source: src/Clight/SwitchAndLabel.ma @ 3046

Last change on this file since 3046 was 3046, checked in by campbell, 6 years ago

Main part of combined switch removal and labelling proof.

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1include "Clight/switchRemoval.ma".
2include "Clight/labelSimulation.ma".
3
4axiom switch_final_related : ∀ge1,s1,s2.
5  switch_state_sim ge1 s1 s2 →
6  is_final s1 = is_final s2.
7
8lemma after_aux_result : ∀avs,n,s,tr,P.
9  after_aux avs n s tr P →
10  ∃tr',s'. P tr' s' ∧ after_aux avs n s tr (λtr'',s''. 〈tr'',s''〉 = 〈tr',s'〉).
11#avs #n elim n
12[ #s #tr #P #A whd in A; %{tr} %{s} % [ @A | % ]
13| #n' #IH #s #tr #P
14  whd in ⊢ (% → ?);
15  lapply (refl ? (is_final … s))
16  cases (is_final … s) in ⊢ (???% → %);
17  [ #F whd in ⊢ (% → ?);
18    lapply (refl ? (step … s))
19    cases (step … s) in ⊢ (???% → %);
20    [ #o #k #_ *
21    | * #tr1 #s1 #ST whd in ⊢ (% → ?);
22      cases n' in IH ⊢ %;
23      [ #_ whd in ⊢ (% → ?); #p % [2: % [2: %{p} whd >F whd >ST whd % | skip ] | skip ]
24      | #n'' #IH lapply (refl ? (avs_inv avs s1)) cases (avs_inv avs s1) in ⊢ (???% → %);
25        [ #INV #AF
26          cases (IH … AF) #tr' * #s' * #p #AF'
27         % [2: % [2: %{p} whd >F whd >ST whd >INV @AF' | skip ] | skip ]
28        | #_ *
29        ]
30      ]
31    | #m #_ *
32    ]
33  | #r #F *
34  ]
35] qed.
36
37lemma after_n_result : ∀n,O,I,exec,g,s,P,inv.
38  after_n_steps n O I exec g s inv P →
39  ∃tr',s'.
40    P tr' s' ∧
41    after_n_steps n O I exec g s inv (λtr'',s''. 〈tr'',s''〉 = 〈tr',s'〉).
42#n #O #I #exec #g #s #P #inv #A
43cases (after_aux_result … A) #tr * #s' * #p #A'
44%{tr} %{s'} %{p} @A'
45qed.
46
47
48lemma after_1_of_n_steps' : ∀n,O,I,exec,g,tr,s',s.
49  after_n_steps (S n) O I exec g s (λ_.true) (λtrx,sx. 〈trx,sx〉 = 〈tr,s'〉) →
50  ∃tr1,tr2,s1.
51  is_final … exec g s = None ? ∧
52  step … exec g s = Value … 〈tr1,s1〉 ∧
53  tr = tr1⧺tr2 ∧
54  after_n_steps n O I exec g s1 (λ_.true) (λtrx,sx. 〈trx,sx〉 = 〈tr2,s'〉).
55#n #O #I #exec #g #tr #s' #s #A
56cases (after_1_of_n_steps … A)
57#tr1 * #s1 * * * #F #ST #_ #A'
58cases (after_n_result … A')
59#tr'' * #s'' * #E #A'' destruct
60% [2: % [2: % [2: % [ % [% [ @F | @ST ] | % ] | @A'' ] | skip ] | skip ] | skip ]
61qed.
62
63theorem steps_with_labels : ∀ge,ge'.
64  related_globals_gen … label_fundef ge ge' →
65  ∀n,s1,s1',tr,s2.
66  state_with_labels s1 s1' →
67  after_n_steps (S n) … clight_exec ge s1 (λ_.true) (λtr',s'.〈tr',s'〉 = 〈tr,s2〉) →
68   ∃m. after_n_steps (S m) … clight_exec ge' s1' (λ_.true)
69   (λtr',s2'. trace_with_extra_labels tr tr' ∧
70              state_with_labels s2 s2').
