source: LTS/Lang_meas.ma @ 3540

Last change on this file since 3540 was 3540, checked in by piccolo, 5 years ago

closed some daemons

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1(**************************************************************************)
2(*       ___                                                              *)
3(*      ||M||                                                             *)
4(*      ||A||       A project by Andrea Asperti                           *)
5(*      ||T||                                                             *)
6(*      ||I||       Developers:                                           *)
7(*      ||T||         The HELM team.                                      *)
8(*      ||A||         http://helm.cs.unibo.it                             *)
9(*      \   /                                                             *)
10(*       \ /        This file is distributed under the terms of the       *)
11(*        v         GNU General Public License Version 2                  *)
12(*                                                                        *)
13(**************************************************************************)
14
15include "Language.ma".
16
17
18lemma top_move_source_is_empty : ∀p,p',prog.
19let 〈t_prog,m,keep〉 ≝ trans_prog … prog in
20∀l.
21∀s1,s2,s1' : state p.∀abs_top,abs_tail.
22execute_l … p' (env … prog) l s1 s2 →
23is_costlabelled_act l →
24(is_call_act l → is_call_post_label (operational_semantics p p' prog) … s1) →
25state_rel … m keep abs_top abs_tail s1 s1' → abs_top = nil ? ∧
26(is_call_act l → is_call_post_label (operational_semantics p p' t_prog) … s1').
27#p #p' #prog @pair_elim * #t_prog #m #keep #EQt_prog normalize nodelta
28#l #s1 #s2 #s1' #abs_top #abs_tail #H elim H -s1 -s2 -l
29[ #s1 #s2 * #lab #code_cont #stack #EQcode #EQcont #EQ1 #EQ2 destruct #EQstore
30  #Hio1 #Hio2 #no_ret_lab #cost_lab whd in match state_rel; normalize nodelta
31  inversion(check_continuations ?????) [ #_ #_ *] ** #Hcont #a_top #a_tail #EQcheck #_
32  normalize nodelta **** #HHcont #EQcall_post #EQstore #EQio #EQ destruct
33  >EQcont in EQcheck; whd in ⊢ (??%? → ?); inversion(cont … s1') normalize nodelta
34  [ #_ #EQ destruct] * #lab' #code_cont' #stack' #_ #EQcont'
35    change with (check_continuations ?????) in match (foldr2 ???????);
36    inversion(check_continuations ?????) [ #_ normalize #EQ destruct]
37    ** #H2 #a_top1 #a_tail1 #EQcheck >m_return_bind normalize nodelta
38    inversion(call_post_clean ?????) normalize nodelta [ #_ whd in ⊢ (??%% → ?); #EQ destruct cases HHcont]
39    * #labels #code_cleaned #EQclean normalize nodelta
40    >EQcode in EQcall_post; inversion(code ? s1')
41        [2,3,4,5,6,7:
42          [ #ret | #seq #opt_l #instr #_ | #cond #ltrue #i_true #lfalse #i_false #instr #_ #_ #_|
43            #cond #ltrue #itrue #lfalse #ifalse #_ #_| #f #pars #opt_l #instr #_ | #lin #io #lout #instr #_ ]
44          #_ whd in ⊢ (??%% → ?);
45          [ | cases(call_post_clean ?????); normalize nodelta [2: #x cases opt_l normalize nodelta [| #label cases(?==?) normalize nodelta]]
46          | cases(call_post_clean ?????); normalize nodelta
47            [2: #x whd in ⊢ (??%? → ?); cases(call_post_clean ?????) normalize nodelta
48               [|#y whd in ⊢ (??%? → ?); cases(call_post_clean ?????) normalize nodelta
49                 [| #z cases(?∧?) normalize nodelta
50                 ]
51               ]
52            ]
53          | cases(call_post_clean ?????) normalize nodelta
54            [| #x whd in ⊢ (??%? → ?); cases(call_post_clean ?????) normalize nodelta
55               [| #y cases(?∧?) normalize nodelta
56               ]
57            ]
58          | cases(call_post_clean ?????) normalize nodelta
59             [| #x cases opt_l normalize nodelta [| #lbls cases(memb ???) normalize nodelta
60                [ cases (?==?) normalize nodelta ]
61             ]]
62          | cases(call_post_clean ?????) normalize nodelta
63             [| #x cases(? ∧ ?) normalize nodelta ]
64          ]
65          whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct
66        | #EQcode_s1' whd in ⊢ (??%% → ?); #EQ destruct cases lab' normalize nodelta
67          [ #f #c whd in ⊢ (??%% → ?); #EQ destruct %{(refl …)} cases HHcont
68          | * normalize nodelta [| #Lab cases(ret_costed_abs ??) normalize nodelta [| #xxx]] whd in ⊢ (??%% → ?);
69            #EQ destruct %{(refl …)} cases HHcont * [3: *] #EQ destruct #_ #_ [ #_] * #f * #c #EQ destruct
70          | * [|#LAB] normalize nodelta whd in ⊢ (??%% → ?); #EQ destruct % // cases HHcont * #EQ destruct
71            try cases cost_lab #_ #_ cases EQ -EQ #EQ destruct #_ * #f * #c #EQ destruct
72          ]
73        ]
74| #s1 #s2 #i #instr #st * [|#lbl] #EQcode_s1 #EQst #EQ destruct #EQcont #EQ destruct #Hios1 #Hios2 *
75  whd in match state_rel; normalize nodelta cases(check_continuations ?????) normalize nodelta [ #_ *] **
76  #H1 #a_top #a_tail normalize nodelta #_ **** #HH1 >EQcode_s1 inversion(code ? s1')
77  [1,2,4,5,6,7:
78    [ | #ret | #cond #ltrue #i_true #lfalse #i_false #instr #_ #_ #_|
79            #cond #ltrue #itrue #lfalse #ifalse #_ #_| #f #pars #opt_l #instr #_ | #lin #io #lout #instr #_ ]
80          #_ whd in ⊢ (??%% → ?);
81          [ |
82          | cases(call_post_clean ?????); normalize nodelta
83            [2: #x whd in ⊢ (??%? → ?); cases(call_post_clean ?????) normalize nodelta
84               [|#y whd in ⊢ (??%? → ?); cases(call_post_clean ?????) normalize nodelta
85                 [| #z cases(?∧?) normalize nodelta
86                 ]
87               ]
88            ]
89          | cases(call_post_clean ?????) normalize nodelta
90            [| #x whd in ⊢ (??%? → ?); cases(call_post_clean ?????) normalize nodelta
91               [| #y cases(?∧?) normalize nodelta
92               ]
93            ]
94          | cases(call_post_clean ?????) normalize nodelta
95             [| #x cases opt_l normalize nodelta [| #lbls cases(memb ???) normalize nodelta
96                [ cases (?==?) normalize nodelta ]
97             ]]
98          | cases(call_post_clean ?????) normalize nodelta
99             [| #x cases(? ∧ ?) normalize nodelta ]
100          ]
101          whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct
102  | #i' * [|#lb] #instr' #_ #EQcode_s1' whd in ⊢ (??%% → ?); cases(call_post_clean ?????) normalize nodelta
103    [1,3: #EQ destruct] #x whd in ⊢ (??%% → ?); [ #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct ]
104    cases(get_element ???) normalize nodelta [#EQ destruct] #LAB #ll normalize nodelta cases(?==?) normalize nodelta
105    [2:  #EQ destruct] whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct #_ #_ #_ % // * #f * #c #EQ4 destruct
106 ]
107| #s1 #s2 #exp #ltrue #i_true #lfalse #i_false #instrs #new_m #EQcode_s1 #EQeval #EQcont_s2 #EQ destruct #EQ destruct
108  #Hio1 #Hio2 #_ #_ whd in match state_rel; normalize nodelta inversion(check_continuations ?????) normalize nodelta
109  [ #_ *] ** #H1 #a_top #a_tail #EQcheck normalize nodelta **** #HH1 >EQcode_s1 inversion(code ? s1')
110  [1,2,3,5,6,7:
111    [ | #ret | #seq #opt_l #instr #_ | #cond #ltrue #itrue #lfalse #ifalse #_ #_| #f #pars #opt_l #instr #_
112       | #lin #io #lout #instr #_ ]
113          #_ whd in ⊢ (??%% → ?);
114          [ |
115          | cases(call_post_clean ?????); normalize nodelta [2: #x cases opt_l normalize nodelta [| #label cases(?==?) normalize nodelta]]
116          | cases(call_post_clean ?????) normalize nodelta
117            [| #x whd in ⊢ (??%? → ?); cases(call_post_clean ?????) normalize nodelta
118               [| #y cases(?∧?) normalize nodelta
119               ]
120            ]
121          | cases(call_post_clean ?????) normalize nodelta
122             [| #x cases opt_l normalize nodelta [| #lbls cases(memb ???) normalize nodelta
123                [ cases (?==?) normalize nodelta ]
124             ]]
125          | cases(call_post_clean ?????) normalize nodelta
126             [| #x cases(? ∧ ?) normalize nodelta ]
127          ]
128          whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct
129  ]
130  cases daemon (*TODO*)
131| cases daemon
132| cases daemon
133| cases daemon
134| #s1 #s2 #lin #io #lout #instr #men #EQcode  #EQeval #EQcodes2 #EQcont_s2 #EQ destruct #Hio * whd in match state_rel;
135  normalize nodelta cases(check_continuations ?????) normalize nodelta [ #_ *] **
136  #H1 #a_top #a_tail normalize nodelta #_ **** #HH1 >EQcode inversion(code ? s1')
137  [1,2,3,4,5,6: cases daemon (* ASSURDI*)
138  | #lin' #io' #lout' #i' #_ #EQcode_s1' whd in ⊢ (??%% → ?); cases(call_post_clean ?????) normalize nodelta
139    [ #EQ destruct] #x
140    cases(get_element ???) normalize nodelta [#EQ destruct] #LAB #ll normalize nodelta cases(?∧?) normalize nodelta
141    [2:  #EQ destruct] whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct #_ #_ #_ % // * #f * #c #EQ destruct
142 ]
143| #s1 #s2 #f #act_p #lbl #instr #men #env_it #EQcode #EQenv_it  #EQeval #EQ destruct #EQcodes2 #EQ destruct #_ #_ *
144  whd in match state_rel; normalize nodelta cases(check_continuations ?????) normalize nodelta [ #_ *] **
145  #H1 #a_top #a_tail normalize nodelta #Hcall **** #HH1 >EQcode inversion(code ? s1')
146  [1,2,3,4,5,7: cases daemon (* ASSURDI*)
147  |*: #f' #act_p' #lbl' #instr' #_ #EQcode_s1' whd in ⊢ (??%% → ?); cases(call_post_clean ?????) normalize nodelta
148    [ #EQ destruct] #x cases lbl' in EQcode_s1'; normalize nodelta [ #_ #EQ destruct] #lbl'' #EQcode_s1'
149    inversion(?∈?) normalize nodelta
150    [#_ cases(get_element ???) normalize nodelta [#EQ destruct] #LAB #ll normalize nodelta
151      cases(?==?) normalize nodelta
152    [2:  #EQ destruct] whd in ⊢ (??%% → ?); #EQ1 lapply(eq_to_jmeq ??? EQ1) -EQ1 #EQ1 destruct #_#_ #EQ1 destruct % //
153    #_ whd in match (is_call_post_label ??); >EQcode_s1' %
154    ]
155    #ABS whd in ⊢ (??%% → ?);  #EQ1 lapply(eq_to_jmeq ??? EQ1) -EQ1 #EQ1 destruct lapply(Hcall ?) [ %{f} %{(f_lab … env_it)} %]
156    whd in ⊢ (% → ?); whd in match (is_call_post_label ??); normalize nodelta >EQcode *
157 ]
158| #s1 #s2 #r_t #mem #stack * [|#lbl] #i #EQcode #EQcont #EQ destruct #_ #_ #EQeval #EQ1 #EQ2 destruct * #_
159  whd in match state_rel; normalize nodelta inversion(check_continuations ?????) normalize nodelta [ #_ *]
160  ** #H1 #a_top #a_tail #EQcheck normalize nodelta **** #HH1 >EQcode inversion(code … s1')
161  [1,3,4,5,6,7: cases daemon (*ASSURDI*)
162  | #r_t' #EQcode_s1' whd in ⊢ (??%% → ?); #EQ1 lapply(eq_to_jmeq ??? EQ1) -EQ1 #EQ1 destruct
163    >EQcont in EQcheck; inversion (cont ? s1') [ #_ whd in ⊢ (??%% → ?); #EQ destruct]
164    * #LAB #instrs #stack' #_ #EQcont_s1' whd in match check_continuations; normalize nodelta
165    whd in match (foldr2 ???????); change with (check_continuations ?????) in match (foldr2 ???????);
166    cases(check_continuations ?????) normalize nodelta [#EQ destruct] ** #H2 #a_top1 #a_tail1
167    normalize nodelta   whd in ⊢ (??%? → ?); cases(call_post_clean ?????) normalize nodelta [ #EQ destruct cases HH1]
168    * #ret_top #cleaned cases LAB in EQcont_s1';
169    [ #f #c #_ normalize nodelta #EQ destruct cases HH1
170    | *
171      [ #EQcont_s1' normalize nodelta #EQ destruct cases HH1 * #EQ destruct
172      | #lbl1 #EQcont_s1' whd in match ret_costed_abs; normalize nodelta inversion(? ∈ ?) normalize nodelta
173        [ #EQkeep #EQ destruct cases HH1 ** #EQ destruct #HH2 #EQ destruct #_ #_ #_ #_ % // * #f * #c #EQ destruct
174        | #EQkeep #EQ destruct #_ #_ #_ % // * #f * #c #EQ destruct
175        ]
176      ]
177    | * [|#lbl] #EQcont_s1' normalize nodelta  #EQ destruct cases HH1 *  #EQ destruct cases EQ -EQ #EQ destruct
178    ]
179  ]
180]
181qed.
182
183lemma bool_true : ∀b : bool.b=true → b. * // #EQ destruct qed.
184(*
185axiom actionlabel_of_cost_is_not_empty : ∀l : ActionLabel.
186is_costlabelled_act l → ∃c.actionlabel_to_costlabel l = [c].
187*)
188lemma labelled_action_is_correct :
189∀p,p',prog.
190no_duplicates_labels … prog →
191let 〈t_prog,m,keep〉 ≝ trans_prog … prog in
192∀s1,s2,s1' : state p.∀abs_top,abs_tail.
193∀l.is_costlabelled_act l →
194∀prf :execute_l p p' (env … prog) l s1 s2.
195state_rel … m keep abs_top abs_tail s1 s1' →
196∃abs_top'.∃s2'.
197execute_l p p' (env … t_prog) l s1' s2' ∧
198state_rel  … m keep abs_top' abs_tail s2 s2' ∧
199map_labels_on_trace … m … (actionlabel_to_costlabel l) =
200actionlabel_to_costlabel l @ abs_top'.
201#p #p' #prog  #no_dup @pair_elim * #t_prog #m #keep #EQt_prog normalize nodelta
202#s1 #s2 #s1' #abs_top #abs_tail #l #Hl #H lapply Hl -Hl elim H -s1 -s2 -l
203[ #s1 #s2 * #lbl #i #stack #EQcode_s1 #EQcont_s1 #EQ destruct #EQ destruct #EQstore #Hio1 #Hio2
204  #Hlbl #cost_lbl whd in match state_rel in ⊢ (% → ?); normalize nodelta inversion(check_continuations ?????)
205  normalize nodelta [ #_ *] ** #H1 #a_top #a_tail >EQcont_s1 inversion(cont … s1') [ #_ whd in ⊢ (??%% → ?); #EQ destruct]
206  * #lbl' #i' #stack' #_ #EQcont_s1' whd in ⊢ (??%% → ?); change with (check_continuations ?????) in match (foldr2 ???????);
207  inversion(check_continuations ?????) normalize nodelta [ #_ #EQ destruct] ** #H2 #a_top1 #a_tail1 #EQcheck
208  normalize nodelta inversion(call_post_clean ?????) normalize nodelta [#_ whd in ⊢ (??%% → ?); #EQ destruct ***** ]
209  * #gen_labels #cleaned_i' #EQclean cases lbl' in EQcont_s1'; -lbl' normalize nodelta
210  [ #f #c #_ whd in ⊢ (??%% → ?); #EQ destruct *****
211  | * [|#lbl'] normalize nodelta #_ whd in ⊢ (??%% → ?); [2: cases(ret_costed_abs ??) normalize nodelta [|#lbl'']]
212    #EQ destruct ****** [2: *] #EQ destruct @⊥ cases Hlbl #H @H %
213  | * [|#lbl'] #EQcont_s1' normalize nodelta whd in ⊢ (??%% → ?); #EQ destruct ****** [2: *] #EQ destruct
214    cases cost_lbl #HH2 #EQ destruct #EQget #EQ_cleaned #EQstore #EQio #EQ destruct
215    >EQcode_s1 in EQ_cleaned; inversion(code ? s1')
216    [2,3,4,5,6,7: cases daemon (*ASSURDI*)] #EQcode_s1' whd in ⊢ (??%% → ?); #EQ destruct
217    %{gen_labels} %{(mk_state … i' stack' (store … s1') false)} %
218    [% [ @(empty … EQcode_s1' EQcont_s1') // #_ %{lbl'} %]
219       whd >EQcheck in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?]); normalize nodelta % // % // % // % // >EQclean %]
220   whd in match actionlabel_to_costlabel; normalize nodelta whd in ⊢ (??%%); >EQget >append_nil %
221 ]
222| #s1 #s2 #seq #i #store * [|#lbl] #EQcode_s1 #EQeval #EQ destruct #EQcont #EQ destruct #Hio1 #Hio2 *
223  whd in match state_rel in ⊢ (% → ?); normalize nodelta inversion(check_continuations ?????) normalize nodelta
224  [#_ *] ** #H1 #a_top #a_tail #EQcheck normalize nodelta **** #HH1 >EQcode_s1 inversion(code ? s1')
225  [1,2,4,5,6,7: cases daemon (*ASSURDI*)] #seq' #opt_l' #i' #_ #EQcode_s1' whd in ⊢ (??%% → ?);
226  inversion(call_post_clean ?????) normalize nodelta [ #_ #EQ destruct] * #genlab #cleaned #EQcleaned
227  cases opt_l' in EQcode_s1'; normalize nodelta [|#lbl'] #EQcode_s1' [| inversion(?==?) normalize nodelta #EQget]
228  whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct #EQst #EQioinfo #EQ destruct
229  %{genlab} %{(mk_state … i' (cont … s1') (store … s2) false)} % [ %
230  [ @(seq_sil … EQcode_s1') // ]
231  whd <EQcont >EQcheck normalize nodelta % // % // % // % // >EQcleaned %]
232  whd in match actionlabel_to_costlabel; normalize nodelta whd in ⊢ (??%%); >(\P EQget) >append_nil %
233| cases daemon
234| cases daemon
235| cases daemon
236| cases daemon
237| #s1 #s2 #lin #io #lout #i #store #EQcode_s1 #EQeval #EQcode_s2 #EQcont_s2. #EQ destruct #EQio_s2 *
238  whd in match state_rel in ⊢ (% → ?); normalize nodelta inversion(check_continuations ?????) normalize nodelta [#_ *]
239  ** #H1 #a_top #a_tail #EQcheck normalize nodelta **** #HH1 >EQcode_s1 inversion(code ? s1')
240  [1,2,3,4,5,6: cases daemon (*ASSURDI*) ]
241  #lin' #io' #lout' #i' #_ #EQcode_s1' whd in ⊢ (??%% → ?); inversion(call_post_clean ?????) normalize nodelta [ #_ #EQ destruct]
242  * #genlab #cleaned #EQcleaned inversion(?∧?) normalize nodelta #EQget whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) -EQ
243  #EQ destruct #EQstore_s1_s1' #EQioinfo #EQ destruct %{[]}
244  %{(mk_state … (EMPTY …) (〈cost_act (Some ? lout),i'〉::(cont … s1')) (store … s2) true)} % [%
245  [ @(io_in … EQcode_s1') // ]
246  whd >EQcont_s2 whd in match (check_continuations ?????); change with (check_continuations ?????) in match (foldr2 ???????);
247  >EQcheck normalize nodelta >EQcleaned normalize nodelta % // % // % // % [2: >EQcode_s2 %] %
248  [2: cases(andb_Prop_true … (bool_true … EQget)) #H #_ >(\P (bool_true … H)) % ] % // % //]
249  cases(andb_Prop_true … (bool_true … EQget)) #_ #EQget'
250  whd in match actionlabel_to_costlabel; normalize nodelta whd in ⊢ (??%%); >append_nil >append_nil inversion(? == ?) in EQget';
251  [2: #_ *] #EQget' #_ >(\P EQget') %
252|*: cases daemon
253]
254qed.
255
256include "Vm.ma".
257
258theorem correctness_theorem : ∀p,p',prog.
259no_duplicates_labels … prog →
260let 〈t_prog,m,keep〉 ≝ trans_prog … prog in
261∀si,s1,s2,sn,si' : state p.∀abs_top,abs_tail.
262∀t : raw_trace (operational_semantics … p' prog) si sn.
263measurable (operational_semantics … p' prog) … s1 s2 … t →
264state_rel … m keep abs_top abs_tail si si' →
265∃abs_top',abs_tail'.∃sn'.∃t' : raw_trace (operational_semantics … p' t_prog) si' sn'.
266state_rel  … m keep abs_top' abs_tail' sn sn' ∧
267is_permutation ? (map_labels_on_trace m (chop … (get_costlabels_of_trace … t))) 
268                  (chop … (get_costlabels_of_trace … t')) ∧
269                  ∃s1',s2'.
270measurable (operational_semantics … p' t_prog)  … s1' s2' … t'.
271#p #p' #prog #no_dup @pair_elim * #t_prog #m #keep #EQt_prog normalize nodelta
272#si #s1 #s3 #sn #si' #abs_top #abs_tail #t *  #s0 * #s2 * #ti0 * #t12 * #t3n * #l1 * #l2
273* #prf1 * #prf2 ******* #pre_t12 #EQ destruct #Hl1 #Hl2 #Hcall_fin #sil_ti0 #sil_t3n #Hcall_init #Rsi_si'
274lapply(silent_in_silent … p' prog no_dup) >EQt_prog normalize nodelta #Hsilent
275cases(Hsilent … sil_ti0 … Rsi_si')
276#abs_top1 * #s0' * #t_i0' * #Rs0_s0' #sil_ti0'
277lapply(labelled_action_is_correct … p' prog no_dup) >EQt_prog normalize nodelta #Hmove
278cases(Hmove … prf1 … Rs0_s0') //
279#abs_top2 * #s1' ** #prf1' #Rs1_s1' #EQmap_l1
280lapply(correctness_lemma … p' prog no_dup) >EQt_prog normalize nodelta #H
281cases(H … pre_t12 … Rs1_s1') -H
282#abs_top3 * #s2' * #t12' ***** #Rs2_s2' #Hperm #_ #_ #pre_t12' #_
283cases(Hmove … prf2 … Rs2_s2') -Hmove //
284#abs_top4 * #s3' ** #prf2' #Rs3_s3' #EQmap_l2
285cases(Hsilent … sil_t3n … Rs3_s3')
286#abs_top5 * #sn' * #t3n' * #Rsn_sn' #sil_t3n'
287%{abs_top5} %{abs_tail} %{sn'}
288%{(t_i0' @ (t_ind … prf1' (t12' @ (t_ind … prf2' t3n'))))} %
289[ %{Rsn_sn'} >get_cost_label_append >get_cost_label_append >(get_cost_label_silent_is_empty … sil_ti0)
290  >(get_cost_label_silent_is_empty … sil_ti0') >(get_cost_label_append … (t_ind … (t_base …)))
291  >(get_cost_label_append … (t_ind … (t_base …))) in ⊢ (???%); >get_cost_label_append
292  >get_cost_label_append whd in match (get_costlabels_of_trace … (t_ind … t3n));
293  whd in match (get_costlabels_of_trace … (t_ind … t3n'));
294  >(get_cost_label_silent_is_empty … sil_t3n) >(get_cost_label_silent_is_empty … sil_t3n')
295  >append_nil cases(actionlabel_ok … Hl2) #c2 #EQc2 >EQc2 <associative_append <associative_append
296  <associative_append >chop_append_singleton <associative_append <associative_append >chop_append_singleton
297  >append_nil whd in match (get_costlabels_of_trace ??? (t_ind (operational_semantics p p' t_prog) … prf1' (t_base …)));
298  >append_nil >map_labels_on_trace_append >EQmap_l1 >associative_append @is_permutation_append_left_cancellable
299  lapply(top_move_source_is_empty … p' prog) >EQt_prog normalize nodelta #H
300  >(proj1 … (H … prf2 Hl2 … Rs2_s2')) in Hperm; [2: assumption] >append_nil @permute_equal_right
301| %{s1'} %{s3'} whd %{s0'} %{s2'} %{t_i0'} %{t12'} %{t3n'} %{l1} %{l2} %{prf1'} %{prf2'}
302  % [2: lapply(top_move_source_is_empty … p' prog) >EQt_prog normalize nodelta #H
303  @(proj2 … (H … prf1 Hl1 … Rs0_s0')) assumption ] % // % // %
304  [2: lapply(top_move_source_is_empty … p' prog) >EQt_prog normalize nodelta #H
305  @(proj2 … (H … prf2 Hl2 … Rs2_s2')) assumption ] % // % // % //
306]
307qed.
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