source: src/Clight/SwitchAndLabel.ma @ 3195

include "Clight/switchRemoval.ma".
include "Clight/labelSimulation.ma".

(* Combine the results about switch removal and labelling to provide a single
   result linking the input Clight program with the cost labelled Clight
   program. *)

(* First lift the step result for labelling to a finite run of steps. *)

lemma steps_with_labels : ∀ge,ge'.
  related_globals_gen … label_fundef ge ge' →
  ∀n,s1,s1',tr,s2.
  state_with_labels s1 s1' →
  after_n_steps (S n) … clight_exec ge s1 (λ_.true) (λtr',s'.〈tr',s'〉 = 〈tr,s2〉) →
   ∃m. after_n_steps (S m) … clight_exec ge' s1' (λ_.true)
   (λtr',s2'. trace_with_extra_labels tr tr' ∧
              state_with_labels s2 s2').
#ge #ge' #RG
#n elim n
[ #s1 #s1' #tr #s2 #SWL #AFTER
  cases (after_1_step … AFTER) #tr' * #s' * * #F1 #ST1 #E destruct
  @(step_with_labels … RG … ST1) //
| #n #IH #s1 #s1' #tr #s2 #SWL #AFTER
  cases (after_1_of_n_steps' … AFTER) #tr1 * #tr2 * #s' * * * #F1 #ST1 #E #AFTER'
  destruct 
  cases (step_with_labels … RG … SWL ST1) #m1 #AF1
  cases (after_n_result … AF1) #tr2 * #s2' * * #TWEL #SWL' #AF''
  cases (IH … SWL' AFTER')
  #m2 #AF2
  %{(m1+S m2)}
  @(after_n_m (S m1) (S m2) … AF2) //
  @(after_n_covariant … AF'')
  
  #tr #s #E destruct %{(refl ??)}
  #tr'' #s'' * #TWEL #SWL % /2/
] qed.

lemma interactive_switch_step : ∀ge,ge'.
  related_globals … fundef_switch_removal ge ge' →
  ∀s1,s1',o,k.
  exec_step ge s1 = Interact … o k →
  switch_state_sim ge s1 s1' →
  ∃k'.
      exec_step ge' s1' = Interact … o k' ∧
  ∀i. ∃tr,s2.
    k i = Value … 〈tr,s2〉 ∧
    ∃s2'.
      k' i = Value … 〈tr,s2'〉 ∧
      switch_state_sim ge s2 s2'.
#ge #ge' #RG #s1 #s1' #o #k #EX
cases (exec_step_interaction … EX)
#vf * #fn * #argtys * #retty * #vargs * #k' * #m #E
destruct
#S inversion S
[1,3,4: #H1 #H2 #H3 #H4 destruct
  #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 destruct
  #H13 #H14 #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 #H29 destruct
]
#s_vf #s_fd #s_args #s_k #s_k_ext #s_m #s_m_ext #s_writeable #s_me #s_H
#E1 #E2 #E3 destruct
whd in EX:(??%?);
@(bindIO_res_interact ?????????? EX) -EX
#vs #CHECK #EX whd in EX:(??%?); destruct
% [2: %
[ whd in ⊢ (??%?);
  >CHECK in ⊢ (??%?);
  whd in ⊢ (??%?);
  @refl
| #i
  % [2: % [2: %
  [ @refl
  | % [2: %
    [ @refl
    | @(sws_returnstate … s_me) // ]
    | skip ] ] | skip ]
  ] ]| skip
] qed.

lemma Value_eq_l : ∀tr,s,tr',s'.
  Value io_out io_in (trace×state) 〈tr,s〉 = Value io_out io_in (trace×state) 〈tr',s'〉 →
  tr = tr'.
#tr #s #tr' #s' #E destruct %
qed.

lemma Value_eq_r : ∀tr,s,tr',s'.
  Value io_out io_in (trace×state) 〈tr,s〉 = Value io_out io_in (trace×state) 〈tr',s'〉 →
  s = s'.
#tr #s #tr' #s' #E destruct %
qed.

(* Now build the coinduction simulation, taking one step to many after switch
   removal, then use steps_with_labels to take those to the cost labelled
   version. *)

