source: LTS/Language.ma @ 3447

(**************************************************************************)
(*       ___                                                              *)
(*      ||M||                                                             *)
(*      ||A||       A project by Andrea Asperti                           *)
(*      ||T||                                                             *)
(*      ||I||       Developers:                                           *)
(*      ||T||         The HELM team.                                      *)
(*      ||A||         http://helm.cs.unibo.it                             *)
(*      \   /                                                             *)
(*       \ /        This file is distributed under the terms of the       *)
(*        v         GNU General Public License Version 2                  *)
(*                                                                        *)
(**************************************************************************)

include "basics/types.ma".
include "Traces.ma".
include "basics/lists/list.ma".
include "../src/utilities/option.ma".
include "basics/jmeq.ma".

discriminator option.

record instr_params : Type[1] ≝ 
{ seq_instr : DeqSet
; io_instr : DeqSet
; cond_instr : DeqSet
; loop_instr : DeqSet
; act_params_type : DeqSet
; return_type : DeqSet
}.


inductive Instructions (p : instr_params) : Type[0] ≝
 | EMPTY : Instructions p
 | RETURN : return_type p → Instructions p
 | SEQ : (seq_instr p) → option NonFunctionalLabel → Instructions p → Instructions p
 | COND : (cond_instr p) → NonFunctionalLabel → Instructions p → 
                 NonFunctionalLabel → Instructions p → Instructions p → 
                       Instructions p
 | LOOP : (loop_instr p) → NonFunctionalLabel → Instructions p → 
                  NonFunctionalLabel → Instructions p → Instructions p
 | CALL : FunctionName → (act_params_type p) → option ReturnPostCostLabel → 
            Instructions p → Instructions p
 | IO : NonFunctionalLabel → (io_instr p) → NonFunctionalLabel → Instructions p → 
             Instructions p.

let rec eq_instructions (p : instr_params) (i : Instructions p) 
 on i : (Instructions p) → bool ≝ 
match i with 
[ EMPTY ⇒ λi'.match i' with [ EMPTY ⇒ true | _ ⇒ false ]
| RETURN x ⇒ λi'.match i' with [ RETURN y ⇒ x == y | _ ⇒ false ]
| SEQ x lab instr ⇒ λi'.match i' with 
                      [ SEQ y lab' instr' ⇒ x == y ∧ eq_instructions … instr instr' ∧
                              match lab with [ None ⇒ match lab' with [ None ⇒ true | _ ⇒ false ]
                                             | Some l1 ⇒ match lab' with [Some l2 ⇒ eq_nf_label l1 l2 | _ ⇒ false]
                                             ]
                      | _ ⇒ false
                      ]
| COND x ltrue i1 lfalse i2 i3 ⇒ λi'.match i' with
                          [ COND y ltrue' i1' lfalse' i2' i3' ⇒ 
                             x == y ∧ eq_nf_label ltrue ltrue' ∧ 
                             eq_instructions … i1 i1' ∧ eq_nf_label lfalse lfalse' ∧
                             eq_instructions … i2 i2' ∧ eq_instructions … i3 i3'
                         | _ ⇒ false
                         ]
| LOOP x ltrue i1 lfalse i2 ⇒ λi'.match i' with
              [ LOOP y ltrue' i1' lfalse' i2' ⇒ x == y ∧ 
                      eq_instructions … i1 i1' ∧ eq_nf_label ltrue ltrue' ∧
                      eq_instructions … i2 i2'
              | _ ⇒ false
              ]
| CALL f act_p r_lb i1 ⇒ λi'.match i' with
             [ CALL f' act_p' r_lb' i1' ⇒ eq_function_name f f' ∧ 
                       act_p == act_p' ∧ eq_instructions … i1 i1' ∧ 
                       match r_lb with [ None ⇒ match r_lb' with [None ⇒ true | _ ⇒ false]
                                       | Some z ⇒ match r_lb' with [Some w ⇒  eq_return_cost_lab z w | _ ⇒ false ]
                                       ]
            | _ ⇒ false
            ]
| IO lin io lout i1 ⇒ λi'.match i' with
             [ IO lin' io' lout' i1' ⇒ eq_nf_label lin lin' ∧ io == io' ∧
                                       eq_nf_label lout lout' ∧ eq_instructions … i1 i1'
             | _ ⇒ false                                       
             ]
].

(*
lemma eq_instructions_elim : ∀ P : bool → Prop.∀p,i1,i2.(i1 = i2 → P true) → 
(i1 ≠ i2 → P false) → P (eq_instructions p i1 i2).
#P #p #i1 elim i1
[* normalize [/2/ ] #x [|*: #y #z [2,3: #w1 #w2 [#w3] |4,5: #w1]] #_ #H2 @H2 % #EQ
  lapply (eq_to_jmeq ??? EQ) #EQ' destruct(EQ')
| #rt * normalize [2: #rt' cases (dec_eq … rt rt') #H [>(\b H) | >(\bf H) ] /2/
  #_ #K @K % #abs lapply (eq_to_jmeq ??? abs) #abs' destruct(abs') @(absurd ?? H) //]
  [|*: #x #y #z [2,3: #w1 #w2 [#w3] |4,5: #w1]] #_ #H2 @H2 % #EQ
  lapply (eq_to_jmeq ??? EQ) #EQ' destruct(EQ')
| cases daemon (*TODO*)
qed.
*)

record env_params : Type[1] ≝ 
{ form_params_type : Type[0]
}.

record signature (p : env_params) (p' : instr_params) : Type[0] ≝
{ f_name : FunctionName
; f_pars : form_params_type p
; f_ret : return_type p'
}.

record env_item (p : env_params) (p' : instr_params) : Type[0] ≝
{ f_sig :> signature p p'
; f_lab : CallCostLabel
; f_body : Instructions p'
}.

record state_params : Type[1] ≝ 
{ i_pars :> instr_params
; e_pars :> env_params
; store_type : DeqSet
}.

record state (p : state_params) : Type[0] ≝ 
{ code : Instructions p
; cont : list (ActionLabel × (Instructions p))
; store : store_type p
; io_info : bool
}.

definition is_io : ∀p.state p → Prop ≝ λp,st.io_info … st = true.

record sem_state_params (p : state_params) : Type[0] ≝ 
{ eval_seq : seq_instr p → (store_type p) → option (store_type p)
; eval_io : io_instr p → store_type p → option (store_type p)
; eval_cond_cond : cond_instr p → store_type p → option (bool × (store_type p))
; eval_loop_cond : loop_instr p → store_type p → option (bool × (store_type p))
; eval_call : signature p p → act_params_type p → store_type p → option (store_type p)
; eval_after_return : return_type p → store_type p → option (store_type p)
; init_store : store_type p
}.


let rec lookup (p : env_params) (p' : instr_params) (l : list (env_item p p')) 
 on l : FunctionName → option (env_item p p') ≝ 
match l with
[ nil ⇒ λ_.None ?
| cons x xs ⇒  λf.if (eq_function_name f (f_name … x)) 
                  then Some ? x
                  else lookup … xs f
].

definition is_ret_act : ActionLabel → Prop ≝ 
λa.match a with [ret_act _ ⇒ True | _ ⇒ False ].

inductive execute_l (p : state_params) (p' : sem_state_params p) (env : list (env_item p p)) : 
                                         ActionLabel → relation (state p) ≝ 
| empty : ∀st,st',hd,tl.(code ? st) = (EMPTY p)→ (cont ? st) = hd :: tl → 
           (code ? st') = \snd hd → (cont … st') = tl → (store … st) = (store … st') → 
           (io_info … st = true → is_non_silent_cost_act (\fst hd)) →  (io_info … st') = false → ¬ is_ret_act (\fst hd) →  execute_l … (\fst hd) st st'
| seq_sil : ∀st,st',i,cd,s,opt_l.(code ? st) = SEQ … i opt_l cd → 
             eval_seq … p' i (store … st) = return s → (code ? st') = cd → 
             (cont … st) = (cont … st') → (store … st') = s → 
             io_info … st = false →  io_info ? st' = false → execute_l … (cost_act opt_l) st st'
| cond_true : ∀st,st',exp,ltrue,i_true,lfalse,i_false,cd,new_m.
   (code ? st) = COND … exp ltrue i_true lfalse i_false cd → eval_cond_cond … p' exp (store … st) = return 〈true,new_m〉 → 
   cont ? st' = 〈cost_act (None ?),cd〉 ::(cont … st) → code … st' = i_true → store … st' = new_m → 
   io_info … st = false →  io_info … st' = false → execute_l … (cost_act (Some ? ltrue)) st st'
| cond_false : ∀st,st',exp,ltrue,i_true,lfalse,i_false,cd,new_m.
   (code ? st) = COND … exp ltrue i_true lfalse i_false cd → eval_cond_cond … p' exp (store … st) = return 〈false,new_m〉 → 
   cont ? st' = 〈cost_act (None ?),cd〉 ::(cont … st) → code … st' = i_false → store … st' = new_m → 
   io_info … st = false →  io_info … st' = false → execute_l … (cost_act (Some ? lfalse)) st st'
| loop_true : ∀st,st',exp,ltrue,i_true,lfalse,i_false,new_m.
   code ? st = LOOP … exp ltrue i_true lfalse i_false → eval_loop_cond … p' exp (store … st) = return 〈true,new_m〉 → 
   cont ? st' = 〈cost_act (None ?),LOOP … exp ltrue i_true lfalse i_false〉 :: (cont … st) → 
   code … st' = i_true → store … st' = new_m → io_info … st = false →  io_info … st' = false → 
   execute_l … (cost_act (Some ? ltrue)) st st'
| loop_false : ∀st,st',exp,ltrue,i_true,lfalse,i_false,new_m.
   code ? st = LOOP … exp ltrue i_true lfalse i_false → eval_loop_cond … p' exp (store … st) = return 〈false,new_m〉 → 
   cont ? st' = cont … st → code … st' = i_false → store … st' = new_m → 
   io_info … st = false →  io_info … st' = false → execute_l … (cost_act (Some ? lfalse)) st st'
| io_in : ∀st,st',lin,io,lout,cd,mem.(code ? st) = IO … lin io lout cd → 
    eval_io … p' io (store … st) = return mem → code ? st' = EMPTY p → 
    cont … st' = 〈cost_act (Some ? lout),cd〉 :: (cont … st) → store … st' = mem →
    io_info … st' = true → execute_l … (cost_act (Some ? lin)) st st'
| call : ∀st,st',f,act_p,r_lb,cd,mem,env_it.(code ? st) = CALL … f act_p r_lb cd → 
    lookup … env f = return env_it → 
    eval_call ? p' env_it act_p (store … st) = return mem → 
    store ? st' = mem → code … st' = f_body … env_it → 
     cont … st' = 
       〈(ret_act r_lb),cd〉 :: (cont … st) →  
    io_info … st = false →  (io_info … st') = false → 
    execute_l … (call_act f (f_lab ?? env_it)) st st'
| ret_instr : ∀st,st',r_t,mem,tl',rb,cd.code ? st = RETURN … r_t → 
   cont … st = 〈ret_act rb,cd〉 :: tl' → cont ? st' = tl' → 
   io_info … st = false →  io_info ? st' = false → 
   eval_after_return … p' r_t (store … st) = return mem → code … st' = cd → 
   store … st' = mem → execute_l … (ret_act rb) st st'.
   
let rec get_labels_of_code (p : instr_params) (i : Instructions p) on i : list CostLabel ≝ 
match i with
[ EMPTY ⇒ [ ]
| RETURN x ⇒ [ ]
| SEQ x lab instr ⇒ let ih ≝ get_labels_of_code … instr in
  match lab with [ None ⇒ ih | Some lbl ⇒ a_non_functional_label lbl :: ih ]
| COND x ltrue i1 lfalse i2 i3 ⇒ 
   let ih3 ≝ get_labels_of_code … i3 in
   let ih2 ≝ get_labels_of_code … i2 in
   let ih1 ≝ get_labels_of_code … i1 in
   ltrue :: lfalse :: (ih1 @ ih2 @ih3)
| LOOP x ltrue i1 lfalse i2 ⇒ 
   let ih2 ≝ get_labels_of_code … i2 in
   let ih1 ≝ get_labels_of_code … i1 in
   a_non_functional_label ltrue :: a_non_functional_label lfalse :: (ih1 @ ih2)
| CALL f act_p r_lb i1 ⇒ 
   let ih1 ≝ get_labels_of_code … i1 in
   match r_lb with [ None ⇒ ih1 | Some lbl ⇒ a_return_post lbl :: ih1]
| IO lin io lout i1 ⇒ 
   let ih1 ≝ get_labels_of_code … i1 in
   a_non_functional_label lin :: a_non_functional_label lout :: ih1
].

include "basics/lists/listb.ma".
include "../src/utilities/hide.ma".

let rec no_duplicates (A : DeqSet) (l : list A) on l : Prop ≝ 
match l with
[ nil ⇒ True 
| cons x xs ⇒ ¬ (bool_to_Prop (x ∈ xs)) ∧ no_duplicates … xs
].

lemma no_duplicates_append_r : ∀A : DeqSet.∀l1,l2 : list A.no_duplicates … (l1 @ l2) → 
no_duplicates … l2.
#A #l1 elim l1 // #x #xs normalize #IH #l2 * /2/
qed.

lemma no_duplicates_append_l : ∀A : DeqSet.∀l1,l2 : list A.no_duplicates … (l1 @ l2) → 
no_duplicates … l1.
#A #l1 elim l1 // #x #xs normalize #IH #l2 * #H1 #H2 % [2: /2/ ]
inversion(x ∈ xs @l2) in H1; normalize [ #_ * #H @⊥ @H %] #H1 #_
% inversion(x ∈ xs) normalize [2: //] #H3 #_ >(memb_append_l1 … H3) in H1;
#EQ destruct(EQ)
qed.
   
record Program (p : env_params) (p' : instr_params) : Type[0] ≝ 
{ env : list (env_item p p')
; main : Instructions p'
}.


definition no_duplicates_labels : ∀p,p'.Program p p' → Prop ≝ 
λp,p',prog.
   no_duplicates …
    (foldr … (λitem,acc.((a_call (f_lab … item)) :: get_labels_of_code … (f_body … item)) @ acc) (get_labels_of_code … (main … prog)) (env … prog)).

lemma no_duplicates_domain_of_fun:
 ∀p,p',prog.no_duplicates_labels … prog → 
 ∀f,env_it.lookup p p' (env … prog) f = return env_it → 
 no_duplicates … (get_labels_of_code … (f_body … env_it)).
#p #p' * #env elim env [ #main normalize #_ #f #env_it #EQ destruct(EQ)]
#x #xs #IH #main whd in ⊢ (% → ?); whd in match (foldr ?????); #H #f #env_it
whd in ⊢ (??%? → ?); @eq_function_name_elim normalize nodelta
[ whd in ⊢ (? → ???% → ?); #EQ1 #EQ2 destruct(EQ1 EQ2) cases H #_ /2/ ]
#H1 #EQenv_it @IH cases H /2/
qed.


definition is_synt_succ : ∀p.relation (state p) ≝ λp,s1,s2.cont … s1 = cont … s2 ∧
 match (code … s1) with
 [ CALL f act_p r_lb i1 ⇒ code … s2 = i1
 | _ ⇒ False
 ].

definition operational_semantics : ∀p : state_params.∀p'.Program p p → abstract_status ≝ 
λp,p',prog.mk_abstract_status 
                (state p) 
                (execute_l ? p' (env … prog)) 
                (is_synt_succ …)
                (λs.match (code … s) with 
                    [ COND _ _ _ _ _ _ ⇒ cl_jump 
                    | LOOP _ _ _ _ _ ⇒ cl_jump
                    | EMPTY ⇒ if io_info … s then cl_io else cl_other
                    | _ ⇒ cl_other
                    ]) 
                (λs.match (code … s) with
                    [CALL _ _ m _ ⇒ match m with [ Some _ ