018 | mk_trace_result ge d e s2 t2 TM2 T2 l2 |
---|
[1574] | 1019 | (new_state … r) |
---|
| 1020 | (remainder … r) |
---|
| 1021 | (cost_labelled … r) |
---|
[1594] | 1022 | trace |
---|
[1681] | 1023 | SP |
---|
[1719] | 1024 | ? |
---|
| 1025 | (*(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] | 1026 | match d' with [ O ⇒ True | S d'' ⇒ ΣTM.l2 > will_return_length ge d'' (new_state … r) (remainder … r) TM] with |
---|
[1719] | 1027 | [O ⇒ λ_. I | _ ⇒ λTM. «pi1 … TM, ?» ] (terminates ???????? r))*) |
---|
| 1028 | . |
---|
| 1029 | cases e in r ⊢ %; |
---|
| 1030 | [ <TME -TME * cases d in TM1 TM2 ⊢ %; |
---|
| 1031 | [ #TM1 #TM2 #ns #rem #WCLS #T1NS #SP whd in ⊢ (% → %); #TMS @TMS |
---|
| 1032 | | #d' #TM1 #TM2 #ns #rem #WCLS #T1NS #SP whd in ⊢ (% → %); * #TMa * #L1 #TME |
---|
| 1033 | %{TMa} % // @(transitive_le … lGE) @L1 |
---|
| 1034 | ] |
---|
| 1035 | | <TME -TME * #ns #rem #WCLS #T1NS #SP whd in ⊢ (% → %); |
---|
| 1036 | * #TMa * #L1 #TME |
---|
| 1037 | %{TMa} % // @(transitive_le … lGE) @L1 |
---|
| 1038 | ] qed. |
---|
[1574] | 1039 | |
---|
[2044] | 1040 | definition replace_sub_trace : ∀ge,d.∀s1,s2:RTLabs_state ge.∀t1,t2,TM1,TM2,T1,T2,l1,l2. l2 ≥ l1 → |
---|
[1719] | 1041 | ∀r:sub_trace_result ge d s1 t1 TM1 T1 l1. |
---|
| 1042 | will_return_end … TM1 = will_return_end … TM2 → |
---|
[1712] | 1043 | T2 (ends … r) (new_state … r) → |
---|
| 1044 | stack_preserved (ends … r) s2 (new_state … r) → |
---|
[1719] | 1045 | sub_trace_result ge d s2 t2 TM2 T2 l2 ≝ |
---|
| 1046 | λge,d,s1,s2,t1,t2,TM1,TM2,T1,T2,l1,l2,lGE,r,TME,trace,SP. |
---|
| 1047 | mk_sub_trace_result ge d s2 t2 TM2 T2 l2 |
---|
[1637] | 1048 | (ends … r) |
---|
[1719] | 1049 | (replace_trace … lGE … r TME trace SP). |
---|
[1637] | 1050 | |
---|
[1638] | 1051 | (* Small syntax hack to avoid ambiguous input problems. *) |
---|
[1637] | 1052 | definition myge : nat → nat → Prop ≝ ge. |
---|
| 1053 | |
---|
[2044] | 1054 | let rec make_label_return ge depth (s:RTLabs_state ge) |
---|
[1565] | 1055 | (trace: flat_trace io_out io_in ge s) |
---|
| 1056 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
[1574] | 1057 | (STATE_COSTLABELLED: well_cost_labelled_state s) (* functions in the state *) |
---|
[1583] | 1058 | (STATEMENT_COSTLABEL: RTLabs_cost s = true) (* current statement is a cost label *) |
---|
[1596] | 1059 | (TERMINATES: will_return ge depth s trace) |
---|
[1637] | 1060 | (TIME: nat) |
---|
| 1061 | (TERMINATES_IN_TIME: myge TIME (plus 2 (times 3 (will_return_length … TERMINATES)))) |
---|
[1719] | 1062 | on TIME : trace_result ge depth ends_with_ret s trace TERMINATES |
---|
[1638] | 1063 | (trace_label_return (RTLabs_status ge) s) |
---|
[2044] | 1064 | (will_return_length … TERMINATES) ≝ |
---|
[1638] | 1065 | |
---|
[1637] | 1066 | match TIME return λTIME. TIME ≥ ? → ? with |
---|
| 1067 | [ O ⇒ λTERMINATES_IN_TIME. ⊥ |
---|
| 1068 | | S TIME ⇒ λTERMINATES_IN_TIME. |
---|
[1638] | 1069 | |
---|
| 1070 | let r ≝ make_label_label ge depth s |
---|
| 1071 | trace |
---|
| 1072 | ENV_COSTLABELLED |
---|
| 1073 | STATE_COSTLABELLED |
---|
| 1074 | STATEMENT_COSTLABEL |
---|
| 1075 | TERMINATES |
---|
| 1076 | TIME ? in |
---|
[1719] | 1077 | match ends … r return λx. trace_result ge depth x s trace TERMINATES (trace_label_label (RTLabs_status ge) x s) ? → |
---|
| 1078 | trace_result ge depth ends_with_ret s trace TERMINATES (trace_label_return (RTLabs_status ge) s) (will_return_length … TERMINATES) with |
---|
[1596] | 1079 | [ ends_with_ret ⇒ λr. |
---|
[1712] | 1080 | replace_trace … r ? (tlr_base (RTLabs_status ge) s (new_state … r) (new_trace … r)) (stack_ok … r) |
---|
[1596] | 1081 | | doesnt_end_with_ret ⇒ λr. |
---|
| 1082 | let r' ≝ make_label_return ge depth (new_state … r) |
---|
[1638] | 1083 | (remainder … r) |
---|
| 1084 | ENV_COSTLABELLED |
---|
| 1085 | (cost_labelled … r) ? |
---|
| 1086 | (pi1 … (terminates … r)) TIME ? in |
---|
[1712] | 1087 | replace_trace … r' ? |
---|
[1638] | 1088 | (tlr_step (RTLabs_status ge) s (new_state … r) |
---|
[1681] | 1089 | (new_state … r') (new_trace … r) (new_trace … r')) ? |
---|
[1596] | 1090 | ] (trace_res … r) |
---|
[1638] | 1091 | |
---|
[1637] | 1092 | ] TERMINATES_IN_TIME |
---|
[1574] | 1093 | |
---|
[1638] | 1094 | |
---|
[2044] | 1095 | and make_label_label ge depth (s:RTLabs_state ge) |
---|
[1574] | 1096 | (trace: flat_trace io_out io_in ge s) |
---|
| 1097 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
| 1098 | (STATE_COSTLABELLED: well_cost_labelled_state s) (* functions in the state *) |
---|
[1583] | 1099 | (STATEMENT_COSTLABEL: RTLabs_cost s = true) (* current statement is a cost label *) |
---|
[1596] | 1100 | (TERMINATES: will_return ge depth s trace) |
---|
[1637] | 1101 | (TIME: nat) |
---|
| 1102 | (TERMINATES_IN_TIME: myge TIME (plus 1 (times 3 (will_return_length … TERMINATES)))) |
---|
[1719] | 1103 | on TIME : sub_trace_result ge depth s trace TERMINATES |
---|
[1638] | 1104 | (λends. trace_label_label (RTLabs_status ge) ends s) |
---|
| 1105 | (will_return_length … TERMINATES) ≝ |
---|
| 1106 | |
---|
[1637] | 1107 | match TIME return λTIME. TIME ≥ ? → ? with |
---|
| 1108 | [ O ⇒ λTERMINATES_IN_TIME. ⊥ |
---|
| 1109 | | S TIME ⇒ λTERMINATES_IN_TIME. |
---|
[1638] | 1110 | |
---|
[1637] | 1111 | let r ≝ make_any_label ge depth s trace ENV_COSTLABELLED STATE_COSTLABELLED TERMINATES TIME ? in |
---|
[1712] | 1112 | replace_sub_trace … r ? |
---|
[1960] | 1113 | (tll_base (RTLabs_status ge) (ends … r) s (new_state … r) (new_trace … r) ?) (stack_ok … r) |
---|
[1638] | 1114 | |
---|
[1637] | 1115 | ] TERMINATES_IN_TIME |
---|
[1574] | 1116 | |
---|
[1638] | 1117 | |
---|
[2044] | 1118 | and make_any_label ge depth (s0:RTLabs_state ge) |
---|
| 1119 | (trace: flat_trace io_out io_in ge s0) |
---|
[1574] | 1120 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
[2044] | 1121 | (STATE_COSTLABELLED: well_cost_labelled_state s0) (* functions in the state *) |
---|
| 1122 | (TERMINATES: will_return ge depth s0 trace) |
---|
[1637] | 1123 | (TIME: nat) |
---|
| 1124 | (TERMINATES_IN_TIME: myge TIME (times 3 (will_return_length … TERMINATES))) |
---|
[2044] | 1125 | on TIME : sub_trace_result ge depth s0 trace TERMINATES |
---|
| 1126 | (λends. trace_any_label (RTLabs_status ge) ends s0) |
---|
[1638] | 1127 | (will_return_length … TERMINATES) ≝ |
---|
[1637] | 1128 | |
---|
| 1129 | match TIME return λTIME. TIME ≥ ? → ? with |
---|
| 1130 | [ O ⇒ λTERMINATES_IN_TIME. ⊥ |
---|
| 1131 | | S TIME ⇒ λTERMINATES_IN_TIME. |
---|
[2044] | 1132 | match s0 return λs:RTLabs_state ge. ∀trace:flat_trace io_out io_in ge s. |
---|
| 1133 | well_cost_labelled_state s → |
---|
| 1134 | ∀TM:will_return ??? trace. |
---|
| 1135 | myge ? (times 3 (will_return_length ??? trace TM)) → |
---|
| 1136 | sub_trace_result ge depth s trace TM (λends. trace_any_label (RTLabs_status ge) ends s) (will_return_length … TM) |
---|
| 1137 | with [ mk_RTLabs_state s stk mtc0 ⇒ λtrace. |
---|
| 1138 | match trace return λs,trace. ∀mtc:Ras_Fn_Match ge s stk. |
---|
| 1139 | well_cost_labelled_state s → |
---|
[1719] | 1140 | ∀TM:will_return ??? trace. |
---|
| 1141 | myge ? (times 3 (will_return_length ??? trace TM)) → |
---|
[2044] | 1142 | sub_trace_result ge depth (mk_RTLabs_state ge s stk mtc) trace TM (λends. trace_any_label (RTLabs_status ge) ends (mk_RTLabs_state ge s stk mtc)) (will_return_length … TM) with |
---|
[1638] | 1143 | [ ft_stop st FINAL ⇒ |
---|
[2044] | 1144 | λmtc,STATE_COSTLABELLED,TERMINATES,TERMINATES_IN_TIME. ⊥ |
---|
[1638] | 1145 | |
---|
[2044] | 1146 | | ft_step start tr next EV trace' ⇒ λmtc,STATE_COSTLABELLED,TERMINATES,TERMINATES_IN_TIME. |
---|
| 1147 | let start' ≝ mk_RTLabs_state ge start stk mtc in |
---|
| 1148 | let next' ≝ next_state ? start' ?? EV in |
---|
| 1149 | match RTLabs_classify start return λx. RTLabs_classify start = x → sub_trace_result ge depth ??? (λends. trace_any_label (RTLabs_status ge) ends ?) (will_return_length … TERMINATES) with |
---|
[1583] | 1150 | [ cl_other ⇒ λCL. |
---|
[2044] | 1151 | match RTLabs_cost next return λx. RTLabs_cost next = x → sub_trace_result ge depth ??? (λends. trace_any_label (RTLabs_status ge) ends ?) (will_return_length … TERMINATES) with |
---|
[1638] | 1152 | (* We're about to run into a label. *) |
---|
[1960] | 1153 | [ true ⇒ λCS. |
---|
[2044] | 1154 | mk_sub_trace_result ge depth start' ? TERMINATES (λends. trace_any_label (RTLabs_status ge) ends start') ? |
---|
[1596] | 1155 | doesnt_end_with_ret |
---|
[2044] | 1156 | (mk_trace_result ge … next' trace' ? |
---|
| 1157 | (tal_base_not_return (RTLabs_status ge) start' next' ?? (proj1 … (RTLabs_costed ge next') CS)) ??) |
---|
[1638] | 1158 | (* An ordinary step, keep going. *) |
---|
[1583] | 1159 | | false ⇒ λCS. |
---|
[2044] | 1160 | let r ≝ make_any_label ge depth next' trace' ENV_COSTLABELLED ? (will_return_notfn … TERMINATES) TIME ? in |
---|
| 1161 | replace_sub_trace ????????????? r ? |
---|
[1638] | 1162 | (tal_step_default (RTLabs_status ge) (ends … r) |
---|
[2044] | 1163 | start' next' (new_state … r) ? (new_trace … r) ? (RTLabs_not_cost ? next' CS)) ? |
---|
[1583] | 1164 | ] (refl ??) |
---|
[1638] | 1165 | |
---|
[1586] | 1166 | | cl_jump ⇒ λCL. |
---|
[2044] | 1167 | mk_sub_trace_result ge depth start' ? TERMINATES (λends. trace_any_label (RTLabs_status ge) ends start') ? |
---|
[1596] | 1168 | doesnt_end_with_ret |
---|
[2044] | 1169 | (mk_trace_result ge … next' trace' ? |
---|
| 1170 | (tal_base_not_return (RTLabs_status ge) start' next' ???) ??) |
---|
[1638] | 1171 | |
---|
[1595] | 1172 | | cl_call ⇒ λCL. |
---|
[2044] | 1173 | let r ≝ make_label_return ge (S depth) next' trace' ENV_COSTLABELLED ?? (will_return_call … CL TERMINATES) TIME ? in |
---|
| 1174 | 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] | 1175 | (* We're about to run into a label, use base case for call *) |
---|
| 1176 | [ true ⇒ λCS. |
---|
[2044] | 1177 | mk_sub_trace_result ge depth start' ? TERMINATES (λends. trace_any_label (RTLabs_status ge) ends start') ? |
---|
[1654] | 1178 | doesnt_end_with_ret |
---|
[1719] | 1179 | (mk_trace_result ge … |
---|
[2044] | 1180 | (tal_base_call (RTLabs_status ge) start' next' (new_state … r) |
---|
| 1181 | ? CL ? (new_trace … r) ((proj1 … (RTLabs_costed …)) … CS)) ??) |
---|
[1654] | 1182 | (* otherwise use step case *) |
---|
| 1183 | | false ⇒ λCS. |
---|
| 1184 | let r' ≝ make_any_label ge depth |
---|
| 1185 | (new_state … r) (remainder … r) ENV_COSTLABELLED ? |
---|
| 1186 | (pi1 … (terminates … r)) TIME ? in |
---|
[1712] | 1187 | replace_sub_trace … r' ? |
---|
[1654] | 1188 | (tal_step_call (RTLabs_status ge) (ends … r') |
---|
[2044] | 1189 | start' next' (new_state … r) (new_state … r') ? CL ? |
---|
[1681] | 1190 | (new_trace … r) (RTLabs_not_cost … CS) (new_trace … r')) ? |
---|
[1654] | 1191 | ] (refl ??) |
---|
[1638] | 1192 | |
---|
[1594] | 1193 | | cl_return ⇒ λCL. |
---|
[2044] | 1194 | mk_sub_trace_result ge depth start' ? TERMINATES (λends. trace_any_label (RTLabs_status ge) ends start') ? |
---|
[1596] | 1195 | ends_with_ret |
---|
[1719] | 1196 | (mk_trace_result ge … |
---|
[2044] | 1197 | next' |
---|
[1596] | 1198 | trace' |
---|
| 1199 | ? |
---|
[2044] | 1200 | (tal_base_return (RTLabs_status ge) start' next' ? CL) |
---|
[1681] | 1201 | ? |
---|
[1596] | 1202 | ?) |
---|
[1583] | 1203 | ] (refl ? (RTLabs_classify start)) |
---|
[1638] | 1204 | |
---|
[2044] | 1205 | | ft_wrong start m NF EV ⇒ λmtc,STATE_COSTLABELLED,TERMINATES. ⊥ |
---|
[1638] | 1206 | |
---|
[2044] | 1207 | ] mtc0 ] trace STATE_COSTLABELLED TERMINATES TERMINATES_IN_TIME |
---|
[1637] | 1208 | ] TERMINATES_IN_TIME. |
---|
[1574] | 1209 | |
---|
[1637] | 1210 | [ cases (not_le_Sn_O ?) [ #H @H @TERMINATES_IN_TIME ] |
---|
| 1211 | | // |
---|
[1712] | 1212 | | // |
---|
[1719] | 1213 | | cases r #H1 #H2 #H3 #H4 #H5 * #H7 * #GT #_ @(le_S_to_le … GT) |
---|
| 1214 | | cases r #H1 #H2 #H3 #H4 #H5 * #H7 * #_ #EEQ // |
---|
[1681] | 1215 | | @(stack_preserved_join … (stack_ok … r)) // |
---|
[2044] | 1216 | | @(proj2 … (RTLabs_costed ge …)) @(trace_label_label_label … (new_trace … r)) |
---|
[1719] | 1217 | | cases r #H1 #H2 #H3 #H4 #H5 * #H7 * #LT #_ |
---|
[1637] | 1218 | @(le_plus_to_le … 1) @(transitive_le … TERMINATES_IN_TIME) |
---|
| 1219 | @(transitive_le … (3*(will_return_length … TERMINATES))) |
---|
| 1220 | [ >commutative_times change with ((S ?) * 3 ≤ ?) >commutative_times |
---|
[1681] | 1221 | @(monotonic_le_times_r 3 … LT) |
---|
[1637] | 1222 | | @le_S @le_S @le_n |
---|
| 1223 | ] |
---|
| 1224 | | @le_S_S_to_le @TERMINATES_IN_TIME |
---|
| 1225 | | cases (not_le_Sn_O ?) [ #H @H @TERMINATES_IN_TIME ] |
---|
| 1226 | | @le_n |
---|
[1712] | 1227 | | // |
---|
[2044] | 1228 | | @(proj1 … (RTLabs_costed …)) // |
---|
[1637] | 1229 | | @le_S_S_to_le @TERMINATES_IN_TIME |
---|
| 1230 | | @(wrl_nonzero … TERMINATES_IN_TIME) |
---|
[1713] | 1231 | | (* We can't reach the final state because the function terminates with a |
---|
| 1232 | return *) |
---|
| 1233 | inversion TERMINATES |
---|
| 1234 | [ #H214 #H215 #H216 #H217 #H218 #H219 #H220 #H221 #H222 #H223 #H224 #H225 #_ -TERMINATES -TERMINATES destruct |
---|
| 1235 | | #H228 #H229 #H230 #H231 #H232 #H233 #H234 #H235 #H236 #H237 #H238 #H239 #H240 -TERMINATES -TERMINATES destruct |
---|
| 1236 | | #H242 #H243 #H244 #H245 #H246 #H247 #H248 #H249 #H250 #H251 #H252 #H253 #H254 -TERMINATES -TERMINATES destruct |
---|
| 1237 | | #H256 #H257 #H258 #H259 #H260 #H261 #H262 #H263 #H264 #H265 -TERMINATES -TERMINATES destruct |
---|
| 1238 | ] |
---|
[1637] | 1239 | | @(will_return_return … CL TERMINATES) |
---|
[2044] | 1240 | | @(stack_preserved_return … EV) // |
---|
| 1241 | | %{tr} %{EV} @refl |
---|
[1586] | 1242 | | @(well_cost_labelled_state_step … EV) // |
---|
[1596] | 1243 | | whd @(will_return_notfn … TERMINATES) %2 @CL |
---|
[2044] | 1244 | | @(stack_preserved_step … EV) /2/ |
---|
| 1245 | | %{tr} %{EV} % |
---|
[1654] | 1246 | | %1 whd @CL |
---|
[2044] | 1247 | | @(proj1 … (RTLabs_costed …)) @(well_cost_labelled_jump … EV) // |
---|
[1594] | 1248 | | @(well_cost_labelled_state_step … EV) // |
---|
[1719] | 1249 | | whd cases (terminates ???????? r) #TMr * #LTr #EQr %{TMr} % |
---|
| 1250 | [ @(transitive_lt … LTr) cases (will_return_call … CL TERMINATES) |
---|
| 1251 | #TMx * #LT' #_ @LT' |
---|
| 1252 | | <EQr cases (will_return_call … CL TERMINATES) |
---|
| 1253 | #TM' * #_ #EQ' @EQ' |
---|
| 1254 | ] |
---|
[1682] | 1255 | | @(stack_preserved_call … EV (stack_ok … r)) // |
---|
[2044] | 1256 | | %{tr} %{EV} % |
---|
| 1257 | | @(RTLabs_after_call … next') [2: @EV | skip | // ] |
---|
[1719] | 1258 | | @(cost_labelled … r) |
---|
| 1259 | | skip |
---|
| 1260 | | cases r #ns #rm #WS #TLR #SP * #TM * #LT #_ @le_S_to_le |
---|
| 1261 | @(transitive_lt … LT) |
---|
| 1262 | cases (will_return_call … CL TERMINATES) #TM' * #LT' #_ @LT' |
---|
| 1263 | | cases r #ns #rm #WS #TLR #SP * #TM * #_ #EQ <EQ |
---|
[2044] | 1264 | cases (will_return_call … CL TERMINATES) #TM' * #_ #EQ' @sym_eq @EQ' |
---|
| 1265 | | @(RTLabs_after_call … next') [2: @EV | skip | // ] |
---|
| 1266 | | %{tr} %{EV} % |
---|
[1682] | 1267 | | @(stack_preserved_join … (stack_ok … r')) @(stack_preserved_call … EV (stack_ok … r)) // |
---|
[1595] | 1268 | | @(cost_labelled … r) |
---|
[1719] | 1269 | | cases r #H72 #H73 #H74 #H75 #HX * #HY * #GT #H78 |
---|
[1637] | 1270 | @(le_plus_to_le … 1) @(transitive_le … TERMINATES_IN_TIME) |
---|
[1719] | 1271 | cases (will_return_call … TERMINATES) in GT; |
---|
| 1272 | #X * #Y #_ #Z |
---|
[1637] | 1273 | @(transitive_le … (monotonic_lt_times_r 3 … Y)) |
---|
| 1274 | [ @(transitive_le … (monotonic_lt_times_r 3 … Z)) // |
---|
| 1275 | | // |
---|
| 1276 | ] |
---|
[1596] | 1277 | | @(well_cost_labelled_state_step … EV) // |
---|
| 1278 | | @(well_cost_labelled_call … EV) // |
---|
[1638] | 1279 | | cases (will_return_call … TERMINATES) |
---|
[1719] | 1280 | #TM * #GT #_ @le_S_S_to_le |
---|
[1637] | 1281 | >commutative_times change with ((S ?) * 3 ≤ ?) >commutative_times |
---|
| 1282 | @(transitive_le … TERMINATES_IN_TIME) |
---|
| 1283 | @(monotonic_le_times_r 3 … GT) |
---|
[1596] | 1284 | | whd @(will_return_notfn … TERMINATES) %1 @CL |
---|
[1682] | 1285 | | @(stack_preserved_step … EV) /2/ |
---|
[2044] | 1286 | | %{tr} %{EV} % |
---|
[1654] | 1287 | | %2 whd @CL |
---|
[1596] | 1288 | | @(well_cost_labelled_state_step … EV) // |
---|
[1719] | 1289 | | cases (will_return_notfn … TERMINATES) #TM * #GT #_ @(le_S_to_le … GT) |
---|
[2044] | 1290 | | cases (will_return_notfn … TERMINATES) #TM * #_ #EQ @sym_eq @EQ |
---|
[1594] | 1291 | | @CL |
---|
[2044] | 1292 | | %{tr} %{EV} % |
---|
[1682] | 1293 | | @(stack_preserved_join … (stack_ok … r)) @(stack_preserved_step … EV) /2/ |
---|
[1594] | 1294 | | @(well_cost_labelled_state_step … EV) // |
---|
[1638] | 1295 | | %1 @CL |
---|
[1719] | 1296 | | cases (will_return_notfn … TERMINATES) #TM * #GT #_ |
---|
[1637] | 1297 | @le_S_S_to_le |
---|
| 1298 | @(transitive_le … (monotonic_lt_times_r … GT) TERMINATES_IN_TIME) |
---|
| 1299 | // |
---|
[1574] | 1300 | | inversion TERMINATES |
---|
[1637] | 1301 | [ #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 #H13 -TERMINATES -TERMINATES destruct |
---|
| 1302 | | #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 #H29 -TERMINATES -TERMINATES destruct |
---|
| 1303 | | #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 #H42 #H43 #H44 #H45 -TERMINATES -TERMINATES destruct |
---|
| 1304 | | #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 -TERMINATES -TERMINATES destruct |
---|
[1583] | 1305 | ] |
---|
[1713] | 1306 | ] qed. |
---|
[1583] | 1307 | |
---|
[1638] | 1308 | (* We can initialise TIME with a suitably large value based on the length of the |
---|
| 1309 | termination proof. *) |
---|
[2044] | 1310 | let rec make_label_return' ge depth (s:RTLabs_state ge) |
---|
[1637] | 1311 | (trace: flat_trace io_out io_in ge s) |
---|
| 1312 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
| 1313 | (STATE_COSTLABELLED: well_cost_labelled_state s) (* functions in the state *) |
---|
| 1314 | (STATEMENT_COSTLABEL: RTLabs_cost s = true) (* current statement is a cost label *) |
---|
| 1315 | (TERMINATES: will_return ge depth s trace) |
---|
[1719] | 1316 | : trace_result ge depth ends_with_ret s trace TERMINATES (trace_label_return (RTLabs_status ge) s) (will_return_length … TERMINATES) ≝ |
---|
[1637] | 1317 | make_label_return ge depth s trace ENV_COSTLABELLED STATE_COSTLABELLED STATEMENT_COSTLABEL TERMINATES |
---|
| 1318 | (2 + 3 * will_return_length … TERMINATES) ?. |
---|
| 1319 | @le_n |
---|
| 1320 | qed. |
---|
[1574] | 1321 | |
---|
[1713] | 1322 | (* Tail-calls would not be handled properly (which means that if we try to show the |
---|
[1617] | 1323 | full version with non-termination we'll fail because calls and returns aren't |
---|
| 1324 | balanced. |
---|
[1651] | 1325 | *) |
---|
| 1326 | |
---|
| 1327 | inductive inhabited (T:Type[0]) : Prop ≝ |
---|
| 1328 | | witness : T → inhabited T. |
---|
| 1329 | |
---|
| 1330 | |
---|
[1705] | 1331 | (* Define a notion of sound labellings of RTLabs programs. *) |
---|
[1675] | 1332 | |
---|
[1705] | 1333 | definition actual_successor : state → option label ≝ |
---|
| 1334 | λs. match s with |
---|
| 1335 | [ State f fs m ⇒ Some ? (next f) |
---|
| 1336 | | Callstate _ _ _ fs _ ⇒ match fs with [ cons f _ ⇒ Some ? (next f) | _ ⇒ None ? ] |
---|
| 1337 | | Returnstate _ _ _ _ ⇒ None ? |
---|
[1713] | 1338 | | Finalstate _ ⇒ None ? |
---|
[1705] | 1339 | ]. |
---|
| 1340 | |
---|
| 1341 | lemma nth_opt_Exists : ∀A,n,l,a. |
---|
| 1342 | nth_opt A n l = Some A a → |
---|
| 1343 | Exists A (λa'. a' = a) l. |
---|
| 1344 | #A #n elim n |
---|
| 1345 | [ * [ #a #E normalize in E; destruct | #a #l #a' #E normalize in E; destruct % // ] |
---|
| 1346 | | #m #IH * |
---|
| 1347 | [ #a #E normalize in E; destruct |
---|
| 1348 | | #a #l #a' #E %2 @IH @E |
---|
| 1349 | ] |
---|
| 1350 | ] qed. |
---|
| 1351 | |
---|
| 1352 | lemma eval_successor : ∀ge,f,fs,m,tr,s'. |
---|
| 1353 | eval_statement ge (State f fs m) = Value ??? 〈tr,s'〉 → |
---|
| 1354 | RTLabs_classify s' = cl_return ∨ |
---|
| 1355 | ∃l. actual_successor s' = Some ? l ∧ Exists ? (λl0. l0 = l) (successors (lookup_present … (f_graph (func f)) (next f) (next_ok f))). |
---|
| 1356 | #ge * #func #locals #next #next_ok #sp #dst #fs #m #tr #s' |
---|
| 1357 | whd in ⊢ (??%? → ?); |
---|
| 1358 | generalize in ⊢ (??(?%)? → ?); cases (lookup_present ??? next next_ok) |
---|
| 1359 | [ #l #LP whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1360 | | #cl #l #LP whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
[1960] | 1361 | | #ty #r #c #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1362 | | #ty #ty' #op #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #v' #Ev' @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1363 | | #ty1 #ty2 #ty' #op #r1 #r2 #r3 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #v' #Ev' @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1364 | | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1365 | | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #m' #Em whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1366 | | #id #rs #r #l #LP whd in ⊢ (??%? → ?); @bind_res_value #b #Eb @bind_ok #fd #Efd @bind_ok #vs #Evs whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1367 | | #r #rs #r' #l #LP whd in ⊢ (??%? → ?); @bind_res_value #fv #Efv @bind_ok #fd #Efd @bind_ok #vs #Evs whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
| 1368 | | #r #l1 #l2 #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #b #Eb whd in ⊢ (??%? → ?); #E destruct %2 cases b [ %{l1} | %{l2} ] % // [ % | %2 %] // |
---|
| 1369 | | #r #ls #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev |
---|
[2184] | 1370 | cases v [ #E normalize in E; destruct | #sz #i | #f #E normalize in E; destruct | #E normalize in E; destruct | #p #E normalize in E; destruct ] |
---|
[1705] | 1371 | whd in ⊢ (??%? → ?); |
---|
| 1372 | generalize in ⊢ (??(?%)? → ?); |
---|
| 1373 | cases (nth_opt label (nat_of_bitvector (bitsize_of_intsize sz) i) ls) in ⊢ (???% → ??(match % with [ _ ⇒ ? | _ ⇒ ? ] ?)? → ?); |
---|
| 1374 | [ #e #E normalize in E; destruct |
---|
| 1375 | | #l #El whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // @(nth_opt_Exists … El) |
---|
| 1376 | ] |
---|
[1960] | 1377 | | #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev whd in ⊢ (??%? → ?); #E destruct %1 @refl |
---|
[1705] | 1378 | ] qed. |
---|
| 1379 | |
---|
[1707] | 1380 | (* |
---|
[1706] | 1381 | lemma steps_to_label_bound_inv : ∀g,l,n. |
---|
[1705] | 1382 | steps_to_label_bound g l n → |
---|
[1706] | 1383 | ∀H. let stmt ≝ lookup_present … g l H in |
---|
| 1384 | ∃n'. n = steps_for_statement stmt + n' ∧ |
---|
[1705] | 1385 | (∀l'. Exists label (λl0. l0 = l') (successors stmt) → |
---|
[1706] | 1386 | (∃H'. bool_to_Prop (is_cost_label (lookup_present … g l' H'))) ∨ |
---|
| 1387 | steps_to_label_bound g l' n'). |
---|
| 1388 | #g #l0 #n0 #S inversion S #l #n #H #IH #E1 #E2 #_ destruct #H' |
---|
| 1389 | % [2: % [ @refl | #l' #EX cases (IH l' EX) /2/ ] | skip ] |
---|
| 1390 | qed. |
---|
[1707] | 1391 | *) |
---|
[1719] | 1392 | |
---|
[1707] | 1393 | (* |
---|
[1705] | 1394 | definition soundly_labelled_pc ≝ λg,l. ∃n. steps_to_label_bound g l n. |
---|
| 1395 | |
---|
| 1396 | let rec soundly_labelled_fn (fn : internal_function) : Prop ≝ |
---|
| 1397 | soundly_labelled_pc (f_graph fn) (f_entry fn). |
---|
| 1398 | |
---|
| 1399 | |
---|
[1675] | 1400 | definition soundly_labelled_frame : frame → Prop ≝ |
---|
| 1401 | λf. soundly_labelled_pc (f_graph (func f)) (next f). |
---|
| 1402 | |
---|
| 1403 | definition soundly_labelled_state : state → Prop ≝ |
---|
| 1404 | λs. match s with |
---|
| 1405 | [ State f _ _ ⇒ soundly_labelled_frame f |
---|
| 1406 | | Callstate _ _ _ stk _ ⇒ match stk with [ nil ⇒ False | cons f _ ⇒ soundly_labelled_frame f ] |
---|
| 1407 | | Returnstate _ _ stk _ ⇒ match stk with [ nil ⇒ False | cons f _ ⇒ soundly_labelled_frame f ] |
---|
| 1408 | ]. |
---|
[1707] | 1409 | *) |
---|
| 1410 | definition frame_bound_on_steps_to_cost : frame → nat → Prop ≝ |
---|
| 1411 | λf. bound_on_steps_to_cost (f_graph (func f)) (next f). |
---|
| 1412 | definition frame_bound_on_steps_to_cost1 : frame → nat → Prop ≝ |
---|
| 1413 | λf. bound_on_steps_to_cost1 (f_graph (func f)) (next f). |
---|
[1705] | 1414 | |
---|
[1707] | 1415 | inductive state_bound_on_steps_to_cost : state → nat → Prop ≝ |
---|
| 1416 | | 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 |
---|
| 1417 | | 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) |
---|
| 1418 | | 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] | 1419 | . |
---|
| 1420 | |
---|
[1707] | 1421 | lemma state_bound_on_steps_to_cost_zero : ∀s. |
---|
| 1422 | ¬ state_bound_on_steps_to_cost s O. |
---|
[1705] | 1423 | #s % #H inversion H |
---|
[1707] | 1424 | [ #H46 #H47 #H48 #H49 #H50 #H51 #H52 #H53 destruct |
---|
| 1425 | whd in H50; @(bound_on_steps_to_cost1_inv_ind … H50) (* XXX inversion H50*) |
---|
| 1426 | #H55 #H56 #H57 #H58 #H59 #H60 #H61 normalize in H60; destruct |
---|
[1705] | 1427 | | #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 destruct |
---|
| 1428 | | #H13 #H14 #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 destruct |
---|
| 1429 | ] qed. |
---|
| 1430 | |
---|
| 1431 | lemma eval_steps : ∀ge,f,fs,m,tr,s'. |
---|
| 1432 | eval_statement ge (State f fs m) = Value ??? 〈tr,s'〉 → |
---|
| 1433 | steps_for_statement (lookup_present ?? (f_graph (func f)) (next f) (next_ok f)) = |
---|
[1713] | 1434 | match s' with [ State _ _ _ ⇒ 1 | Callstate _ _ _ _ _ ⇒ 2 | Returnstate _ _ _ _ ⇒ 2 | Finalstate _ ⇒ 1 ]. |
---|
[1705] | 1435 | #ge * #func #locals #next #next_ok #sp #dst #fs #m #tr #s' |
---|
| 1436 | whd in ⊢ (??%? → ?); |
---|
| 1437 | generalize in ⊢ (??(?%)? → ?); cases (lookup_present ??? next next_ok) |
---|
| 1438 | [ #l #LP whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1439 | | #cl #l #LP whd in ⊢ (??%? → ?); #E destruct @refl |
---|
[1960] | 1440 | | #ty #r #c #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1441 | | #ty #ty' #op #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #v' #Ev' @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1442 | | #ty1 #ty2 #ty' #op #r1 #r2 #r3 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #v' #Ev' @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1443 | | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1444 | | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #m' #Em whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1445 | | #id #rs #r #l #LP whd in ⊢ (??%? → ?); @bind_res_value #b #Eb @bind_ok #fd #Efd @bind_ok #vs #Evs whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1446 | | #r #rs #r' #l #LP whd in ⊢ (??%? → ?); @bind_res_value #fv #Efv @bind_ok #fd #Efd @bind_ok #vs #Evs whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1447 | | #r #l1 #l2 #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #b #Eb whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1448 | | #r #ls #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev |
---|
[2184] | 1449 | cases v [ #E normalize in E; destruct | #sz #i | #f #E normalize in E; destruct | #E normalize in E; destruct | #p #E normalize in E; destruct ] |
---|
[1705] | 1450 | whd in ⊢ (??%? → ?); |
---|
| 1451 | generalize in ⊢ (??(?%)? → ?); |
---|
| 1452 | cases (nth_opt label (nat_of_bitvector (bitsize_of_intsize sz) i) ls) in ⊢ (???% → ??(match % with [ _ ⇒ ? | _ ⇒ ? ] ?)? → ?); |
---|
| 1453 | [ #e #E normalize in E; destruct |
---|
| 1454 | | #l #El whd in ⊢ (??%? → ?); #E destruct @refl |
---|
| 1455 | ] |
---|
[1960] | 1456 | | #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev whd in ⊢ (??%? → ?); #E destruct @refl |
---|
[1705] | 1457 | ] qed. |
---|
| 1458 | |
---|
[2044] | 1459 | lemma bound_after_call : ∀ge.∀s,s':RTLabs_state ge.∀n. |
---|
[1736] | 1460 | state_bound_on_steps_to_cost s (S n) → |
---|
| 1461 | ∀CL:RTLabs_classify s = cl_call. |
---|
| 1462 | as_after_return (RTLabs_status ge) «s, CL» s' → |
---|
| 1463 | RTLabs_cost s' = false → |
---|
| 1464 | state_bound_on_steps_to_cost s' n. |
---|
[2044] | 1465 | #ge * #s #stk #mtc * #s' #stk' #mtc' #n #H #CL whd in ⊢ (% → ?); -stk -stk' lapply CL -CL inversion H |
---|
[1736] | 1466 | [ #f #fs #m #n' #S #E1 #E2 #_ #CL @⊥ cases (rtlabs_call_inv … CL) |
---|
| 1467 | #fn * #args * #dst * #stk * #m' #E destruct |
---|
| 1468 | | #fd #args #dst #f #fs #m #n' #S #E1 #E2 #_ destruct |
---|
| 1469 | whd in S; #CL cases s' |
---|
| 1470 | [ #f' #fs' #m' * #N #F #CS |
---|
| 1471 | %1 whd |
---|
| 1472 | inversion S |
---|
| 1473 | [ #l #n #P #CS' #E1 #E2 #_ destruct @⊥ |
---|
| 1474 | change with (is_cost_label ?) in CS:(??%?); >N in P CS'; >F >CS #P * |
---|
| 1475 | | #l #n #B #E1 #E2 #_ destruct <N <F @B |
---|
| 1476 | ] |
---|
| 1477 | | #fd' #args' #dst' #fs' #m' * |
---|
| 1478 | | #rv #dst' #fs' #m' * |
---|
| 1479 | | #r #E normalize in E; destruct |
---|
| 1480 | ] |
---|
| 1481 | | #rtv #dst #f #fs #m #n' #S #E1 #E2 #E3 destruct #CL normalize in CL; destruct |
---|
| 1482 | ] qed. |
---|
| 1483 | |
---|
[1707] | 1484 | lemma bound_after_step : ∀ge,s,tr,s',n. |
---|
| 1485 | state_bound_on_steps_to_cost s (S n) → |
---|
[1705] | 1486 | eval_statement ge s = Value ??? 〈tr, s'〉 → |
---|
[1706] | 1487 | RTLabs_cost s' = false → |
---|
[1705] | 1488 | (RTLabs_classify s' = cl_return ∨ RTLabs_classify s = cl_call) ∨ |
---|
[1707] | 1489 | state_bound_on_steps_to_cost s' n. |
---|
| 1490 | #ge #s #tr #s' #n #BOUND1 inversion BOUND1 |
---|
[1705] | 1491 | [ #f #fs #m #m #FS #E1 #E2 #_ destruct |
---|
| 1492 | #EVAL cases (eval_successor … EVAL) |
---|
| 1493 | [ /3/ |
---|
| 1494 | | * #l * #S1 #S2 #NC %2 |
---|
[1707] | 1495 | (* |
---|
| 1496 | cases (bound_on_steps_to_cost1_inv … FS ?) [2: @(next_ok f) ] |
---|
| 1497 | *) |
---|
| 1498 | @(bound_on_steps_to_cost1_inv_ind … FS) #next #n' #next_ok #IH #E1 #E2 #E3 destruct |
---|
[2025] | 1499 | inversion (eval_preserves … EVAL) |
---|
[1707] | 1500 | [ #ge0 #f0 #f' #fs' #m0 #m' #F #E4 #E5 #E6 #_ destruct |
---|
| 1501 | >(eval_steps … EVAL) in E2; #En normalize in En; |
---|
| 1502 | inversion F #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 destruct |
---|
| 1503 | %1 inversion (IH … S2) |
---|
| 1504 | [ #lx #nx #LPx #CSx #E1x #E2x @⊥ destruct |
---|
| 1505 | change with (RTLabs_cost (State (mk_frame H1 H7 lx LPx H5 H6) fs' m')) in CSx:(?%); |
---|
| 1506 | whd in S1:(??%?); destruct >NC in CSx; * |
---|
| 1507 | | whd in S1:(??%?); destruct #H71 #H72 #H73 #H74 #H75 #H76 destruct @H73 |
---|
[1706] | 1508 | ] |
---|
[1707] | 1509 | | #ge0 #f0 #fs' #m0 #fd #args #f' #dst #F #b #FFP #E4 #E5 #E6 #_ destruct |
---|
| 1510 | >(eval_steps … EVAL) in E2; #En normalize in En; |
---|
| 1511 | inversion F #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 destruct |
---|
| 1512 | %2 @IH normalize in S1; destruct @S2 |
---|
| 1513 | | #H14 #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 |
---|
[1705] | 1514 | destruct |
---|
[1707] | 1515 | | #H31 #H32 #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 destruct |
---|
[1705] | 1516 | normalize in S1; destruct |
---|
[1707] | 1517 | | #H44 #H45 #H46 #H47 #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 destruct |
---|
[1713] | 1518 | | #H267 #H268 #H269 #H270 #H271 #H272 #H273 #H274 destruct |
---|
[1705] | 1519 | ] |
---|
| 1520 | ] |
---|
| 1521 | | #H58 #H59 #H60 #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 #H70 destruct |
---|
| 1522 | /3/ |
---|
| 1523 | | #rtv #dst #f #fs #m #n' #FS #E1 #E2 #_ destruct |
---|
[2025] | 1524 | #EVAL #NC %2 inversion (eval_preserves … EVAL) |
---|
[1705] | 1525 | [ #H72 #H73 #H74 #H75 #H76 #H77 #H78 #H79 #H80 #H81 #H82 destruct |
---|
| 1526 | | #H84 #H85 #H86 #H87 #H88 #H89 #H90 #H91 #H92 #H93 #H94 #H95 #H96 #H97 #H98 destruct |
---|
| 1527 | | #H100 #H101 #H102 #H103 #H104 #H105 #H106 #H107 #H108 #H109 #H110 #H111 #H112 #H113 #H114 destruct |
---|
| 1528 | | #H116 #H117 #H118 #H119 #H120 #H121 #H122 #H123 #H124 #H125 #H126 destruct |
---|
| 1529 | | #ge' #f' #fs' #rtv' #dst' #f'' #m' #F #E1 #E2 #E3 #_ destruct |
---|
| 1530 | %1 whd in FS ⊢ %; |
---|
[1736] | 1531 | inversion (stack_preserved_return … EVAL) [ @refl | 2,4,5: #H141 #H142 #H143 #H144 #H145 #H146 #H147 try #H148 try #H149 destruct ] |
---|
| 1532 | #s1 #f1 #f2 #fs #m #FE #FR #SS1 #_ #E1 #E2 #_ destruct <FE |
---|
[1705] | 1533 | 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 ] |
---|
| 1534 | inversion F #func #locals #next #next_ok #sp #retdst #locals' #next' #next_ok' #E1 #E2 #_ destruct |
---|
[1707] | 1535 | inversion FS |
---|
| 1536 | [ #lx #nx #LPx #CSx #E1x #E2x @⊥ destruct |
---|
| 1537 | change with (RTLabs_cost (State (mk_frame func locals' lx ? sp retdst) fs m0)) in CSx:(?%); |