let corec build_exec
  (ge1,ge2,ge3:genv)
  (SRG:related_globals … fundef_switch_removal ge1 ge2)
  (RG:related_globals_gen … label_fundef ge2 ge3)
  (s1,s2,s3:state)
  (S1:switch_state_sim ge1 s1 s2)
  (S2:state_with_labels s2 s3)
  (NW:not_wrong state (exec_inf_aux … clight_fullexec ge1 (exec_step ge1 s1)))
: sim_with_labels (exec_inf_aux … clight_fullexec ge1 (exec_step ge1 s1)) (exec_inf_aux … clight_fullexec ge3 (exec_step ge3 s3)) ≝ 
match NW return λe,NW. exec_inf_aux … clight_fullexec ?? = e → sim_with_labels e ? with
[ nw_stop tr i s ⇒ ?
| nw_step tr1 s1' e1 NW1 ⇒ ?
| nw_interact o k NWk ⇒ ?
] (refl ? (exec_inf_aux … clight_fullexec ge1 (exec_step ge1 s1))).
[ #E1
  cases (exec_inf_stops_at_final ?? clight_fullexec … E1)
  #EX1 #F1
  cases (switch_removal_correction … SRG … S1 EX1)
  #n1 #A1
  cases (after_n_result … A1) #tr1' * #s1' * * #E destruct #S1' #A1'
  cases (steps_with_labels … RG … S2 A1')
  #n2 #A2 cases (after_inv clight_fullexec ????? A2) #tr' * #s' * * #TWL #S'
  lapply F1 whd in ⊢ (??%? → ?); >(switch_final_related … S1') >(final_related … S') #F3
    whd in match (is_final … s'); >F3 *
    #tr2' #S2
    @(swl_stop … S2 TWL)
| #E1
  cases (extract_step ?? clight_fullexec … E1)
  #EX1 #E1'
  cases (switch_removal_correction … SRG … S1 EX1)
  #n1 #A1
  cases (after_n_result … A1) #tr1' * #s2' * * #E @(match E with [refl ⇒ ?]) #S1' #A1'
  cases (steps_with_labels … RG … S2 A1')
  #n #AF cases (after_inv clight_fullexec ????? AF) #tr' * #s' * * #TWL #S'
  whd in match (is_final … s'); lapply (refl ? (is_final s'))
  cases (is_final s') in ⊢ (???% → %);
  [ #NF #H whd in H; @(swl_steps … H) //
    @(match sym_eq … E1' return λe,E1'. sim_with_labels e ? with [refl ⇒ ?])
    @(build_exec … SRG RG … S1' S') >E1' in NW1; //
  | #r <(final_related … S') <(switch_final_related … S1') change with (is_final … clight_fullexec ge1 s1') in ⊢ (??%? → ?);
    >(exec_e_step_not_final ?? clight_fullexec … E1) #E' destruct
  ]
| #E1 
  cases (extract_interact ?? clight_fullexec … E1)
  #k' * #EX1 #Ek lapply NWk -NWk @(match sym_eq … Ek with [refl ⇒ ?]) #NWk
  cases (interactive_switch_step … SRG … EX1 S1) #k2' * #EX2 #H2
  cases (interactive_step_with_labels … RG … EX2 S2)
  #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 ⊢ (??%);
  @swl_interact
  #i cases (H3 i) #tr2 * #s2' * #K2 * #tr3 * #s3' * * #K3 #TR2 #S2'
  cases (H2 i) #tr1 * #s1' * #K1 * #s2x' * @(match sym_eq … K2 with [refl ⇒ ?]) #Ex
  @(match (Value_eq_r … Ex) with [refl ⇒ ?]) #S1'
  lapply (NWk i)
  @(match sym_eq … K1 with [refl ⇒ ?]) @(match sym_eq … (exec_inf_aux_unfold …) with [refl ⇒ ?]) whd in ⊢ (??% → ?%%); whd in match (is_final ?????);
  @(match sym_eq … (switch_final_related … S1') with [refl ⇒ ?]) @(match sym_eq … (final_related … S2') with [refl ⇒ ?])
  @(match sym_eq … K2 with [refl ⇒ ?]) @(match sym_eq … (exec_inf_aux_unfold ?? clight_fullexec ge3 …) with [refl ⇒ ?]) whd in ⊢ (? → ??%);
  @(match sym_eq … K3 with [refl ⇒ ?]) whd in ⊢ (? → ??%); whd in match (is_final ?????);
  cases (is_final s3')
  [ whd in ⊢ (??% → ?%%); #NW' @(swl_steps … (steps_step …))
    [ destruct //
    | @build_exec //
      inversion NW'
      [ #H1 #H2 #H3 #H4 #H5 destruct
      | #H7 #H8 #H9 #H10 #H11 #H12 destruct //
      | #H14 #H15 #H16 #H17 #H18 destruct
      ]
    ]
  | #r whd in ⊢ (??% → ?%%); #NW' @(swl_stop … (steps_stop …)) destruct //
  ]
] qed.

(* And deal with the start of the program to tie up everything nicely. *)

theorem switch_and_labelling_sim : ∀p.
  let e1 ≝ exec_inf … clight_fullexec p in
  let e2 ≝ exec_inf … clight_fullexec (\fst (clight_label (program_switch_removal p))) in
  not_wrong state e1 →
  sim_with_labels e1 e2.
#p letin p' ≝ (program_switch_removal p)
#NW
cases (not_wrong_init clight_fullexec p NW)
whd in ⊢ (% → ??%? → ?); #s1 #I1
cases (initial_state_in_switch_simulation … I1)
#s' * * change with p' in match (program_switch_removal p); #I' #RG' #S'
cases (initial_state_in_simulation … I')
#s2 * #I2

whd in match (exec_inf ?? clight_fullexec p) in NW ⊢ %;
letin ge1 ≝ (make_global ?? clight_fullexec p) in NW ⊢ %;
change with (make_initial_state p) in match (make_initial_state ?? clight_fullexec p);
>I1

whd in match (exec_inf ????);
letin ge2 ≝ (make_global ?? clight_fullexec (\fst (clight_label p')))
change with (make_initial_state (\fst (clight_label p'))) in match (make_initial_state ?? clight_fullexec (\fst (clight_label p')));
>I2

whd in ⊢ (??% → ? → ?%%);
>exec_inf_aux_unfold
>exec_inf_aux_unfold
whd in ⊢ (??% → ? → ?%%);
whd in match (is_final ?????);
>(initial_state_is_not_final … I1)
whd in match (is_final ?????);
>(initial_state_is_not_final … I2)
whd in ⊢ (??% → ? → ?%%);

#NW #S
@(swl_steps E0 E0)
[ 2: @steps_step | skip | // | @(build_exec … RG' … S' S)
  [ whd in match ge2; >unfold_clight_label
    @transform_program_gen_related
  | inversion NW
    [ #tr #i #s #E1 #E2 destruct
    | #trX #sX #eX #NWX #E1X #E2X destruct //
    | #H1 #H2 #H3 #H4 #H5 destruct
    ]
  ]
] qed.