71#ge #ge' #RG
72#n elim n
73[ #s1 #s1' #tr #s2 #SWL #AFTER
74  cases (after_1_step … AFTER) #tr' * #s' * * #F1 #ST1 #E destruct
75  @(step_with_labels … RG … ST1) //
76| #n #IH #s1 #s1' #tr #s2 #SWL #AFTER
77  cases (after_1_of_n_steps' … AFTER) #tr1 * #tr2 * #s' * * * #F1 #ST1 #E #AFTER'
78  destruct
79  cases (step_with_labels … RG … SWL ST1) #m1 #AF1
80  cases (after_n_result … AF1) #tr2 * #s2' * * #TWEL #SWL' #AF''
81  cases (IH … SWL' AFTER')
82  #m2 #AF2
83  %{(m1+S m2)}
84  @(after_n_m (S m1) (S m2) … AF2) //
85  @(after_n_covariant … AF'')
86 
87  #tr #s #E destruct %{(refl ??)}
88  #tr'' #s'' * #TWEL #SWL % /2/
89] qed.
90
91lemma interactive_switch_step : ∀ge,ge'.
92  switch_removal_globals ? fundef_switch_removal ge ge' →
93  ∀s1,s1',o,k.
94  exec_step ge s1 = Interact … o k →
95  switch_state_sim ge s1 s1' →
96  ∃k'.
97      exec_step ge' s1' = Interact … o k' ∧
98  ∀i. ∃tr,s2.
99    k i = Value … 〈tr,s2〉 ∧
100    ∃s2'.
101      k' i = Value … 〈tr,s2'〉 ∧
102      switch_state_sim ge s2 s2'.
103#ge #ge' #RG #s1 #s1' #o #k #EX
104cases (exec_step_interaction … EX)
105#vf * #fn * #argtys * #retty * #vargs * #k' * #m #E
106destruct
107#S inversion S
108[1,3,4: #H1 #H2 #H3 #H4 destruct
109  #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 destruct
110  #H13 #H14 #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 #H29 destruct
111]
112#s_vf #s_fd #s_args #s_k #s_k_ext #s_m #s_m_ext #s_writeable #s_me #s_H
113#E1 #E2 #E3 destruct
114whd in EX:(??%?);
115@(bindIO_res_interact ?????????? EX) -EX
116#vs #CHECK #EX whd in EX:(??%?); destruct
117% [2: %
118[ whd in ⊢ (??%?);
119  >CHECK in ⊢ (??%?);
120  whd in ⊢ (??%?);
121  @refl
122| #i
123  % [2: % [2: %
124  [ @refl
125  | % [2: %
126    [ @refl
127    | @(sws_returnstate … s_me) // ]
128    | skip ] ] | skip ]
129  ] ]| skip
130] qed.
131
132lemma Value_eq_l : ∀tr,s,tr',s'.
133  Value io_out io_in (trace×state) 〈tr,s〉 = Value io_out io_in (trace×state) 〈tr',s'〉 →
134  tr = tr'.
135#tr #s #tr' #s' #E destruct %
136qed.
137
138lemma Value_eq_r : ∀tr,s,tr',s'.
139  Value io_out io_in (trace×state) 〈tr,s〉 = Value io_out io_in (trace×state) 〈tr',s'〉 →
140  s = s'.
141#tr #s #tr' #s' #E destruct %
142qed.
143
144let corec build_exec
145  (ge1,ge2,ge3:genv)
146  (SRG:switch_removal_globals ? fundef_switch_removal ge1 ge2)
147  (RG:related_globals_gen … label_fundef ge2 ge3)
148  (s1,s2,s3:state)
149  (S1:switch_state_sim ge1 s1 s2)
150  (S2:state_with_labels s2 s3)
151  (NW:not_wrong state (exec_inf_aux … clight_fullexec ge1 (exec_step ge1 s1)))
152: sim_with_labels (exec_inf_aux … clight_fullexec ge1 (exec_step ge1 s1)) (exec_inf_aux … clight_fullexec ge3 (exec_step ge3 s3)) ≝
153match NW return λe,NW. exec_inf_aux … clight_fullexec ?? = e → sim_with_labels e ? with
154[ nw_stop tr i s ⇒ ?
155| nw_step tr1 s1' e1 NW1 ⇒ ?
156| nw_interact o k NWk ⇒ ?
157] (refl ? (exec_inf_aux … clight_fullexec ge1 (exec_step ge1 s1))).
158[ #E1
159  cases (exec_inf_stops_at_final ?? clight_fullexec … E1)
160  #EX1 #F1
161  cases (switch_removal_correction … SRG … S1 EX1)
162  #n1 #A1
163  cases (after_n_result … A1) #tr1' * #s1' * * #E destruct #S1' #A1'
164  cases (steps_with_labels … RG … S2 A1')
165  #n2 #A2 cases (after_inv clight_fullexec ????? A2) #tr' * #s' * * #TWL #S'
166  lapply F1 whd in ⊢ (??%? → ?); >(switch_final_related … S1') >(final_related … S') #F3
167    whd in match (is_final … s'); >F3 *
168    #tr2' #S2
169    @(swl_stop … S2 TWL)
170| #E1
171  cases (extract_step ?? clight_fullexec … E1)
172  #EX1 #E1'
173  cases (switch_removal_correction … SRG … S1 EX1)
174  #n1 #A1
175  cases (after_n_result … A1) #tr1' * #s2' * * #E @(match E with [refl ⇒ ?]) #S1' #A1'
176  cases (steps_with_labels … RG … S2 A1')
177  #n #AF cases (after_inv clight_fullexec ????? AF) #tr' * #s' * * #TWL #S'
178  whd in match (is_final … s'); lapply (refl ? (is_final s'))
179  cases (is_final s') in ⊢ (???% → %);
180  [ #NF #H whd in H; @(swl_steps … H) //
181    @(match sym_eq … E1' return λe,E1'. sim_with_labels e ? with [refl ⇒ ?])
182    @(build_exec … SRG RG … S1' S') >E1' in NW1; //
183  | #r <(final_related … S') <(switch_final_related … S1') change with (is_final … clight_fullexec ge1 s1') in ⊢ (??%? → ?);
184    >(exec_e_step_not_final ?? clight_fullexec … E1) #E' destruct
185  ]
186| #E1
187  cases (extract_interact ?? clight_fullexec … E1)
188  #k' * #EX1 #Ek lapply NWk -NWk @(match sym_eq … Ek with [refl ⇒ ?]) #NWk
189  cases (interactive_switch_step … SRG … EX1 S1) #k2' * #EX2 #H2
190  cases (interactive_step_with_labels … RG … EX2 S2)
191  #k3' * #EX3 #H3 @(match sym_eq … EX3 with [refl ⇒ ?]) @(match sym_eq … (exec_inf_aux_unfold …) with [refl ⇒ ?]) (* XXX why is this necessary? *) whd in ⊢ (??%);
192  @swl_interact
193  #i cases (H3 i) #tr2 * #s2' * #K2 * #tr3 * #s3' * * #K3 #TR2 #S2'
194  cases (H2 i) #tr1 * #s1' * #K1 * #s2x' * @(match sym_eq … K2 with [refl ⇒ ?]) #Ex
195  @(match (Value_eq_r … Ex) with [refl ⇒ ?]) #S1'
196  lapply (NWk i)
197  @(match sym_eq … K1 with [refl ⇒ ?]) @(match sym_eq … (exec_inf_aux_unfold …) with [refl ⇒ ?]) whd in ⊢ (??% → ?%%); whd in match (is_final ?????);
198  @(match sym_eq … (switch_final_related … S1') with [refl ⇒ ?]) @(match sym_eq … (final_related … S2') with [refl ⇒ ?])
199  @(match sym_eq … K2 with [refl ⇒ ?]) @(match sym_eq … (exec_inf_aux_unfold ?? clight_fullexec ge3 …) with [refl ⇒ ?]) whd in ⊢ (? → ??%);
200  @(match sym_eq … K3 with [refl ⇒ ?]) whd in ⊢ (? → ??%); whd in match (is_final ?????);
201  cases (is_final s3')
202  [ whd in ⊢ (??% → ?%%); #NW' @(swl_steps … (steps_step …))
203    [ destruct //
204    | @build_exec //
205      inversion NW'
206      [ #H1 #H2 #H3 #H4 #H5 destruct
207      | #H7 #H8 #H9 #H10 #H11 #H12 destruct //
208      | #H14 #H15 #H16 #H17 #H18 destruct
209      ]
210    ]
211  | #r whd in ⊢ (??% → ?%%); #NW' @(swl_stop … (steps_stop …)) destruct //
212  ]
213] qed.
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