---|
| 1538 | >NC in CSx; * |
---|
| 1539 | | #lx #nx #H #E1x #E2x #_ destruct @H |
---|
| 1540 | ] |
---|
[1713] | 1541 | | #H284 #H285 #H286 #H287 #H288 #H289 #H290 #H291 destruct |
---|
[1705] | 1542 | ] |
---|
| 1543 | ] qed. |
---|
[1806] | 1544 | |
---|
| 1545 | |
---|
| 1546 | |
---|
| 1547 | |
---|
| 1548 | definition soundly_labelled_ge : genv → Prop ≝ |
---|
[2044] | 1549 | λge. ∀b,f. find_funct_ptr … ge b = Some ? (Internal ? f) → soundly_labelled_fn f. |
---|
[1806] | 1550 | |
---|
| 1551 | definition soundly_labelled_state : state → Prop ≝ |
---|
| 1552 | λs. match s with |
---|
| 1553 | [ State f fs m ⇒ soundly_labelled_fn (func f) ∧ All ? (λf. soundly_labelled_fn (func f)) fs |
---|
| 1554 | | Callstate fd _ _ fs _ ⇒ match fd with [ Internal fn ⇒ soundly_labelled_fn fn | External _ ⇒ True ] ∧ |
---|
| 1555 | All ? (λf. soundly_labelled_fn (func f)) fs |
---|
| 1556 | | Returnstate _ _ fs _ ⇒ All ? (λf. soundly_labelled_fn (func f)) fs |
---|
| 1557 | | Finalstate _ ⇒ True |
---|
| 1558 | ]. |
---|
| 1559 | |
---|
| 1560 | lemma steps_from_sound : ∀s. |
---|
| 1561 | RTLabs_cost s = true → |
---|
| 1562 | soundly_labelled_state s → |
---|
| 1563 | ∃n. state_bound_on_steps_to_cost s n. |
---|
| 1564 | * [ #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 ] |
---|
| 1565 | whd in ⊢ (% → ?); * #SLF #_ |
---|
| 1566 | cases (SLF (next f) (next_ok f)) #n #B1 |
---|
| 1567 | %{n} % @B1 |
---|
| 1568 | qed. |
---|
| 1569 | |
---|
| 1570 | lemma soundly_labelled_state_step : ∀ge,s,tr,s'. |
---|
| 1571 | soundly_labelled_ge ge → |
---|
| 1572 | eval_statement ge s = Value ??? 〈tr,s'〉 → |
---|
| 1573 | soundly_labelled_state s → |
---|
| 1574 | soundly_labelled_state s'. |
---|
| 1575 | #ge #s #tr #s' #ENV #EV #S |
---|
[2025] | 1576 | inversion (eval_preserves … EV) |
---|
[1806] | 1577 | [ #ge' #f #f' #fs #m #m' #F #E1 #E2 #E3 #_ destruct |
---|
| 1578 | whd in S ⊢ %; inversion F #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 destruct @S |
---|
| 1579 | | #ge' #f #fs #m #fd #args #f' #dst #F #b #FFP #E1 #E2 #E3 #_ destruct |
---|
| 1580 | whd in S ⊢ %; % |
---|
| 1581 | [ cases fd in FFP ⊢ %; // #fn #FFP @ENV // |
---|
| 1582 | | inversion F #H30 #H31 #H32 #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 destruct @S |
---|
| 1583 | ] |
---|
| 1584 | | #ge' #fn #locals #next #nok #sp #fs #m #args #dst #m' #E1 #E2 #E3 #E4 destruct |
---|
| 1585 | whd in S ⊢ %; @S |
---|
| 1586 | | #ge' #f #fs #m #rtv #dst #m' #E1 #E2 #E3 #E4 destruct |
---|
| 1587 | whd in S ⊢ %; cases S // |
---|
| 1588 | | #ge' #f #fs #rtv #dst #f' #m #F #E1 #E2 #E3 #E4 destruct |
---|
| 1589 | whd in S ⊢ %; inversion F #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 destruct @S |
---|
| 1590 | | #ge' #r #dst #m #E1 #E2 #E3 #E4 destruct @I |
---|
| 1591 | ] qed. |
---|
| 1592 | |
---|
| 1593 | lemma soundly_labelled_state_preserved : ∀s,s'. |
---|
| 1594 | stack_preserved ends_with_ret s s' → |
---|
| 1595 | soundly_labelled_state s → |
---|
| 1596 | soundly_labelled_state s'. |
---|
| 1597 | #s0 #s0' #SP inversion SP |
---|
| 1598 | [ #H73 #H74 #H75 #H76 #H77 #H78 #H79 #H80 #H81 #H82 destruct |
---|
| 1599 | | #s1 #f #f' #fs #m #N #F #S1 #E1 #E2 #E3 #E4 destruct |
---|
| 1600 | inversion S1 |
---|
| 1601 | [ #f1 #fs1 #m1 #E1 #E2 #E3 destruct |
---|
| 1602 | * #_ #S whd in S; |
---|
| 1603 | inversion F #H96 #H97 #H98 #H99 #H100 #H101 #H102 #H103 #H104 #H105 #H106 #H107 |
---|
| 1604 | destruct @S |
---|
| 1605 | | #fd #args #dst #f1 #fs1 #m1 #E1 #E2 #E3 destruct * #_ * #_ #S |
---|
| 1606 | inversion F #H96 #H97 #H98 #H99 #H100 #H101 #H102 #H103 #H104 #H105 #H106 #H107 |
---|
| 1607 | destruct @S |
---|
| 1608 | | #rtv #dst #fs1 #m1 #E1 #E2 #E3 destruct #S |
---|
| 1609 | inversion F #H96 #H97 #H98 #H99 #H100 #H101 #H102 #H103 #H104 #H105 #H106 #H107 |
---|
| 1610 | destruct @S |
---|
| 1611 | ] |
---|
| 1612 | | // |
---|
| 1613 | | // |
---|
| 1614 | ] qed. |
---|
| 1615 | |
---|
[1653] | 1616 | (* When constructing an infinite trace, we need to be able to grab the finite |
---|
| 1617 | portion of the trace for the next [trace_label_diverges] constructor. We |
---|
| 1618 | use the fact that the trace is soundly labelled to achieve this. *) |
---|
| 1619 | |
---|
[2044] | 1620 | record remainder_ok (ge:genv) (s:RTLabs_state ge) (t:flat_trace io_out io_in ge s) : Type[0] ≝ { |
---|
[1805] | 1621 | ro_well_cost_labelled: well_cost_labelled_state s; |
---|
[1806] | 1622 | ro_soundly_labelled: soundly_labelled_state s; |
---|
[1805] | 1623 | ro_no_termination: Not (∃depth. inhabited (will_return ge depth s t)); |
---|
| 1624 | ro_not_undefined: not_wrong … t; |
---|
| 1625 | ro_not_final: RTLabs_is_final s = None ? |
---|
| 1626 | }. |
---|
| 1627 | |
---|
[2044] | 1628 | inductive finite_prefix (ge:genv) : RTLabs_state ge → Prop ≝ |
---|
| 1629 | | fp_tal : ∀s,s':RTLabs_state ge. |
---|
[1653] | 1630 | trace_any_label (RTLabs_status ge) doesnt_end_with_ret s s' → |
---|
[1805] | 1631 | ∀t:flat_trace io_out io_in ge s'. |
---|
| 1632 | remainder_ok ge s' t → |
---|
[1653] | 1633 | finite_prefix ge s |
---|
[2044] | 1634 | | fp_tac : ∀s1,s2,s3:RTLabs_state ge. |
---|
[1806] | 1635 | trace_any_call (RTLabs_status ge) s1 s2 → |
---|
| 1636 | well_cost_labelled_state s2 → |
---|
[2044] | 1637 | as_execute (RTLabs_status ge) s2 s3 → |
---|
[1806] | 1638 | ∀t:flat_trace io_out io_in ge s3. |
---|
| 1639 | remainder_ok ge s3 t → |
---|
| 1640 | finite_prefix ge s1 |
---|
[1653] | 1641 | . |
---|
| 1642 | |
---|
[2044] | 1643 | definition fp_add_default : ∀ge. ∀s,s':RTLabs_state ge. |
---|
[1653] | 1644 | RTLabs_classify s = cl_other → |
---|
| 1645 | finite_prefix ge s' → |
---|
[2044] | 1646 | as_execute (RTLabs_status ge) s s' → |
---|
[1653] | 1647 | RTLabs_cost s' = false → |
---|
| 1648 | finite_prefix ge s ≝ |
---|
| 1649 | λge,s,s',OTHER,fp. |
---|
[2044] | 1650 | match fp return λs1.λfp1:finite_prefix ge s1. as_execute (RTLabs_status ge) ? s1 → RTLabs_cost s1 = false → finite_prefix ge s with |
---|
[1805] | 1651 | [ fp_tal s' sf TAL rem rok ⇒ λEVAL, NOT_COST. fp_tal ge s sf |
---|
[1670] | 1652 | (tal_step_default (RTLabs_status ge) doesnt_end_with_ret s s' sf EVAL TAL OTHER (RTLabs_not_cost … NOT_COST)) |
---|
[1805] | 1653 | rem rok |
---|
[2044] | 1654 | | fp_tac s1 s2 s3 TAC WCL2 EV rem rok ⇒ λEVAL, NOT_COST. fp_tac ge s s2 s3 |
---|
[1806] | 1655 | (tac_step_default (RTLabs_status ge) ??? EVAL TAC OTHER (RTLabs_not_cost … NOT_COST)) |
---|
| 1656 | WCL2 EV rem rok |
---|
[1653] | 1657 | ]. |
---|
[1670] | 1658 | |
---|
[2044] | 1659 | definition fp_add_terminating_call : ∀ge.∀s,s1,s'':RTLabs_state ge. |
---|
| 1660 | as_execute (RTLabs_status ge) s s1 → |
---|
[1653] | 1661 | ∀CALL:RTLabs_classify s = cl_call. |
---|
[1806] | 1662 | finite_prefix ge s'' → |
---|
| 1663 | trace_label_return (RTLabs_status ge) s1 s'' → |
---|
| 1664 | as_after_return (RTLabs_status ge) (mk_Sig ?? s CALL) s'' → |
---|
| 1665 | RTLabs_cost s'' = false → |
---|
[1653] | 1666 | finite_prefix ge s ≝ |
---|
[1806] | 1667 | λge,s,s1,s'',EVAL,CALL,fp. |
---|
[2044] | 1668 | match fp return λs''.λfp:finite_prefix ge s''. trace_label_return (RTLabs_status ge) ? s'' → as_after_return (RTLabs_status ge) ? s'' → RTLabs_cost s'' = false → finite_prefix ge s with |
---|
[1806] | 1669 | [ fp_tal s'' sf TAL rem rok ⇒ λTLR,RET,NOT_COST. fp_tal ge s sf |
---|
| 1670 | (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] | 1671 | rem rok |
---|
[2044] | 1672 | | fp_tac s'' s2 s3 TAC WCL2 EV rem rok ⇒ λTLR,RET,NOT_COST. fp_tac ge s s2 s3 |
---|
[1806] | 1673 | (tac_step_call (RTLabs_status ge) s s'' s2 s1 EVAL CALL RET TLR (RTLabs_not_cost … NOT_COST) TAC) |
---|
| 1674 | WCL2 EV rem rok |
---|
[1653] | 1675 | ]. |
---|
[1670] | 1676 | |
---|
[1765] | 1677 | lemma not_return_to_not_final : ∀ge,s,tr,s'. |
---|
| 1678 | eval_statement ge s = Value ??? 〈tr, s'〉 → |
---|
| 1679 | RTLabs_classify s ≠ cl_return → |
---|
| 1680 | RTLabs_is_final s' = None ?. |
---|
| 1681 | #ge #s #tr #s' #EV |
---|
[2025] | 1682 | inversion (eval_preserves … EV) // |
---|
[1765] | 1683 | #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 #CL |
---|
| 1684 | @⊥ @(absurd ?? CL) @refl |
---|
| 1685 | qed. |
---|
| 1686 | |
---|
[1670] | 1687 | definition termination_oracle ≝ ∀ge,depth,s,trace. |
---|
[1671] | 1688 | inhabited (will_return ge depth s trace) ∨ ¬ inhabited (will_return ge depth s trace). |
---|
[1670] | 1689 | |
---|
[2044] | 1690 | let rec finite_segment ge (s:RTLabs_state ge) n trace |
---|
[1670] | 1691 | (ORACLE: termination_oracle) |
---|
[1805] | 1692 | (TRACE_OK: remainder_ok ge s trace) |
---|
[1670] | 1693 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
[1806] | 1694 | (ENV_SOUNDLY_LABELLED: soundly_labelled_ge ge) |
---|
[1707] | 1695 | (LABEL_LIMIT: state_bound_on_steps_to_cost s n) |
---|
[2044] | 1696 | on n : finite_prefix ge s ≝ |
---|
[1707] | 1697 | match n return λn. state_bound_on_steps_to_cost s n → finite_prefix ge s with |
---|
[1705] | 1698 | [ O ⇒ λLABEL_LIMIT. ⊥ |
---|
[2044] | 1699 | | S n' ⇒ |
---|
| 1700 | match s return λs:RTLabs_state ge. ∀trace:flat_trace io_out io_in ge s. remainder_ok ge s trace → state_bound_on_steps_to_cost s (S n') → finite_prefix ge s with [ mk_RTLabs_state s0 stk mtc0 ⇒ λtrace'. |
---|
| 1701 | match trace' return λs:state.λtrace:flat_trace io_out io_in ge s. ∀mtc:Ras_Fn_Match ge s stk. remainder_ok ge (mk_RTLabs_state ge s ? mtc) trace → state_bound_on_steps_to_cost s (S n') → finite_prefix ge (mk_RTLabs_state ge s ? mtc) with |
---|
| 1702 | [ ft_stop st FINAL ⇒ λmtc,TRACE_OK,LABEL_LIMIT. ⊥ |
---|
| 1703 | | ft_step start tr next EV trace' ⇒ λmtc,TRACE_OK,LABEL_LIMIT. |
---|
| 1704 | let start' ≝ mk_RTLabs_state ge start stk mtc in |
---|
| 1705 | let next' ≝ next_state ge start' next tr EV in |
---|
[1670] | 1706 | match RTLabs_classify start return λx. RTLabs_classify start = x → ? with |
---|
| 1707 | [ cl_other ⇒ λCL. |
---|
[1805] | 1708 | let TRACE_OK' ≝ ? in |
---|
[1670] | 1709 | match RTLabs_cost next return λx. RTLabs_cost next = x → ? with |
---|
| 1710 | [ true ⇒ λCS. |
---|
[2044] | 1711 | fp_tal ge start' next' (tal_base_not_return (RTLabs_status ge) start' next' ?? ((proj1 … (RTLabs_costed ge next')) … CS)) trace' TRACE_OK' |
---|
[1670] | 1712 | | false ⇒ λCS. |
---|
[2044] | 1713 | let fs ≝ finite_segment ge next' n' trace' ORACLE TRACE_OK' ENV_COSTLABELLED ENV_SOUNDLY_LABELLED ? in |
---|
| 1714 | fp_add_default ge start' next' CL fs ? CS |
---|
[1670] | 1715 | ] (refl ??) |
---|
| 1716 | | cl_jump ⇒ λCL. |
---|
[2044] | 1717 | fp_tal ge start' next' (tal_base_not_return (RTLabs_status ge) start' next' ?? ?) trace' ? |
---|
[1707] | 1718 | | cl_call ⇒ λCL. |
---|
[2044] | 1719 | match ORACLE ge O next trace' return λ_. finite_prefix ge start' with |
---|
[1671] | 1720 | [ or_introl TERMINATES ⇒ |
---|
| 1721 | match TERMINATES with [ witness TERMINATES ⇒ |
---|
[2044] | 1722 | let tlr ≝ make_label_return' ge O next' trace' ENV_COSTLABELLED ?? TERMINATES in |
---|
[1805] | 1723 | let TRACE_OK' ≝ ? in |
---|
[2044] | 1724 | match RTLabs_cost (new_state … tlr) return λx. RTLabs_cost (new_state … tlr) = x → finite_prefix ge start' with |
---|
| 1725 | [ true ⇒ λCS. fp_tal ge start' (new_state … tlr) (tal_base_call (RTLabs_status ge) start' next' (new_state … tlr) ? CL ? (new_trace … tlr) ((proj1 … (RTLabs_costed ge ?)) … CS)) (remainder … tlr) TRACE_OK' |
---|
[1707] | 1726 | | false ⇒ λCS. |
---|
[1812] | 1727 | let fs ≝ finite_segment ge (new_state … tlr) n' (remainder … tlr) ORACLE TRACE_OK' ENV_COSTLABELLED ENV_SOUNDLY_LABELLED ? in |
---|
[1707] | 1728 | fp_add_terminating_call … fs (new_trace … tlr) ? CS |
---|
[1671] | 1729 | ] (refl ??) |
---|
| 1730 | ] |
---|
| 1731 | | or_intror NO_TERMINATION ⇒ |
---|
[2044] | 1732 | fp_tac ge start' start' next' (tac_base (RTLabs_status ge) start' CL) ?? trace' ? |
---|
[1707] | 1733 | ] |
---|
[1670] | 1734 | | cl_return ⇒ λCL. ⊥ |
---|
| 1735 | ] (refl ??) |
---|
[2044] | 1736 | | ft_wrong start m NF EV ⇒ λmtc,TRACE_OK,LABEL_LIMIT. ⊥ |
---|
| 1737 | ] mtc0 |
---|
| 1738 | ] trace TRACE_OK |
---|
[1705] | 1739 | ] LABEL_LIMIT. |
---|
[1707] | 1740 | [ cases (state_bound_on_steps_to_cost_zero s) /2/ |
---|
[1805] | 1741 | | @(absurd … (ro_not_final … TRACE_OK) FINAL) |
---|
| 1742 | | @(absurd ?? (ro_no_termination … TRACE_OK)) |
---|
[1670] | 1743 | %{0} % @wr_base // |
---|
[2044] | 1744 | | @(proj1 … (RTLabs_costed …)) @(well_cost_labelled_jump … EV) [ @(ro_well_cost_labelled … TRACE_OK) | // ] |
---|
| 1745 | | 5,6,9,10,11: /3/ |
---|
| 1746 | | cases TRACE_OK #H1 #H2 #H3 #H4 #H5 |
---|
| 1747 | % [ @(well_cost_labelled_state_step … EV) // |
---|
| 1748 | | @(soundly_labelled_state_step … EV) // |
---|
[1805] | 1749 | | @(not_to_not … (ro_no_termination … TRACE_OK)) * #depth * #TM1 %{depth} % @wr_step /2/ |
---|
[2044] | 1750 | | @(still_not_wrong … EV) // |
---|
[1805] | 1751 | | @(not_return_to_not_final … EV) >CL % #E destruct |
---|
| 1752 | ] |
---|
[2044] | 1753 | | @(RTLabs_after_call ge start' next' … EV (stack_ok … tlr)) |
---|
| 1754 | | @(RTLabs_after_call ge start' next' … EV (stack_ok … tlr)) |
---|
| 1755 | | @(bound_after_call ge start' (new_state … tlr) ? LABEL_LIMIT CL ? CS) |
---|
| 1756 | @(RTLabs_after_call ge start' next' … EV (stack_ok … tlr)) |
---|
[1805] | 1757 | | % [ /2/ |
---|
[1806] | 1758 | | @(soundly_labelled_state_preserved … (stack_ok … tlr)) |
---|
[2044] | 1759 | @(soundly_labelled_state_step … EV) /2/ @(ro_soundly_labelled … TRACE_OK) |
---|
[1805] | 1760 | | @(not_to_not … (ro_no_termination … TRACE_OK)) * #depth * #TM1 %{depth} % |
---|
| 1761 | @wr_call // |
---|
| 1762 | @(will_return_prepend … TERMINATES … TM1) |
---|
| 1763 | cases (terminates … tlr) // |
---|
| 1764 | | @(will_return_not_wrong … TERMINATES) |
---|
[2044] | 1765 | [ @(still_not_wrong … EV) @(ro_not_undefined … TRACE_OK) |
---|
[1805] | 1766 | | cases (terminates … tlr) // |
---|
| 1767 | ] |
---|
| 1768 | | (* By stack preservation we cannot be in the final state *) |
---|
| 1769 | inversion (stack_ok … tlr) |
---|
| 1770 | [ #H101 #H102 #H103 #H104 #H105 #H106 #H107 #H108 #H109 destruct |
---|
| 1771 | | #s1 #f #f' #fs #m #N #F #S #E1 #E2 #E3 #E4 -TERMINATES destruct @refl |
---|
[2044] | 1772 | | #s1 #r #S #E1 #E2 #E3 #E4 -TERMINATES destruct whd in S:(??%); -next' -s0 |
---|
[1805] | 1773 | cases (rtlabs_call_inv … CL) #fd * #args * #dst * #stk * #m #E destruct |
---|
[2025] | 1774 | inversion (eval_preserves … EV) |
---|
[2044] | 1775 | [ 1,2,4,5,6: #H111 #H112 #H113 #H114 #H115 #H116 #H117 #H118 try #H119 try #H120 try #H121 try #H122 try #H123 @⊥ -next destruct ] |
---|
| 1776 | #ge' #fn #locals #nextx #nok #sp #fs #m' #args' #dst' #m'' #E1 #E2 #E3 #E4 -TRACE_OK destruct |
---|
[1805] | 1777 | inversion S |
---|
| 1778 | [ #f #fs0 #m #E1 #E2 #E3 destruct | *: #H123 #H124 #H125 #H126 #H127 #H128 #H129 try #H130 destruct ] |
---|
| 1779 | (* state_bound_on_steps_to_cost needs to know about the current stack frame, |
---|
| 1780 | so we can use it as a witness that at least one frame exists *) |
---|
| 1781 | inversion LABEL_LIMIT |
---|
| 1782 | #H141 #H142 #H143 #H144 #H145 #H146 #H147 #H148 try #H150 destruct |
---|
| 1783 | | #H173 #H174 #H175 #H176 #H177 #H178 #H179 #H180 #H181 destruct |
---|
| 1784 | ] |
---|
| 1785 | ] |
---|
[2044] | 1786 | | @(well_cost_labelled_state_step … EV) /2/ @(ro_well_cost_labelled … TRACE_OK) |
---|
| 1787 | | @(well_cost_labelled_call … EV) [ @(ro_well_cost_labelled … TRACE_OK) | // ] |
---|
[1806] | 1788 | | /2/ |
---|
[2044] | 1789 | | %{tr} %{EV} % |
---|
| 1790 | | cases TRACE_OK #H1 #H2 #H3 #H4 #H5 |
---|
| 1791 | % [ @(well_cost_labelled_state_step … EV) /2/ |
---|
[1806] | 1792 | | @(soundly_labelled_state_step … EV) /2/ |
---|
| 1793 | | @(not_to_not … NO_TERMINATION) * #depth * #TM % |
---|
| 1794 | @(will_return_lower … TM) // |
---|
| 1795 | | @(still_not_wrong … EV) /2/ |
---|
| 1796 | | @(not_return_to_not_final … EV) >CL % #E destruct |
---|
| 1797 | ] |
---|
[2044] | 1798 | | %2 @CL |
---|
| 1799 | | 21,22: %{tr} %{EV} % |
---|
[1707] | 1800 | | cases (bound_after_step … LABEL_LIMIT EV ?) |
---|
[1805] | 1801 | [ * [ #TERMINATES @⊥ @(absurd ?? (ro_no_termination … TRACE_OK)) %{0} % @wr_step [ %1 // | |
---|
[1707] | 1802 | inversion trace' |
---|
[1805] | 1803 | [ #s0 #FINAL #E1 #E2 -TRACE_OK' destruct @⊥ |
---|
[1765] | 1804 | @(absurd ?? FINAL) @(not_return_to_not_final … EV) |
---|
| 1805 | % >CL #E destruct |
---|
[1805] | 1806 | | #s1 #tr1 #s2 #EVAL' #trace'' #E1 #E2 -TRACE_OK' destruct |
---|
[1765] | 1807 | @wr_base // |
---|
[1805] | 1808 | | #H99 #H100 #H101 #H102 #H103 -TRACE_OK' destruct |
---|
| 1809 | inversion (ro_not_undefined … TRACE_OK) |
---|
[1707] | 1810 | [ #H137 #H138 #H139 #H140 #H141 destruct |
---|
[1880] | 1811 | | #H143 #H144 #H145 #H146 #H147 #H148 #H149 #H150 #H151 #H152 destruct |
---|
[1707] | 1812 | inversion H148 |
---|
| 1813 | [ #H153 #H154 #H155 #H156 #H157 destruct |
---|
| 1814 | | #H159 #H160 #H161 #H162 #H163 #H164 #H165 #H166 #H167 destruct |
---|
| 1815 | ] |
---|
| 1816 | ] |
---|
| 1817 | ] |
---|
| 1818 | ] |
---|
| 1819 | | >CL #E destruct |
---|
| 1820 | ] |
---|
| 1821 | | // |
---|
| 1822 | | // |
---|
| 1823 | ] |
---|
[2044] | 1824 | | cases TRACE_OK #H1 #H2 #H3 #H4 #H5 |
---|
| 1825 | % [ @(well_cost_labelled_state_step … EV) // |
---|
| 1826 | | @(soundly_labelled_state_step … EV) // |
---|
[1805] | 1827 | | @(not_to_not … (ro_no_termination … TRACE_OK)) |
---|
| 1828 | * #depth * #TERM %{depth} % @wr_step /2/ |
---|
| 1829 | | @(still_not_wrong … (ro_not_undefined … TRACE_OK)) |
---|
| 1830 | | @(not_return_to_not_final … EV) >CL % #E destruct |
---|
| 1831 | ] |
---|
| 1832 | | inversion (ro_not_undefined … TRACE_OK) |
---|
[1707] | 1833 | [ #H169 #H170 #H171 #H172 #H173 destruct |
---|
| 1834 | | #H175 #H176 #H177 #H178 #H179 #H180 #H181 #H182 #H183 destruct |
---|
| 1835 | ] |
---|
[1765] | 1836 | ] qed. |
---|
[1670] | 1837 | |
---|
[1808] | 1838 | (* NB: This isn't quite what I'd like. Ideally, we'd show the existence of |
---|
| 1839 | a trace_label_diverges value, but I only know how to construct those |
---|
| 1840 | using a cofixpoint in Type[0], which means I can't use the termination |
---|
| 1841 | oracle. |
---|
[1806] | 1842 | *) |
---|
[1784] | 1843 | |
---|
[2044] | 1844 | let corec make_label_diverges ge (s:RTLabs_state ge) |
---|
[1651] | 1845 | (trace: flat_trace io_out io_in ge s) |
---|
[1784] | 1846 | (ORACLE: termination_oracle) |
---|
[1805] | 1847 | (TRACE_OK: remainder_ok ge s trace) |
---|
[1651] | 1848 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
[1784] | 1849 | (ENV_SOUNDLY_LABELLED: soundly_labelled_ge ge) |
---|
[1651] | 1850 | (STATEMENT_COSTLABEL: RTLabs_cost s = true) (* current statement is a cost label *) |
---|
[1808] | 1851 | : trace_label_diverges_exists (RTLabs_status ge) s ≝ |
---|
[1812] | 1852 | match steps_from_sound s STATEMENT_COSTLABEL (ro_soundly_labelled … TRACE_OK) with |
---|
[1784] | 1853 | [ ex_intro n B ⇒ |
---|
[1812] | 1854 | match finite_segment ge s n trace ORACLE TRACE_OK ENV_COSTLABELLED ENV_SOUNDLY_LABELLED B |
---|
[2044] | 1855 | return λs:RTLabs_state ge.λ_. RTLabs_cost s = true → trace_label_diverges_exists (RTLabs_status ge) s |
---|
[1784] | 1856 | with |
---|
[1805] | 1857 | [ fp_tal s s' T t tOK ⇒ λSTATEMENT_COSTLABEL. |
---|
[1812] | 1858 | let T' ≝ make_label_diverges ge s' t ORACLE tOK ENV_COSTLABELLED ENV_SOUNDLY_LABELLED ? in |
---|
[2044] | 1859 | tld_step' (RTLabs_status ge) s s' (tll_base … T ((proj1 … (RTLabs_costed …)) … STATEMENT_COSTLABEL)) T' |
---|
[1808] | 1860 | (* |
---|
[1812] | 1861 | match make_label_diverges ge s' t ORACLE tOK ENV_COSTLABELLED ENV_SOUNDLY_LABELLED ? with |
---|
[1784] | 1862 | [ ex_intro T' _ ⇒ |
---|
| 1863 | ex_intro ?? (tld_step (RTLabs_status ge) s s' (tll_base … T STATEMENT_COSTLABEL) T') I |
---|
[1808] | 1864 | ]*) |
---|
[2044] | 1865 | | fp_tac s s2 s3 T WCL2 EV t tOK ⇒ λSTATEMENT_COSTLABEL. |
---|
[1812] | 1866 | let T' ≝ make_label_diverges ge s3 t ORACLE tOK ENV_COSTLABELLED ENV_SOUNDLY_LABELLED ? in |
---|
[2044] | 1867 | tld_base' (RTLabs_status ge) s s2 s3 (tlc_base … T ((proj1 … (RTLabs_costed …)) … STATEMENT_COSTLABEL)) ?? T' |
---|
[1808] | 1868 | (* |
---|
[1812] | 1869 | match make_label_diverges ge s3 t ORACLE tOK ENV_COSTLABELLED ENV_SOUNDLY_LABELLED ? with |
---|
[1806] | 1870 | [ ex_intro T' _ ⇒ |
---|
| 1871 | ex_intro ?? (tld_base (RTLabs_status ge) s s2 s3 (tlc_base … T STATEMENT_COSTLABEL) ?? T') ? |
---|
[1808] | 1872 | ]*) |
---|
[1784] | 1873 | ] STATEMENT_COSTLABEL |
---|
| 1874 | ]. |
---|
[2044] | 1875 | [ @((proj2 … (RTLabs_costed …))) @(trace_any_label_label … T) |
---|
| 1876 | | @EV |
---|
[1806] | 1877 | | @(trace_any_call_call … T) |
---|
[2044] | 1878 | | cases EV #tr * #EV' #N @(well_cost_labelled_call … EV') // @(trace_any_call_call … T) |
---|
[1812] | 1879 | ] qed. |
---|
| 1880 | |
---|
[1880] | 1881 | lemma after_the_initial_call_is_the_final_state : ∀ge,p,s1,tr,s2,s'. |
---|
| 1882 | make_initial_state p = OK ? s1 → |
---|
| 1883 | eval_statement ge s1 = Value ??? 〈tr,s2〉 → |
---|
| 1884 | stack_preserved ends_with_ret s2 s' → |
---|
| 1885 | RTLabs_is_final s' ≠ None ?. |
---|
| 1886 | #ge #p #s1 #tr #s2 #s' whd in ⊢ (??%? → ?); |
---|
| 1887 | @bind_ok #m #_ |
---|
| 1888 | @bind_ok #b #_ |
---|
| 1889 | @bind_ok #f #_ |
---|
| 1890 | #E destruct |
---|
[2025] | 1891 | #EV #SP inversion (eval_preserves … EV) |
---|
[1880] | 1892 | [ 3: #ge' #fn #locals #next #nok #sp #fs #m1 #args #dst #m2 #E1 #E2 #E3 #_ destruct |
---|
| 1893 | inversion SP |
---|
| 1894 | [ 3: #s1 #r #S #_ #_ #_ #_ % #E whd in E:(??%?); destruct |
---|
| 1895 | | *: #H28 #H29 #H30 #H31 #H32 #H33 #H34 #H35 #H36 try #H38 try #H39 try #H40 destruct |
---|
| 1896 | inversion H35 #H61 #H62 #H63 #H64 #H65 #H66 try #H68 try #H69 try #H70 destruct |
---|
| 1897 | ] |
---|
| 1898 | | *: #H98 #H99 #H100 #H101 #H102 #H103 #H104 #H105 try #H106 try #H107 try #H108 try #H109 try #H110 destruct |
---|
| 1899 | ] qed. |
---|
| 1900 | |
---|
| 1901 | lemma initial_state_is_call : ∀p,s. |
---|
| 1902 | make_initial_state p = OK ? s → |
---|
| 1903 | RTLabs_classify s = cl_call. |
---|
| 1904 | #p #s whd in ⊢ (??%? → ?); |
---|
| 1905 | @bind_ok #m #_ |
---|
| 1906 | @bind_ok #b #_ |
---|
| 1907 | @bind_ok #f #_ |
---|
| 1908 | #E destruct |
---|
| 1909 | @refl |
---|
| 1910 | qed. |
---|
| 1911 | |
---|
[2044] | 1912 | let rec whole_structured_trace_exists ge p (s:RTLabs_state ge) |
---|
[1812] | 1913 | (ORACLE: termination_oracle) |
---|
| 1914 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
| 1915 | (ENV_SOUNDLY_LABELLED: soundly_labelled_ge ge) |
---|
[2044] | 1916 | : ∀trace: flat_trace io_out io_in ge s. |
---|
| 1917 | ∀INITIAL: make_initial_state p = OK state s. |
---|
[1880] | 1918 | ∀NOT_WRONG: not_wrong ??? s trace. |
---|
[1812] | 1919 | ∀STATE_COSTLABELLED: well_cost_labelled_state s. |
---|
| 1920 | ∀STATE_SOUNDLY_LABELLED: soundly_labelled_state s. |
---|
[2044] | 1921 | trace_whole_program_exists (RTLabs_status ge) s ≝ |
---|
| 1922 | match s return λs:RTLabs_state ge. ∀trace:flat_trace io_out io_in ge s. |
---|
| 1923 | make_initial_state p = OK ? s → |
---|
| 1924 | not_wrong ??? s trace → |
---|
| 1925 | well_cost_labelled_state s → |
---|
| 1926 | soundly_labelled_state s → |
---|
| 1927 | trace_whole_program_exists (RTLabs_status ge) s with |
---|
| 1928 | [ mk_RTLabs_state s0 stk mtc0 ⇒ λtrace. |
---|
| 1929 | match trace return λs,trace. ∀mtc:Ras_Fn_Match ge s stk. |
---|
| 1930 | make_initial_state p = OK state s → |
---|
[1880] | 1931 | not_wrong ??? s trace → |
---|
[1812] | 1932 | well_cost_labelled_state s → |
---|
| 1933 | soundly_labelled_state s → |
---|
[2044] | 1934 | trace_whole_program_exists (RTLabs_status ge) (mk_RTLabs_state ge s stk mtc) with |
---|
| 1935 | [ ft_step s tr next EV trace' ⇒ λmtc,INITIAL,NOT_WRONG,STATE_COSTLABELLED,STATE_SOUNDLY_LABELLED. |
---|
[1880] | 1936 | let IS_CALL ≝ initial_state_is_call … INITIAL in |
---|
[2044] | 1937 | let s' ≝ mk_RTLabs_state ge s stk mtc in |
---|
| 1938 | let next' ≝ next_state ge s' next tr EV in |
---|
[1812] | 1939 | match ORACLE ge O next trace' with |
---|
| 1940 | [ or_introl TERMINATES ⇒ |
---|
| 1941 | match TERMINATES with |
---|
| 1942 | [ witness TERMINATES ⇒ |
---|
[2044] | 1943 | let tlr ≝ make_label_return' ge O next' trace' ENV_COSTLABELLED ?? TERMINATES in |
---|
| 1944 | twp_terminating (RTLabs_status ge) s' next' (new_state … tlr) IS_CALL ? (new_trace … tlr) ? |
---|
[1812] | 1945 | ] |
---|
[2044] | 1946 | | or_intror NO_TERMINATION ⇒ |
---|
| 1947 | twp_diverges (RTLabs_status ge) s' next' IS_CALL ? |
---|
| 1948 | (make_label_diverges ge next' trace' ORACLE ? |
---|
[1812] | 1949 | ENV_COSTLABELLED ENV_SOUNDLY_LABELLED ?) |
---|
| 1950 | ] |
---|
[2044] | 1951 | | ft_stop st FINAL ⇒ λmtc,INITIAL,NOT_WRONG. ⊥ |
---|
| 1952 | | ft_wrong start m NF EV ⇒ λmtc,INITIAL,NOT_WRONG. ⊥ |
---|
| 1953 | ] mtc0 ]. |
---|
[1880] | 1954 | [ cases (rtlabs_call_inv … (initial_state_is_call … INITIAL)) #fn * #args * #dst * #stk * #m #E destruct |
---|
[1812] | 1955 | cases FINAL #E @E @refl |
---|
[2044] | 1956 | | %{tr} %{EV} % |
---|
[1880] | 1957 | | @(after_the_initial_call_is_the_final_state … INITIAL EV) |
---|
| 1958 | @(stack_ok … tlr) |
---|
[1812] | 1959 | | @(well_cost_labelled_state_step … EV) // |
---|
| 1960 | | @(well_cost_labelled_call … EV) // |
---|
[2044] | 1961 | | %{tr} %{EV} % |
---|
[1812] | 1962 | | @(well_cost_labelled_call … EV) // |
---|
| 1963 | | % [ @(well_cost_labelled_state_step … EV) // |
---|
| 1964 | | @(soundly_labelled_state_step … EV) // |
---|
| 1965 | | @(not_to_not … NO_TERMINATION) * #d * #TM % /2/ |
---|
| 1966 | | @(still_not_wrong … NOT_WRONG) |
---|
| 1967 | | @(not_return_to_not_final … EV) >IS_CALL % #E destruct |
---|
| 1968 | ] |
---|
| 1969 | | inversion NOT_WRONG #H29 #H30 #H31 #H32 #H33 try #H35 try #H36 try #H37 destruct |
---|
| 1970 | ] qed. |
---|
| 1971 | |
---|
| 1972 | (* |
---|
| 1973 | theorem program_trace_exists : |
---|
| 1974 | termination_oracle → |
---|
| 1975 | ∀p:RTLabs_program. |
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| 1976 | ∀s:state. |
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| 1977 | ∀I: make_initial_state p = OK ? s. |
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| 1978 | |
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| 1979 | let plain_trace ≝ exec_inf io_out io_in RTLabs_fullexec p in |
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| 1980 | |
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| 1981 | ∀NOIO:exec_no_io … plain_trace. |
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| 1982 | |
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| 1983 | let flat_trace ≝ make_whole_flat_trace p s NOIO I in |
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| 1984 | |
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| 1985 | trace_whole_program_exists (RTLabs_status (make_global p)) s. |
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| 1986 | #ORACLE #p #s #I |
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| 1987 | letin plain_trace ≝ (exec_inf io_out io_in RTLabs_fullexec p) |
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| 1988 | #NOIO |
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| 1989 | letin flat_trace ≝ (make_whole_flat_trace p s NOIO I) |
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| 1990 | whd |
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| 1991 | @(whole_structured_trace_exists … flat_trace) |
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| 1992 | // |
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| 1993 | [ whd |
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[1880] | 1994 | *) |
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| 1995 | |
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[2044] | 1996 | lemma simplify_exec : ∀ge.∀s,s':RTLabs_state ge. |
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| 1997 | as_execute (RTLabs_status ge) s s' → |
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| 1998 | ∃tr. eval_statement ge s = Value … 〈tr,s'〉. |
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| 1999 | #ge #s #s' * #tr * #EX #_ %{tr} @EX |
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| 2000 | qed. |
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| 2001 | |
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[1880] | 2002 | (* as_execute might be in Prop, but because the semantics is deterministic |
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| 2003 | we can retrieve the event trace anyway. *) |
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[2044] | 2004 | |
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| 2005 | let rec deprop_execute ge (s,s':state) |
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| 2006 | (X:∃t. eval_statement ge s = Value … 〈t,s'〉) |
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[1880] | 2007 | : Σtr. eval_statement ge s = Value … 〈tr,s'〉 ≝ |
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[2044] | 2008 | match eval_statement ge s return λE. (∃t.E = ?) → Σt.E = Value … 〈t,s'〉 with |
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[1880] | 2009 | [ Value ts ⇒ λY. «fst … ts, ?» |
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| 2010 | | _ ⇒ λY. ⊥ |
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| 2011 | ] X. |
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| 2012 | [ 1,3: cases Y #x #E destruct |
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| 2013 | | cases Y #trP #E destruct @refl |
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| 2014 | ] qed. |
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| 2015 | |
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[2044] | 2016 | let rec deprop_as_execute ge (s,s':RTLabs_state ge) |
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| 2017 | (X:as_execute (RTLabs_status ge) s s') |
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| 2018 | : Σtr. eval_statement ge s = Value … 〈tr,s'〉 ≝ |
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| 2019 | deprop_execute ge s s' ?. |
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| 2020 | cases X #tr * #EX #_ %{tr} @EX |
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| 2021 | qed. |
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| 2022 | |
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[1880] | 2023 | (* A non-empty finite section of a flat_trace *) |
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| 2024 | inductive partial_flat_trace (o:Type[0]) (i:o → Type[0]) (ge:genv) : state → state → Type[0] ≝ |
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| 2025 | | pft_base : ∀s,tr,s'. eval_statement ge s = Value ??? 〈tr,s'〉 → partial_flat_trace o i ge s s' |
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| 2026 | | pft_step : ∀s,tr,s',s''. eval_statement ge s = Value ??? 〈tr,s'〉 → partial_flat_trace o i ge s' s'' → partial_flat_trace o i ge s s''. |
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| 2027 | |
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| 2028 | let rec append_partial_flat_trace o i ge s1 s2 s3 |
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| 2029 | (tr1:partial_flat_trace o i ge s1 s2) |
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| 2030 | on tr1 : partial_flat_trace o i ge s2 s3 → partial_flat_trace o i ge s1 s3 ≝ |
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| 2031 | match tr1 with |
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| 2032 | [ pft_base s tr s' EX ⇒ pft_step … s tr s' s3 EX |
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| 2033 | | pft_step s tr s' s'' EX tr' ⇒ λtr2. pft_step … s tr s' s3 EX (append_partial_flat_trace … tr' tr2) |
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| 2034 | ]. |
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| 2035 | |
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| 2036 | let rec partial_to_flat_trace o i ge s1 s2 |
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| 2037 | (tr:partial_flat_trace o i ge s1 s2) |
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| 2038 | on tr : flat_trace o i ge s2 → flat_trace o i ge s1 ≝ |
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| 2039 | match tr with |
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| 2040 | [ pft_base s tr s' EX ⇒ ft_step … EX |
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| 2041 | | pft_step s tr s' s'' EX tr' ⇒ λtr''. ft_step … EX (partial_to_flat_trace … tr' tr'') |
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| 2042 | ]. |
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| 2043 | |
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| 2044 | (* Extract a flat trace from a structured one. *) |
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[2044] | 2045 | let rec flat_trace_of_label_return ge (s,s':RTLabs_state ge) |
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[1880] | 2046 | (tr:trace_label_return (RTLabs_status ge) s s') |
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| 2047 | on tr : |
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| 2048 | partial_flat_trace io_out io_in ge s s' ≝ |
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| 2049 | match tr with |
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[1960] | 2050 | [ tlr_base s1 s2 tll ⇒ flat_trace_of_label_label ge ends_with_ret s1 s2 tll |
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[1880] | 2051 | | tlr_step s1 s2 s3 tll tlr ⇒ |
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| 2052 | append_partial_flat_trace … |
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[1960] | 2053 | (flat_trace_of_label_label ge doesnt_end_with_ret s1 s2 tll) |
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| 2054 | (flat_trace_of_label_return ge s2 s3 tlr) |
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[1880] | 2055 | ] |
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[2044] | 2056 | and flat_trace_of_label_label ge ends (s,s':RTLabs_state ge) |
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[1880] | 2057 | (tr:trace_label_label (RTLabs_status ge) ends s s') |
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| 2058 | on tr : |
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| 2059 | partial_flat_trace io_out io_in ge s s' ≝ |
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| 2060 | match tr with |
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[1960] | 2061 | [ tll_base e s1 s2 tal _ ⇒ flat_trace_of_any_label ge e s1 s2 tal |
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[1880] | 2062 | ] |
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[2044] | 2063 | and flat_trace_of_any_label ge ends (s,s':RTLabs_state ge) |
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[1880] | 2064 | (tr:trace_any_label (RTLabs_status ge) ends s s') |
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| 2065 | on tr : |
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| 2066 | partial_flat_trace io_out io_in ge s s' ≝ |
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| 2067 | match tr with |
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| 2068 | [ tal_base_not_return s1 s2 EX CL CS ⇒ |
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| 2069 | match deprop_as_execute ge ?? EX with [ mk_Sig tr EX' ⇒ |
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| 2070 | pft_base … EX' ] |
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| 2071 | | tal_base_return s1 s2 EX CL ⇒ |
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| 2072 | match deprop_as_execute ge ?? EX with [ mk_Sig tr EX' ⇒ |
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| 2073 | pft_base … EX' ] |
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| 2074 | | tal_base_call s1 s2 s3 EX CL AR tlr CS ⇒ |
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[1960] | 2075 | let suffix' ≝ flat_trace_of_label_return ge ?? tlr in |
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[1880] | 2076 | match deprop_as_execute ge ?? EX with [ mk_Sig tr EX' ⇒ |
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| 2077 | pft_step … EX' suffix' ] |
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| 2078 | | tal_step_call ends s1 s2 s3 s4 EX CL AR tlr CS tal ⇒ |
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| 2079 | match deprop_as_execute ge ?? EX with [ mk_Sig tr EX' ⇒ |
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| 2080 | pft_step … EX' |
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| 2081 | (append_partial_flat_trace … |
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[1960] | 2082 | (flat_trace_of_label_return ge ?? tlr) |
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| 2083 | (flat_trace_of_any_label ge ??? tal)) |
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[1880] | 2084 | ] |
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| 2085 | | tal_step_default ends s1 s2 s3 EX tal CL CS ⇒ |
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| 2086 | match deprop_as_execute ge ?? EX with [ mk_Sig tr EX' ⇒ |
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[1960] | 2087 | pft_step … EX' (flat_trace_of_any_label ge ??? tal) |
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[1880] | 2088 | ] |
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| 2089 | ]. |
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| 2090 | |
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| 2091 | |
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| 2092 | (* We take an extra step so that we can always return a non-empty trace to |
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| 2093 | satisfy the guardedness condition in the cofixpoint. *) |
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[2044] | 2094 | let rec flat_trace_of_any_call ge (s,s',s'':RTLabs_state ge) et |
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[1880] | 2095 | (tr:trace_any_call (RTLabs_status ge) s s') |
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| 2096 | (EX'':eval_statement ge s' = Value … 〈et,s''〉) |
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| 2097 | on tr : |
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[2044] | 2098 | partial_flat_trace io_out io_in ge s s'' ≝ |
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| 2099 | match tr return λs,s':RTLabs_state ge.λ_. eval_statement ge s' = ? → partial_flat_trace io_out io_in ge s s'' with |
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[1880] | 2100 | [ tac_base s1 CL ⇒ λEX''. pft_base … ge ??? EX'' |
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| 2101 | | tac_step_call s1 s2 s3 s4 EX CL AR tlr CS tac ⇒ λEX''. |
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| 2102 | match deprop_as_execute ge ?? EX with [ mk_Sig et EX' ⇒ |
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| 2103 | pft_step … EX' |
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| 2104 | (append_partial_flat_trace … |
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[1960] | 2105 | (flat_trace_of_label_return ge ?? tlr) |
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| 2106 | (flat_trace_of_any_call ge … tac EX'')) |
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[1880] | 2107 | ] |
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| 2108 | | tac_step_default s1 s2 s3 EX tal CL CS ⇒ λEX''. |
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| 2109 | match deprop_as_execute ge ?? EX with [ mk_Sig tr EX' ⇒ |
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| 2110 | pft_step … EX' |
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[1960] | 2111 | (flat_trace_of_any_call ge … tal EX'') |
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[1880] | 2112 | ] |
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| 2113 | ] EX''. |
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| 2114 | |
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[2044] | 2115 | let rec flat_trace_of_label_call ge (s,s',s'':RTLabs_state ge) et |
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[1880] | 2116 | (tr:trace_label_call (RTLabs_status ge) s s') |
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| 2117 | (EX'':eval_statement ge s' = Value … 〈et,s''〉) |
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| 2118 | on tr : |
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| 2119 | partial_flat_trace io_out io_in ge s s'' ≝ |
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| 2120 | match tr with |
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[1960] | 2121 | [ tlc_base s1 s2 tac CS ⇒ flat_trace_of_any_call … tac |
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[1880] | 2122 | ] EX''. |
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| 2123 | |
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| 2124 | (* Now reconstruct the flat_trace of a diverging execution. Note that we need |
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| 2125 | to take care to satisfy the guardedness condition by witnessing the fact that |
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| 2126 | the partial traces are non-empty. *) |
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[2044] | 2127 | let corec flat_trace_of_label_diverges ge (s:RTLabs_state ge) |
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[1880] | 2128 | (tr:trace_label_diverges (RTLabs_status ge) s) |
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| 2129 | : flat_trace io_out io_in ge s ≝ |
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[2044] | 2130 | match tr return λs:RTLabs_state ge.λtr:trace_label_diverges (RTLabs_status ge) s. flat_trace io_out io_in ge s with |
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[1880] | 2131 | [ tld_step sx sy tll tld ⇒ |
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[2044] | 2132 | match sy in RTLabs_state return λsy:RTLabs_state ge. trace_label_label (RTLabs_status ge) ? sx sy → trace_label_diverges (RTLabs_status ge) sy → flat_trace io_out io_in ge ? with [ mk_RTLabs_state sy' stk mtc0 ⇒ |
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| 2133 | λtll. |
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| 2134 | match flat_trace_of_label_label ge … tll return λs1,s2:state.λ_:partial_flat_trace io_out io_in ge s1 s2. ∀mtc:Ras_Fn_Match ge s2 stk. trace_label_diverges (RTLabs_status ge) (mk_RTLabs_state ge s2 stk mtc) → flat_trace ??? s1 with |
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| 2135 | [ pft_base s1 tr s2 EX ⇒ λmtc,tld. ft_step … EX (flat_trace_of_label_diverges ge ? tld) |
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| 2136 | | pft_step s1 et s2 s3 EX tr' ⇒ λmtc,tld. ft_step … EX (add_partial_flat_trace ge … (mk_RTLabs_state ge s3 stk mtc) tr' tld) |
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| 2137 | ] mtc0 ] tll tld |
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| 2138 | | tld_base s1 s2 s3 tlc EX CL tld ⇒ |
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| 2139 | match s3 in RTLabs_state return λs3:RTLabs_state ge. as_execute (RTLabs_status ge) ? s3 → trace_label_diverges (RTLabs_status ge) s3 → flat_trace io_out io_in ge ? with [ mk_RTLabs_state s3' stk mtc0 ⇒ |
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| 2140 | λEX. match deprop_as_execute ge ?? EX with [ mk_Sig tr EX' ⇒ |
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| 2141 | match flat_trace_of_label_call … tlc EX' return λs1,s3.λ_. ∀mtc:Ras_Fn_Match ge s3 stk. trace_label_diverges (RTLabs_status ge) (mk_RTLabs_state ge s3 stk mtc) → flat_trace ??? s1 with |
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| 2142 | [ pft_base s1 tr s2 EX ⇒ λmtc,tld. ft_step … EX (flat_trace_of_label_diverges ge ? tld) |
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| 2143 | | pft_step s1 et s2 s3 EX tr' ⇒ λmtc,tld. ft_step … EX (add_partial_flat_trace ge … (mk_RTLabs_state ge s3 stk mtc) tr' tld) |
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| 2144 | ] mtc0 |
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[1880] | 2145 | ] |
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[2044] | 2146 | ] EX tld |
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[1880] | 2147 | ] |
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| 2148 | (* Helper to keep adding the partial trace without violating the guardedness |
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| 2149 | condition. *) |
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[2044] | 2150 | and add_partial_flat_trace ge (s:state) (s':RTLabs_state ge) |
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| 2151 | : partial_flat_trace io_out io_in ge s s' → |
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| 2152 | trace_label_diverges (RTLabs_status ge) s' → |
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| 2153 | flat_trace io_out io_in ge s ≝ |
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| 2154 | match s' return λs':RTLabs_state ge. partial_flat_trace io_out io_in ge s s' → trace_label_diverges (RTLabs_status ge) s' → flat_trace io_out io_in ge s with [ mk_RTLabs_state sx stk mtc ⇒ |
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| 2155 | λptr. match ptr return λs,s'.λ_. ∀mtc:Ras_Fn_Match ge s' stk. trace_label_diverges (RTLabs_status ge) (mk_RTLabs_state ge s' ? mtc) → flat_trace io_out io_in ge s with |
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| 2156 | [ pft_base s tr s' EX ⇒ λmtc,tr. ft_step … EX (flat_trace_of_label_diverges ge ? tr) |
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| 2157 | | pft_step s1 et s2 s3 EX tr' ⇒ λmtc,tr. ft_step … EX (add_partial_flat_trace ge s2 (mk_RTLabs_state ge s3 stk mtc) tr' tr) |
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| 2158 | ] mtc ] |
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| 2159 | . |
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[1880] | 2160 | |
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[2044] | 2161 | |
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[1880] | 2162 | coinductive equal_flat_traces (ge:genv) : ∀s. flat_trace io_out io_in ge s → flat_trace io_out io_in ge s → Prop ≝ |
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| 2163 | | eft_stop : ∀s,F. equal_flat_traces ge s (ft_stop … F) (ft_stop … F) |
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| 2164 | | eft_step : ∀s,tr,s',EX,tr1,tr2. equal_flat_traces ge s' tr1 tr2 → equal_flat_traces ge s (ft_step … EX tr1) (ft_step … s tr s' EX tr2) |
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| 2165 | | eft_wrong : ∀s,m,NF,EX. equal_flat_traces ge s (ft_wrong … s m NF EX) (ft_wrong … s m NF EX). |
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| 2166 | |
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| 2167 | (* XXX move to semantics *) |
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| 2168 | lemma final_cannot_move : ∀ge,s. |
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| 2169 | RTLabs_is_final s ≠ None ? → |
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| 2170 | ∃err. eval_statement ge s = Wrong ??? err. |
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| 2171 | #ge * |
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| 2172 | [ #f #fs #m * #F cases (F ?) @refl |
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| 2173 | | #a #b #c #d #e * #F cases (F ?) @refl |
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| 2174 | | #a #b #c #d * #F cases (F ?) @refl |
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| 2175 | | #r #F % [2: @refl | skip ] |
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| 2176 | ] qed. |
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| 2177 | |
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| 2178 | let corec flat_traces_are_determined_by_starting_point ge s tr1 |
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| 2179 | : ∀tr2. equal_flat_traces ge s tr1 tr2 ≝ |
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| 2180 | match tr1 return λs,tr1. flat_trace ??? s → equal_flat_traces ? s tr1 ? with |
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| 2181 | [ ft_stop s F ⇒ λtr2. ? |
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| 2182 | | ft_step s1 tr s2 EX0 tr1' ⇒ λtr2. |
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| 2183 | match tr2 return λs,tr2. ∀EX:eval_statement ge s = ?. equal_flat_traces ? s (ft_step ??? s ?? EX ?) tr2 with |
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| 2184 | [ ft_stop s F ⇒ λEX. ? |
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| 2185 | | ft_step s tr' s2' EX' tr2' ⇒ λEX. ? |
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| 2186 | | ft_wrong s m NF EX' ⇒ λEX. ? |
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| 2187 | ] EX0 |
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| 2188 | | ft_wrong s m NF EX ⇒ λtr2. ? |
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| 2189 | ]. |
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| 2190 | [ inversion tr2 in tr1 F; |
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| 2191 | [ #s #F #_ #_ #tr1 #F' @eft_stop |
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| 2192 | | #s1 #tr #s2 #EX #tr' #E #_ #tr'' #F' @⊥ destruct |
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| 2193 | cases (final_cannot_move ge … F') #err #Er >Er in EX; #E destruct |
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| 2194 | | #s #m #NF #EX #_ #_ #_ #F @⊥ >NF in F; * /2/ |
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| 2195 | ] |
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| 2196 | | @⊥ cases (final_cannot_move ge … F) #err #Er >Er in EX; #E destruct |
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| 2197 | | -EX0 |
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| 2198 | cut (s2 = s2'). >EX in EX'; #E destruct @refl. #E (* Can't use destruct due to cofixpoint guardedness check *) |
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| 2199 | @(match E return λs2',E. ∀tr2':flat_trace ??? s2'. ∀EX':? = Value ??? 〈?,s2'〉. equal_flat_traces ??? (ft_step ????? s2' EX' tr2') with [ refl ⇒ ? ] tr2' EX') |
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| 2200 | -E -EX' -tr2' #tr2' #EX' |
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| 2201 | cut (tr = tr'). >EX in EX'; #E destruct @refl. #E (* Can't use destruct due to cofixpoint guardedness check *) |
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| 2202 | @(match E return λtr',E. ∀EX':? = Value ??? 〈tr',?〉. equal_flat_traces ??? (ft_step ???? tr' ? EX' ?) with [ refl ⇒ ? ] EX') |
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| 2203 | -E -EX' #EX' |
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| 2204 | @eft_step @flat_traces_are_determined_by_starting_point |
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| 2205 | | @⊥ >EX in EX'; #E destruct |
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| 2206 | | inversion tr2 in NF EX; |
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| 2207 | [ #s #F #_ #_ #NF @⊥ >NF in F; * /2/ |
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| 2208 | | #s1 #tr #s2 #EX #tr1 #E1 #_ #NF #EX' @⊥ >EX in EX'; #E destruct |
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| 2209 | | #sx #m' #NF #EX #_ #_ #NF' #EX' cut (m=m'). >EX in EX'; #E destruct @refl. |
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| 2210 | #E destruct |
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| 2211 | @eft_wrong |
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| 2212 | ] |
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| 2213 | ] qed. |
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| 2214 | |
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[2044] | 2215 | let corec diverging_traces_have_unique_flat_trace ge (s:RTLabs_state ge) |
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[1880] | 2216 | (str:trace_label_diverges (RTLabs_status ge) s) |
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| 2217 | (tr:flat_trace io_out io_in ge s) |
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[1960] | 2218 | : equal_flat_traces … (flat_trace_of_label_diverges … str) tr ≝ ?. |
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[1880] | 2219 | @flat_traces_are_determined_by_starting_point |
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| 2220 | qed. |
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| 2221 | |
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[2044] | 2222 | let rec flat_trace_of_whole_program ge (s:RTLabs_state ge) |
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[1880] | 2223 | (tr:trace_whole_program (RTLabs_status ge) s) |
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| 2224 | on tr : flat_trace io_out io_in ge s ≝ |
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[2044] | 2225 | match tr return λs:RTLabs_state ge.λtr. flat_trace io_out io_in ge s with |
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[1880] | 2226 | [ twp_terminating s1 s2 sf CL EX tlr F ⇒ |
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[2044] | 2227 | match deprop_as_execute ge ?? EX with [ mk_Sig tr EX' ⇒ |
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| 2228 | ft_step … EX' (partial_to_flat_trace … (flat_trace_of_label_return … tlr) (ft_stop … F)) |
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[1880] | 2229 | ] |
---|
| 2230 | | twp_diverges s1 s2 CL EX tld ⇒ |
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| 2231 | match deprop_as_execute ge ?? EX with [ mk_Sig tr EX' ⇒ |
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[1960] | 2232 | ft_step … EX' (flat_trace_of_label_diverges … tld) |
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[1880] | 2233 | ] |
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| 2234 | ]. |
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| 2235 | |
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[2044] | 2236 | let corec whole_traces_have_unique_flat_trace ge (s:RTLabs_state ge) |
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[1880] | 2237 | (str:trace_whole_program (RTLabs_status ge) s) |
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| 2238 | (tr:flat_trace io_out io_in ge s) |
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[1960] | 2239 | : equal_flat_traces … (flat_trace_of_whole_program … str) tr ≝ ?. |
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[1880] | 2240 | @flat_traces_are_determined_by_starting_point |
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| 2241 | qed. |
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