source: src/common/StructuredTraces.ma @ 2508

include "basics/types.ma".
include "basics/bool.ma".
include "basics/jmeq.ma".
include "common/CostLabel.ma".
include "utilities/option.ma".
include "basics/lists/listb.ma".
include "ASM/Util.ma".

inductive status_class: Type[0] ≝
  | cl_return: status_class
  | cl_jump: status_class
  | cl_call: status_class
  | cl_other: status_class.

record abstract_status : Type[1] ≝
  { as_status :> Type[0]
  ; as_execute : as_status → as_status → Prop
  ; as_pc : DeqSet
  ; as_pc_of : as_status → as_pc
  ; as_classify : as_status → status_class
  ; as_label_of_pc : as_pc → option costlabel
  ; as_after_return : (Σs:as_status. as_classify s = cl_call) → as_status → Prop
  ; as_final: as_status → Prop
  ; as_call_ident : (Σs:as_status.as_classify s = cl_call) → ident
  }.

definition as_classifier ≝ λS,s,cl.as_classify S s = cl.
definition as_call ≝ λS,s,f.as_call_ident S s = f.

definition as_label ≝ λS : abstract_status. λs : S. as_label_of_pc ? (as_pc_of ? s).

(* temporary alias for backward compatibility *)
(* definition final_abstract_status ≝ abstract_status. *)

definition as_costed : ∀a_s : abstract_status.a_s → Prop ≝
  λa_s,st.as_label ? st ≠ None ?.

definition as_label_safe : ∀a_s : abstract_status.
  (Σs : a_s.as_costed ? s) → costlabel ≝
  λa_s,st_sig.opt_safe … (pi2 … st_sig).

lemma as_costed_exc : ∀S:abstract_status. ∀s:S. (as_costed S s) + (¬as_costed S s).
#S #s whd in match (as_costed S s);
cases (as_label S s) [ %2 % * /2/ | #c %1 % #E destruct ]
qed.

lemma not_costed_no_label : ∀S,st.
  ¬as_costed S st →
  as_label S st = None ?.
#S #st * normalize cases (as_label_of_pc S ?)
[ // | #l #H cases (H ?) % #E destruct ]
qed.

(* cost map generalities *)

definition as_cost_labelled ≝
  λS : abstract_status. λl.∃pc.as_label_of_pc S pc = Some ? l.

definition as_cost_label ≝
  λS : abstract_status. Σl.as_cost_labelled S l. 

definition as_cost_labels ≝
  λS : abstract_status. list (as_cost_label S). 

definition as_cost_get_label ≝
  λS : abstract_status. λl_sig: as_cost_label S. pi1 … l_sig. 

definition as_cost_get_labels ≝
  λS : abstract_status. map … (as_cost_get_label S). 

definition as_cost_map ≝
  λS : abstract_status. (as_cost_label S) → ℕ.
  
definition lift_sigma_map_id :
  ∀A,B : Type[0].∀P_in,P_out : A → Prop.B →
    (∀a.P_out a + ¬ P_out a) →
  ((Σa.P_out a) → B) → (Σa.P_in a) → B ≝ λA,B,P_in,P_out,dflt,dec,m,a_sig.
   match dec a_sig with
   [ inl prf ⇒ m «a_sig, prf»
   | inr _ ⇒ dflt (* labels not present in out code get 0 *)
   ].

lemma lift_sigma_map_id_eq :
  ∀A,B,P_in,P_out,dflt,dec,m,a_in,a_out.
  pi1 ?? a_in = pi1 ?? a_out →
  lift_sigma_map_id A B P_in P_out dflt dec m a_in = m a_out.
#A#B#P_in#P_out#dflt#dec#m#a_in#a_out#EQ
whd in match lift_sigma_map_id; normalize nodelta
cases (dec a_in) normalize nodelta >EQ cases a_out
#a #H #G [ % | @⊥ /2 by absurd/ ]
qed.

notation > "Σ_{ ident i ∈ l } f"
  with precedence 20
  for @{'fold plus 0 (λ${ident i}.true) (λ${ident i}. $f) $l}.
notation < "Σ_{ ident i ∈ l } f"
  with precedence 20
for @{'fold plus 0 (λ${ident i}:$X.true) (λ${ident i}:$Y. $f) $l}.

definition lift_cost_map_id :
  ∀S_in,S_out : abstract_status.? →
  as_cost_map S_out → as_cost_map S_in
  ≝
 λS_in,S_out : abstract_status.
  lift_sigma_map_id costlabel ℕ
    (*λl.∃pc.as_label_of_pc S_in pc = Some ? l*) (as_cost_labelled S_in)
    (*λl.∃pc.as_label_of_pc S_out pc = Some ? l*) (as_cost_labelled S_out)
    0.

lemma lift_cost_map_same_cost :
  ∀S_in, S_out, dec, m_out, trace_in, trace_out.
  map … (pi1 ??) trace_in = map … (pi1 ??) trace_out →
  (Σ_{ l_sig ∈ trace_in } (lift_cost_map_id S_in S_out dec m_out l_sig)) =
  (Σ_{ l_sig ∈ trace_out } (m_out l_sig)).
#S_in #S_out #dec #m_out #trace_in elim trace_in
[2: #hd_in #tl_in #IH] * [2,4: #hd_out #tl_out]
normalize in ⊢ (%→?);
[2,3: #ABS destruct(ABS)
|4: #_ %
|1:
  #EQ destruct
  whd in ⊢(??%%);
  whd in match lift_cost_map_id; normalize nodelta
  >(lift_sigma_map_id_eq ????????? e0)
  >e0 in e1; normalize in ⊢(%→?); #EQ
  >(IH … EQ) %
]
qed.


(* structured traces: down to business *)

inductive trace_ends_with_ret: Type[0] ≝
  | ends_with_ret: trace_ends_with_ret
  | doesnt_end_with_ret: trace_ends_with_ret.

inductive trace_label_return (S:abstract_status) : S → S → Type[0] ≝
  | tlr_base:
      ∀status_before: S.
      ∀status_after: S.
        trace_label_label S ends_with_ret status_before status_after →
        trace_label_return S status_before status_after
  | tlr_step:
      ∀status_initial: S.
      ∀status_labelled: S.
      ∀status_final: S.
        trace_label_label S doesnt_end_with_ret status_initial status_labelled →
        trace_label_return S status_labelled status_final →
          trace_label_return S status_initial status_final
with trace_label_label: trace_ends_with_ret → S → S → Type[0] ≝
  | tll_base:
      ∀ends_flag: trace_ends_with_ret.
      ∀start_status: S.
      ∀end_status: S.
        trace_any_label S ends_flag start_status end_status →
        as_costed S start_status →
          trace_label_label S ends_flag start_status end_status
with trace_any_label: trace_ends_with_ret → S → S → Type[0] ≝
  (* Single steps within a function which reach a label.
     Note that this is the only case applicable for a jump. *)
  | tal_base_not_return:
      ∀start_status: S.
      ∀final_status: S.
        as_execute S start_status final_status →
        (as_classifier S start_status cl_jump ∨
         as_classifier S start_status cl_other) →
        as_costed S final_status →
          trace_any_label S doesnt_end_with_ret start_status final_status
  | tal_base_return:
      ∀start_status: S.
      ∀final_status: S.
        as_execute S start_status final_status →
        as_classifier S start_status cl_return →
          trace_any_label S ends_with_ret start_status final_status
  (* A call followed by a label on return. *)
  | tal_base_call:
      ∀status_pre_fun_call: S.
      ∀status_start_fun_call: S.
      ∀status_final: S.
        as_execute S status_pre_fun_call status_start_fun_call →
        ∀H:as_classifier S status_pre_fun_call cl_call.
          as_after_return S «status_pre_fun_call, H» status_final →
          trace_label_return S status_start_fun_call status_final →
          as_costed S status_final →
            trace_any_label S doesnt_end_with_ret status_pre_fun_call status_final
  (* A call followed by a non-empty trace. *)
  | tal_step_call:
      ∀end_flag: trace_ends_with_ret.
      ∀status_pre_fun_call: S.
      ∀status_start_fun_call: S.
      ∀status_after_fun_call: S.
      ∀status_final: S.
        as_execute S status_pre_fun_call status_start_fun_call →
        ∀H:as_classifier S status_pre_fun_call cl_call.
          as_after_return S «status_pre_fun_call, H» status_after_fun_call →
          trace_label_return S status_start_fun_call status_after_fun_call →
          ¬ as_costed S status_after_fun_call →
          trace_any_label S end_flag status_after_fun_call status_final →
            trace_any_label S end_flag status_pre_fun_call status_final
  | tal_step_default:
      ∀end_flag: trace_ends_with_ret.
      ∀status_pre: S.
      ∀status_init: S.
      ∀status_end: S.
        as_execute S status_pre status_init →
        trace_any_label S end_flag status_init status_end →
        as_classifier S status_pre cl_other →
        ¬ (as_costed S status_init) →
          trace_any_label S end_flag status_pre status_end.

let rec tal_pc_list (S : abstract_status) fl st1 st2 (tal : trace_any_label S fl st1 st2)
  on tal : list (as_pc S) ≝
 match tal with
 [ tal_step_call fl' pre _ st1' st2' _ _ _ _ _ tl ⇒ as_pc_of … pre :: tal_pc_list … tl
 | tal_step_default fl' pre st1' st2' _ tl _ _ ⇒  as_pc_of … pre :: tal_pc_list … tl
 | tal_base_not_return pre _ _ _ _ ⇒ [as_pc_of … pre]
 | tal_base_return pre _ _ _ ⇒ [as_pc_of … pre]
 | tal_base_call pre _ _ _ _ _ _ _ ⇒ [as_pc_of … pre]
 ].

definition as_trace_any_label_length':
    ∀S: abstract_status.
    ∀trace_ends_flag: trace_ends_with_ret.
    ∀start_status: S.
    ∀final_status: S.
    ∀the_trace: trace_any_label S trace_ends_flag start_status final_status.
      nat ≝
  λS: abstract_status.
  λtrace_ends_flag: trace_ends_with_ret.
  λstart_status: S.
  λfinal_status: S.
  λthe_trace: trace_any_label S trace_ends_flag start_status final_status.
    |tal_pc_list … the_trace|.

let rec tlr_unrepeating S st1 st2 (tlr : trace_label_return S st1 st2) on tlr : Prop ≝
  match tlr with
  [ tlr_base st1 st2 tll ⇒ tll_unrepeating … tll
  | tlr_step st1 st2 st3 tll tl ⇒ tll_unrepeating … tll ∧ tlr_unrepeating … tl
  ]
and tll_unrepeating S fl st1 st2 (tll : trace_label_label S fl st1 st2) on tll : Prop ≝
  match tll with
  [ tll_base fl st1 st2 tal _ ⇒ tal_unrepeating … tal
  ]
and tal_unrepeating S fl st1 st2 (tal : trace_any_label S fl st1 st2) on tal : Prop ≝
  match tal with
  [ tal_step_call fl st1 st2 st3 st4 _ _ _ tlr _ tl ⇒
    bool_to_Prop (notb (memb ? (as_pc_of … st1) (tal_pc_list … tl))) ∧
    tal_unrepeating … tl ∧
    tlr_unrepeating … tlr
  | tal_step_default fl st1 st2 st3 _ tl _ _ ⇒
    bool_to_Prop (notb (memb ? (as_pc_of … st1) (tal_pc_list … tl))) ∧
    tal_unrepeating … tl
  | tal_base_call pre _ _ _ _ _ trace _ ⇒ tlr_unrepeating … trace
  | _ ⇒ True
  ].

definition tll_hd_label : ∀S : abstract_status.∀fl,st1,st2.
  trace_label_label S fl st1 st2 → costlabel ≝
  λS,fl,st1,st2,tr.match tr with
  [ tll_base _ st1' _ _ prf ⇒ as_label_safe … «st1', prf» ].

definition tlr_hd_label : ∀S : abstract_status.∀st1,st2.
  trace_label_return S st1 st2 → costlabel ≝
  λS,st1,st2,tr.match tr with
  [ tlr_base st1' st2' tll ⇒ tll_hd_label … tll
  | tlr_step st1' st2' _ tll _ ⇒ tll_hd_label … tll
  ].

let rec tal_unrepeating_uniqueb S fl st1 st2 tal on tal :
  tal_unrepeating S fl st1 st2 tal → bool_to_Prop (uniqueb … (tal_pc_list … tal)) ≝ ?.
cases tal //
#fl' #st1' [#st_fun] #st2' #st3' #H
[ #H0 #H1 #tlr #G #tal | #tal #H0 #G ] whd in ⊢ (% → ?%); [*]*
#A #B [#_] >A normalize nodelta @tal_unrepeating_uniqueb assumption
qed.

lemma tal_pc_list_start : ∀S,fl,s1,s2. ∀tal: trace_any_label S fl s1 s2.
  ∃tl. tal_pc_list … tal = (as_pc_of S s1)::tl.
#S #fl0 #s10 #s20 *
[ #s1 #s2 #EX #CL #CS
| #s1 #s2 #EX #CL
| #s1 #s2 #s3 #EX #CL #AF #tlr #CS
| #fl #s1 #s2 #s3 #s4 #EX #CL #AF #tlr #CS #tal
| #fl #s1 #s2 #s3 #EX #tal #CL #CS
] whd in ⊢ (??(λ_.??%?)); % [ 2,4,6,8,10: % | *: skip ]
qed.

let rec tal_tail_not_costed S fl st1 st2 tal
  (H:Not (as_costed S st1)) on tal :
  All ? (λpc. as_label_of_pc S pc = None ?) (tal_pc_list S fl st1 st2 tal) ≝ ?.
cases tal in H ⊢ %;
[ #start #final #EX #CL #CS #CS' % // @(not_costed_no_label ?? CS')
| #start #final #EX #CL #CS % // @(not_costed_no_label ?? CS)
| #pre #start #final #EX #CL #AF #tlr #CS #CS' % // @(not_costed_no_label ?? CS')
| #fl' #pre #start #after #final #EX #CL #AF #tlr #CS #tal' #CS'
  cases (tal_pc_list_start … tal')
  #hd #E >E
  % [ @(not_costed_no_label ?? CS') | @tal_tail_not_costed assumption ]
| #fl' #pre #init #end #EX #tal' #CL #CS #CS'
  cases (tal_pc_list_start … tal')
  #hd #E >E
  % [ @(not_costed_no_label ?? CS') | @tal_tail_not_costed assumption ]
] qed.


inductive trace_any_call (S:abstract_status) : S → S → Type[0] ≝
  | tac_base:
      ∀status: S.
        as_classifier S status cl_call →
          trace_any_call S status status
  | tac_step_call:
      ∀status_pre_fun_call: S.
      ∀status_after_fun_call: S.
      ∀status_final: S.
      ∀status_start_fun_call: S.
        as_execute S status_pre_fun_call status_start_fun_call →
        ∀H:as_classifier S status_pre_fun_call cl_call.
          as_after_return S (mk_Sig ?? status_pre_fun_call H) status_after_fun_call →
          trace_label_return S status_start_fun_call status_after_fun_call →
          ¬ as_costed S status_after_fun_call →
          trace_any_call S status_after_fun_call status_final →
            trace_any_call S status_pre_fun_call status_final
  | tac_step_default:
      ∀status_pre: S.
      ∀status_end: S.
      ∀status_init: S.
        as_execute S status_pre status_init →
        trace_any_call S status_init status_end →
        as_classifier S status_pre cl_other →
        ¬ (as_costed S status_init) →
          trace_any_call S status_pre status_end.

              
inductive trace_label_call (S:abstract_status) : S → S → Type[0] ≝
  | tlc_base:
      ∀start_status: S.
      ∀end_status: S.
        trace_any_call S start_status end_status →
        as_costed S start_status →
        trace_label_call S start_status end_status
.

definition tlc_hd_label : ∀S : abstract_status.∀st1,st2.
  trace_label_call S st1 st2 → costlabel ≝
  λS,st1,st2,tr.match tr with
  [ tlc_base st1' _ _ prf ⇒ as_label_safe … «st1', prf»
  ].
   
coinductive trace_label_diverges (S:abstract_status) : S → Type[0] ≝
  | tld_step:
      ∀status_initial: S.
      ∀status_labelled: S.
          trace_label_label S doesnt_end_with_ret status_initial status_labelled →
          trace_label_diverges S status_labelled →
            trace_label_diverges S status_initial
  | tld_base:
      ∀status_initial: S.
      ∀status_pre_fun_call: S.
      ∀status_start_fun_call: S.
        trace_label_call S status_initial status_pre_fun_call →
        as_execute S status_pre_fun_call status_start_fun_call →
        ∀H:as_classifier S status_pre_fun_call cl_call.
          trace_label_diverges S status_start_fun_call →
            trace_label_diverges S status_initial.

definition tld_hd_label : ∀S : abstract_status.∀st.
  trace_label_diverges S st → costlabel ≝
  λS,st,tr.match tr with
  [ tld_step st' st'' tll _ ⇒ tll_hd_label … tll
  | tld_base st' st'' _ tlc _ _ _ ⇒ tlc_hd_label … tlc
  ].       

(* Version in Prop for showing existence. *)
coinductive trace_label_diverges_exists (S:abstract_status) : S → Prop ≝
  | tld_step':
      ∀status_initial: S.
      ∀status_labelled: S.
          trace_label_label S doesnt_end_with_ret status_initial status_labelled →
          trace_label_diverges_exists S status_labelled →
            trace_label_diverges_exists S status_initial
  | tld_base':
      ∀status_initial: S.
      ∀status_pre_fun_call: S.
      ∀status_start_fun_call: S.
        trace_label_call S status_initial status_pre_fun_call →
        as_execute S status_pre_fun_call status_start_fun_call →
        ∀H:as_classifier S status_pre_fun_call cl_call.
          trace_label_diverges_exists S status_start_fun_call →
            trace_label_diverges_exists S status_initial.

inductive trace_whole_program (S: abstract_status) : S → Type[0] ≝
  | twp_terminating:
      ∀status_initial: S.
      ∀status_start_fun: S.
      ∀status_final: S.
        as_classifier S status_initial cl_call →
        as_execute S status_initial status_start_fun →
        trace_label_return S status_start_fun status_final →
        as_final S status_final →
        trace_whole_program S status_initial
  | twp_diverges:
      ∀status_initial: S.
      ∀status_start_fun: S.
        as_classifier S status_initial cl_call →
        as_execute S status_initial status_start_fun →
        trace_label_diverges S status_start_fun →
        trace_whole_program S status_initial.

(* Again, an identical version in Prop for existence proofs. *)
inductive trace_whole_program_exists (S: abstract_status) : S → Prop ≝
  | twp_terminating:
      ∀status_initial: S.
      ∀status_start_fun: S.
      ∀status_final: S.
        as_classifier S status_initial cl_call →
        as_execute S status_initial status_start_fun →
        trace_label_return S status_start_fun status_final →
        as_final S status_final →
        trace_whole_program_exists S status_initial
  | twp_diverges:
      ∀status_initial: S.
      ∀status_start_fun: S.
        as_classifier S status_initial cl_call →
        as_execute S status_initial status_start_fun →
        trace_label_diverges_exists S status_start_fun →
        trace_whole_program_exists S status_initial.


let rec trace_any_label_label S s s' f
  (tr:trace_any_label S f s s') on tr : match f with [ doesnt_end_with_ret ⇒ as_costed S s' | ends_with_ret ⇒ True ] ≝
match tr return λf,s,s',tr. match f with [ doesnt_end_with_ret ⇒ as_costed S s' | ends_with_ret ⇒ True ] with 
[ tal_base_not_return start final _ _ C ⇒ C
| tal_base_return _ _  _ _ ⇒ I
| tal_base_call _ _ _ _ _ _ _ C ⇒ C
| tal_step_call f pre start after final X C RET LR C' tr' ⇒ trace_any_label_label … tr'
| tal_step_default f pre init end X tr' C C' ⇒ trace_any_label_label … tr'
].

definition tal_tl_label : ∀S : abstract_status.∀st1,st2.
  trace_any_label S doesnt_end_with_ret st1 st2 → costlabel ≝
  λS,st1,st2,tr.as_label_safe … «st2, trace_any_label_label … tr».

lemma trace_label_label_label : ∀S,s,s',f.
  ∀tr:trace_label_label S f s s'. match f with [ doesnt_end_with_ret ⇒ as_costed S s' | ends_with_ret ⇒ True ].
#S #s #s' #f #tr
cases tr
#f #start #end #tr' #C @(trace_any_label_label … tr')
qed.

definition tll_tl_label : ∀S : abstract_status.∀st1,st2.
  trace_label_label S doesnt_end_with_ret st1 st2 → costlabel ≝
  λS,st1,st2,tr.as_label_safe … «st2, trace_label_label_label … tr».

lemma trace_any_call_call : ∀S,s,s'.
  trace_any_call S s s' → as_classifier S s' cl_call.
#S #s #s' #T elim T [1,3: //]
#H73 #H74 #H75 #H76 #H77 #H78 #H79 #H80 #H81 #H82 #H83 //
qed.

(*
(* an trace of unlabeled and cl_other states, possibly empty *)
inductive trace_no_label_any (S:abstract_status) : S → S → Type[0] ≝
  | tna_base :
      ∀start_status: S.
      ¬as_costed … start_status →
      trace_no_label_any S start_status start_status
  | tna_step :
      ∀status_pre: S.
      ∀status_init: S.
      ∀status_end: S.
        as_execute S status_pre status_init →
        as_classifier S status_pre cl_other →
        ¬as_costed … status_pre →
        trace_no_label_any S status_init status_end →
          trace_no_label_any S status_pre status_end.

let rec tna_append_tna S st1 st2 st3 (taa1 : trace_no_label_any S st1 st2)
  on taa1 :
    trace_no_label_any S st2 st3 →
    trace_no_label_any S st1 st3 ≝
  match taa1 with
  [ tna_base st1' H ⇒ λtaa2.taa2
  | tna_step st1' st2' st3' H G K tl ⇒
    λtaa2.tna_step ???? H G K (tna_append_tna … tl taa2)
  ].

definition tna_non_empty ≝ λS,st1,st2.λtna : trace_no_label_any S st1 st2.
  match tna with
  [ tna_base _ _ ⇒ false
  | tna_step _ _ _ _ _ _ _ ⇒ true
  ].

coercion tna_to_bool : ∀S,st1,st2.∀tna:trace_no_label_any S st1 st2.bool ≝
 tna_non_empty on _tna : trace_no_label_any ??? to bool.

lemma tna_unlabelled : ∀S,st1,st2.trace_no_label_any S st1 st2 → ¬as_costed … st1.
#S #st1 #st2 * [#st #H @H | #st #st' #st'' #_ #_ #H #_ @H] qed.

let rec tna_append_tal S st1 fl st2 st3 (tna : trace_no_label_any S st1 st2)
  on tna :
  trace_any_label S fl st2 st3 →
  if tna then Not (as_costed … st2) else True →
  trace_any_label S fl st1 st3 ≝
  match tna return λst1,st2.λx : trace_no_label_any S st1 st2.
    ∀fl,st3.
    trace_any_label S fl st2 st3 →
    if x then Not (as_costed … st2) else True →
    trace_any_label S fl st1 st3
  with
  [ tna_base st1' H ⇒ λfl,st3,taa2,prf.taa2
  | tna_step st1' st2' st3' H G K tl ⇒ λfl,st3,taa2,prf.
    tal_step_default ????? H (tna_append_tal ????? tl taa2 ?) G (tna_unlabelled … tl)
  ] fl st3.
  cases (tna_non_empty … tl) [@prf|%]
  qed.
*)

inductive trace_any_any (S : abstract_status) : S → S → Type[0] ≝
  | taa_base : ∀st.trace_any_any S st st
  | taa_step : ∀st1,st2,st3.
    as_execute S st1 st2 →
    as_classifier S st1 cl_other →
    ¬as_costed S st2 →
    trace_any_any S st2 st3 →
    trace_any_any S st1 st3.

definition taa_non_empty ≝ λS,st1,st2.λtaa : trace_any_any S st1 st2.
  match taa with
  [ taa_base _ ⇒ false
  | taa_step _ _ _ _ _ _ _ ⇒ true
  ].

coercion taa_to_bool : ∀S,st1,st2.∀taa:trace_any_any S st1 st2.bool ≝
 taa_non_empty on _taa : trace_any_any ??? to bool.

let rec taa_append_tal S st1 fl st2 st3
  (taa : trace_any_any S st1 st2) on taa :
  trace_any_label S fl st2 st3 →
  trace_any_label S fl st1 st3 ≝
  match taa return λst1,st2.λx : trace_any_any S st1 st2.
    ∀fl,st3.
    trace_any_label S fl st2 st3 →
    trace_any_label S fl st1 st3
  with
  [ taa_base st1' ⇒ λfl,st3,tal2.tal2
  | taa_step st1' st2' st3' H G K tl ⇒ λfl,st3,tal2.
    tal_step_default ????? H (taa_append_tal ????? tl tal2) G K
  ] fl st3.

interpretation "trace any any label append" 'append taa tal = (taa_append_tal ????? taa tal).

let rec tal_collapsable S fl s1 s2 (tal : trace_any_label S fl s1 s2) on tal : Prop ≝
match tal with
[ tal_base_not_return _ _ _ _ _ ⇒ True
| tal_step_default fl1 _ st1' st1'' _ tl1 _ _ ⇒ tal_collapsable … tl1
| _ ⇒ False
].

let rec tlr_rel S1 st1 st1' S2 st2 st2'
  (tlr1 : trace_label_return S1 st1 st1')
  (tlr2 : trace_label_return S2 st2 st2') on tlr1 : Prop ≝
match tlr1 with
  [ tlr_base st1 st1' tll1 ⇒
    match tlr2 with
    [ tlr_base st2 st2' tll2 ⇒ tll_rel … tll1 tll2
    | _ ⇒ False
    ]
  | tlr_step st1 st1' st1'' tll1 tl1 ⇒
    match tlr2 with
    [ tlr_step st2 st2' st2'' tll2 tl2 ⇒
      tll_rel … tll1 tll2 ∧ tlr_rel … tl1 tl2
    | _ ⇒ False
    ]
  ]
and tll_rel S1 fl1 st1 st1' S2 fl2 st2 st2'
 (tll1 : trace_label_label S1 fl1 st1 st1')
 (tll2 : trace_label_label S2 fl2 st2 st2') on tll1 : Prop ≝
  match tll1 with
  [ tll_base fl1 st1 st1' tal1 H ⇒
    match tll2 with
    [ tll_base fl2 st2 st2 tal2 G ⇒
      as_label_safe … («?, H») = as_label_safe … («?, G») ∧
      tal_rel … tal1 tal2
    ]
  ]
and tal_rel S1 fl1 st1 st1' S2 fl2 st2 st2'
 (tal1 : trace_any_label S1 fl1 st1 st1')
 (tal2 : trace_any_label S2 fl2 st2 st2')
   on tal1 : Prop ≝
  match tal1 with
  [ tal_base_not_return st1 st1' _ _ _ ⇒
    fl2 = doesnt_end_with_ret ∧
    ∃st2mid,taa,H,G,K.
    tal2 ≃ taa_append_tal ? st2 ??? taa
      (tal_base_not_return ? st2mid st2' H G K)
  | tal_base_return st1 st1' _ _ ⇒
    fl2 = ends_with_ret ∧
    ∃st2mid,taa,H,G.
    tal2 ≃ taa_append_tal ? st2 ? st2mid st2' taa
      (tal_base_return ? st2mid st2' H G)
  | tal_base_call st1 st1' st1'' _ prf _ tlr1 _ ⇒
    fl2 = doesnt_end_with_ret ∧
    ∃st2mid,G.as_call_ident S2 («st2mid, G») = as_call_ident ? «st1, prf» ∧
    ∃taa : trace_any_any ? st2 st2mid.∃st2mid',H.
    (* we must allow a tal_base_call to be similar to a call followed
      by a collapsable trace (trace_any_any followed by a base_not_return;
      we cannot use trace_any_any as it disallows labels in the end as soon
      as it is non-empty) *)
    (∃K.∃tlr2 : trace_label_return ? st2mid' st2'.∃L.
      tal2 ≃ taa @ (tal_base_call … H G K tlr2 L) ∧ tlr_rel … tlr1 tlr2) ∨
    ∃st2mid'',K.∃tlr2 : trace_label_return ? st2mid' st2mid''.∃L.
    ∃tl2 : trace_any_label … doesnt_end_with_ret st2mid'' st2'.
      tal2 ≃ taa @ (tal_step_call … H G K tlr2 L tl2) ∧
      tlr_rel … tlr1 tlr2 ∧ tal_collapsable … tl2
  | tal_step_call fl1 st1 st1' st1'' st1''' _ prf _ tlr1 _ tl1 ⇒
    ∃st2mid,G.as_call_ident S2 («st2mid, G») = as_call_ident ? «st1, prf» ∧
    ∃taa : trace_any_any ? st2 st2mid.∃st2mid',H.
    (fl2 = doesnt_end_with_ret ∧ ∃K.∃tlr2 : trace_label_return ? st2mid' st2'.∃L.
      tal2 ≃ taa @ tal_base_call … H G K tlr2 L ∧
      tal_collapsable … tl1 ∧ tlr_rel … tlr1 tlr2) ∨
    ∃st2mid'',K.∃tlr2 : trace_label_return ? st2mid' st2mid''.∃L.
    ∃tl2 : trace_any_label ? fl2 st2mid'' st2'.
      tal2 ≃ taa @ (tal_step_call … H G K tlr2 L tl2) ∧
      tal_rel … tl1 tl2 ∧ tlr_rel … tlr1 tlr2
  | tal_step_default fl1 st1 st1' st1'' _ tl1 _ _ ⇒
    tal_rel … tl1 tal2 (* <- this makes it many to many *)
  ].

interpretation "trace any label rel" 'napart t1 t2 = (tal_rel ???????? t1 t2).
interpretation "trace label label rel" 'napart t1 t2 = (tll_rel ???????? t1 t2).
interpretation "trace label return rel" 'napart t1 t2 = (tlr_rel ?????? t1 t2).

let rec tal_collapsable_eq_fl S1 fl1 s1 s1'
  (tal1 : trace_any_label S1 fl1 s1 s1') on tal1 :
  tal_collapsable … tal1 → fl1 = doesnt_end_with_ret ≝ ?.
cases tal1 -fl1 -s1 -s1' //
[ #s1 #s1' #H #I *
| #s1 #s1' #s1'' #s1''' #s1'''' #H #I #J #tlr #K #tl *
| #fl1 #s1 #s1' #s1'' #H #tl #I #J @(tal_collapsable_eq_fl … tl)
]
qed.

let rec tal_rel_eq_fl S1 fl1 s1 s1' S2 fl2 s2 s2'
  (tal1 : trace_any_label S1 fl1 s1 s1') on tal1 :
  ∀tal2 : trace_any_label S2 fl2 s2 s2'.tal_rel … tal1 tal2 → fl1 = fl2 ≝
  match tal1 return λfl1,s1,s1',tal1.? with
  [ tal_base_not_return st1 st1' _ _ _ ⇒ let BASE_NR ≝ 0 in ?
  | tal_base_return st1 st1' _ _ ⇒ let BASE_R ≝ 0 in ?
  | tal_base_call st1 st1' st1'' _ _ _ tlr1 _ ⇒ let BASE_C ≝ 0 in ?
  | tal_step_call flg1 st1 st1' st1'' st1''' _ _ _ tlr1 _ tl1 ⇒ let STEP_C ≝ 0 in ?
  | tal_step_default flg1 st1 st1' st1'' _ tl1 _ _ ⇒ let STEP ≝ 0 in ?
  ].
-fl1 -s1 -s1'
[1,2,3: -tal_rel_eq_fl #tal2 * //
| #tal2 * #s2_mid * #G2 * #call * #taa2 * #s2' *#H2 *
  [ * #EQ1 *#K2 *#tlr2 *#L2 ** #_ #coll #_ >(tal_collapsable_eq_fl … coll) //
  | * #s2_mid' *#K2 *#tlr2 *#L2 *#tl2 ** #_ #step #_
     @(tal_rel_eq_fl … step)
  ]
| #tal2 whd in ⊢ (%→?); #step @(tal_rel_eq_fl … step)
]
qed.

let rec taa_rel_inv S1 fl1 st1 st1mid st1' S2 fl2 st2 st2'
  (taa1 : trace_any_any S1 st1 st1mid) on taa1 :
  ∀tal1 : trace_any_label S1 fl1 st1mid st1'.
  ∀tal2 : trace_any_label S2 fl2 st2 st2'.
  tal_rel … (taa1 @ tal1) tal2 →
  tal_rel … tal1 tal2 ≝ ?.
cases taa1 -taa1
[ -taa_rel_inv //
| #st #st' #st'' #H #G #K #tl #tal1 #tal2 whd in ⊢ (%→?);
  @(taa_rel_inv … tl)
]
qed.

lemma taa_append_collapsable : ∀S,s1,fl,s2,s3.
  ∀taa,tal.tal_collapsable S fl s2 s3 tal → tal_collapsable S fl s1 s3 (taa@tal).
  #S #s1 #fl #s2 #s3 #taa elim taa -s1 -s2 /2/
qed.

let rec tal_rel_collapsable S1 fl1 s1 s1' S2 fl2 s2 s2'
  (tal1 : trace_any_label S1 fl1 s1 s1') on tal1 :
  ∀tal2 : trace_any_label S2 fl2 s2 s2'.tal_collapsable … tal1 → tal_rel … tal1 tal2 →
  tal_collapsable … tal2 ≝
  match tal1 return λfl1,s1,s1',tal1.? with
  [ tal_base_not_return st1 st1' _ _ _ ⇒ let BASE_NR ≝ 0 in ?
  | tal_base_return st1 st1' _ _ ⇒ let BASE_R ≝ 0 in ?
  | tal_base_call st1 st1' st1'' _ _ _ tlr1 _ ⇒ let BASE_C ≝ 0 in ?
  | tal_step_call flg1 st1 st1' st1'' st1''' _ _ _ tlr1 _ tl1 ⇒ let STEP_C ≝ 0 in ?
  | tal_step_default flg1 st1 st1' st1'' _ tl1 _ _ ⇒ let STEP ≝ 0 in ?
  ].
-fl1 -s1 -s1'
[1,2,3: -tal_rel_collapsable #tal2 * *
  #EQ * #s2 * #taa2 *#H *#G *#K #EQ' destruct @taa_append_collapsable %
| #tal2 *
| #tal2 #tal2 whd in ⊢ (%→?); #step @(tal_rel_collapsable … step) assumption
]
qed.

let rec flatten_trace_label_label
  (S: abstract_status) (trace_ends_flag: trace_ends_with_ret)
    (start_status: S) (final_status: S)
      (the_trace: trace_label_label S trace_ends_flag start_status final_status)
        on the_trace: list (as_cost_label S) (*Σl: costlabel. ∃pc. as_label_of_pc S pc = Some … l*) ≝
  match the_trace with
  [ tll_base ends_flag initial final given_trace labelled_proof ⇒
      let label ≝
        match as_label … initial return λx: option costlabel. x ≠ None ? → ? with
        [ None ⇒ λabs. ⊥
        | Some l ⇒ λ_. l
        ] labelled_proof
      in
        (mk_Sig … label ?)::flatten_trace_any_label S ends_flag initial final given_trace
  ]
and flatten_trace_any_label
  (S: abstract_status) (trace_ends_flag: trace_ends_with_ret)
    (start_status: S) (final_status: S)
      (the_trace: trace_any_label S trace_ends_flag start_status final_status)
        on the_trace: list (as_cost_label S) (*Σl: costlabel. ∃pc. as_label_of_pc S pc = Some … l*) ≝
  match the_trace with
  [ tal_base_not_return the_status _ _ _ _ ⇒ [ ]
  | tal_base_call pre_fun_call start_fun_call final _ _ _ call_trace _ ⇒
      flatten_trace_label_return … call_trace
  | tal_base_return the_status _ _ _ ⇒ [ ]
  | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final
    _ _ _ call_trace _ final_trace ⇒
    let call_cost_trace ≝ flatten_trace_label_return … call_trace in
    let final_cost_trace ≝ flatten_trace_any_label … end_flag … final_trace in
        call_cost_trace @ final_cost_trace
  | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒
      flatten_trace_any_label … tail_trace
  ]
and flatten_trace_label_return
  (S: abstract_status)
    (start_status: S) (final_status: S)
      (the_trace: trace_label_return S start_status final_status)
        on the_trace: list (as_cost_label S) (*Σl: costlabel. ∃pc. as_label_of_pc S pc = Some … l*) ≝
  match the_trace with
  [ tlr_base before after trace_to_lift ⇒
      flatten_trace_label_label … trace_to_lift
  | tlr_step initial labelled final labelled_trace ret_trace ⇒
    let labelled_cost ≝ flatten_trace_label_label … doesnt_end_with_ret … labelled_trace in
    let return_cost ≝ flatten_trace_label_return … ret_trace in
        labelled_cost @ return_cost
  ].
  [2:
    cases abs -abs #abs @abs %
  |1:
    %{(as_pc_of … initial)} whd in match label;
    change with (as_label ?? = ?)
    generalize in match labelled_proof; whd in ⊢ (% → ?);
    cases (as_label S initial)
    [1:
      #absurd @⊥ cases absurd -absurd #absurd @absurd %
    |2:
      #costlabel normalize nodelta #_ %
    ]
  ]
qed.

(* JHM: base case now passes the termination checker *)
let rec taa_append_tal_same_flatten
  S st1 fl st2 st3 (taa : trace_any_any S st1 st2) on taa :
  ∀tal : trace_any_label S fl st2 st3.
    flatten_trace_any_label … (taa @ tal) =
      flatten_trace_any_label … tal ≝ ?.
cases taa -st1 -st2
[ #st #tal normalize in ⊢ (??%?); //
| #st_pre #st_init #st2 #H #G #K #taa' #tal
  whd in match (? @ ?);
  whd in ⊢ (??%?); //
]
qed.

let rec tal_collapsable_flatten S fl st1 st2 tal
  on tal :
  tal_collapsable S fl st1 st2 tal → flatten_trace_any_label … tal = [ ] ≝
match tal
return λfl,st1,st2,tal.tal_collapsable S fl st1 st2 tal → flatten_trace_any_label … tal = [ ]
with
[ tal_base_not_return the_status _ _ _ _ ⇒ λ_.refl ??
| tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒
    tal_collapsable_flatten ???? tail_trace
| _ ⇒ Ⓧ
].

let rec tll_rel_to_traces_same_flatten
  (S: abstract_status) (S': abstract_status)
    (trace_ends_flag_l: trace_ends_with_ret) (trace_ends_flag_r: trace_ends_with_ret)
    (start_status_l: S) (final_status_l: S) (start_status_r: S') (final_status_r: S')
      (the_trace_l: trace_label_label S trace_ends_flag_l start_status_l final_status_l)
        (the_trace_r: trace_label_label S' trace_ends_flag_r start_status_r final_status_r)
          on the_trace_l:
            tll_rel … the_trace_l the_trace_r →
              map … (pi1 …) (flatten_trace_label_label … the_trace_l) =
                map … (pi1 …) (flatten_trace_label_label … the_trace_r) ≝
  match the_trace_l with
  [ tll_base fl1 st1 st1' tal1 H ⇒
    match the_trace_r with
    [ tll_base fl2 st2 st2 tal2 G ⇒ ?
    ]
  ]
and tal_rel_to_traces_same_flatten
  (S: abstract_status) (S': abstract_status) (trace_ends_flag_l: trace_ends_with_ret)
    (trace_ends_flag_r: trace_ends_with_ret)
      (start_status_l: S) (final_status_l: S) (start_status_r: S') (final_status_r: S')
        (the_trace_l: trace_any_label S trace_ends_flag_l start_status_l final_status_l)
          (the_trace_r: trace_any_label S' trace_ends_flag_r start_status_r final_status_r)
          on the_trace_l:
            tal_rel … the_trace_l the_trace_r →
              map … (pi1 …) (flatten_trace_any_label … the_trace_l) =
                map … (pi1 …) (flatten_trace_any_label … the_trace_r) ≝
  match the_trace_l with
  [ tal_base_not_return st1 st1' H G K ⇒ ?
  | tal_base_return st1 st1' H G ⇒ ?
  | tal_base_call st1 st1' st1'' H G K tlr1 L ⇒ ?
  | tal_step_call fl1 st1 st1' st1'' st1''' H G K tlr1 L tl1 ⇒ ?
  | tal_step_default fl1 st1 st1' st1'' H tl1 G K ⇒ ?
  ]
and tlr_rel_to_traces_same_flatten
  (S: abstract_status) (S': abstract_status) (start_status_l: S) (final_status_l: S)
    (start_status_r: S') (final_status_r: S')
      (the_trace_l: trace_label_return S start_status_l final_status_l)
        (the_trace_r: trace_label_return S' start_status_r final_status_r)
        on the_trace_l:
          tlr_rel … the_trace_l the_trace_r →
            map … (pi1 …) (flatten_trace_label_return … the_trace_l) =
              map … (pi1 …) (flatten_trace_label_return … the_trace_r) ≝
  match the_trace_l with
  [ tlr_base before after tll_l ⇒ ?
  | tlr_step initial labelled final tll_l tlr_l ⇒ ?
  ].  
[ * whd in match as_label_safe; normalize nodelta
  @opt_safe_elim #l1 #EQ1
  @opt_safe_elim #l2 #EQ2
  #EQ destruct(EQ) #H_tal
  change with (? :: ? = ? :: ?) lapply H -H lapply G -G
  whd in match as_costed; normalize nodelta
  >EQ1 >EQ2 normalize nodelta #_ #_
  >(tal_rel_to_traces_same_flatten … H_tal) @refl
|2,3,4,5,6: 
  [1,2,3: * #EQ destruct(EQ)]
  [1,2,3,4: * #st_mid [1,2:|*: * #G' * #call ] * #taa
    [ *#H' *#G' *#K' #EQ
    | *#H' *#G' #EQ
    | *#st_mid' *#H' * [|*#st2_mid''] *#K' *#tlr2 *#L'
      [|*#tl2 *] * #EQ #H_tlr [| #H_tl]
    | *#st_fun *#H' *
      [*#fl_EQ destruct(fl_EQ) |* #st2_mid ] *#K' *#tlr2 *#L'
      [| *#tl2] ** #EQ #H_tl #H_tlr
    ] >EQ >taa_append_tal_same_flatten
  | whd in ⊢ (%→??(????%)?);
    @tal_rel_to_traces_same_flatten
  ]
  [1,2: %
  |3: @(tlr_rel_to_traces_same_flatten … H_tlr)
  |4,5: <map_append
    >(tal_collapsable_flatten … H_tl) >append_nil
    >(tlr_rel_to_traces_same_flatten … H_tlr) %
  |6: <map_append
    >(tlr_rel_to_traces_same_flatten … H_tlr)
    >(tal_rel_to_traces_same_flatten … H_tl)
    @map_append
  ]
|*: cases the_trace_r
  [1,3: #st_before_r #st_after_r #tll_r
    [ @tll_rel_to_traces_same_flatten | * ]
  |*: #st_init_r #st_labld_r #st_fin_r #tll_r #tlr_r *
    #H_tll #H_tlr
    <map_append
    >(tll_rel_to_traces_same_flatten … H_tll)
    >(tlr_rel_to_traces_same_flatten … H_tlr)
    @map_append
  ]
]
qed.

(* cost maps specifics: summing over flattened traces *)

lemma lift_cost_map_same_cost_tal :
  ∀S_in, S_out, dec, m_out, f_in, f_out, start_in, start_out, end_in, end_out.
  ∀the_trace_in : trace_any_label S_in f_in start_in end_in.
  ∀the_trace_out : trace_any_label S_out f_out start_out end_out.
  tal_rel … the_trace_in the_trace_out →
  (Σ_{l ∈ flatten_trace_any_label … the_trace_in}
    (lift_cost_map_id … dec m_out l)) =
  (Σ_{l ∈ flatten_trace_any_label … the_trace_out} (m_out l)).
#S_in #S_out #dec #m_out #f_in #f_out #start_in #start_out #end_in #end_out
#tal_in #tal_out #H_tal
@(lift_cost_map_same_cost … (tal_rel_to_traces_same_flatten … H_tal))
qed.

lemma lift_cost_map_same_cost_tll :
  ∀S_in, S_out, dec, m_out, f_in, f_out, start_in, start_out, end_in, end_out.
  ∀the_trace_in : trace_label_label S_in f_in start_in end_in.
  ∀the_trace_out : trace_label_label S_out f_out start_out end_out.
  tll_rel … the_trace_in the_trace_out →
  (Σ_{l ∈ flatten_trace_label_label … the_trace_in}
    (lift_cost_map_id … dec m_out l)) =
  (Σ_{l ∈ flatten_trace_label_label … the_trace_out} (m_out l)).
#S_in #S_out #dec #m_out #f_in #f_out #start_in #start_out #end_in #end_out
#tll_in #tll_out #H_tll
@(lift_cost_map_same_cost … (tll_rel_to_traces_same_flatten … H_tll))
qed.

lemma lift_cost_map_same_cost_tlr :
  ∀S_in, S_out, dec, m_out, start_in, start_out, end_in, end_out.
  ∀the_trace_in : trace_label_return S_in start_in end_in.
  ∀the_trace_out : trace_label_return S_out start_out end_out.
  tlr_rel … the_trace_in the_trace_out →
  (Σ_{l ∈ flatten_trace_label_return … the_trace_in}
    (lift_cost_map_id … dec m_out l)) =
  (Σ_{l ∈ flatten_trace_label_return … the_trace_out} (m_out l)).
#S_in #S_out #dec #m_out #start_in #start_out #end_in #end_out
#tlr_in #tlr_out #H_tlr
@(lift_cost_map_same_cost … (tlr_rel_to_traces_same_flatten … H_tlr))
qed.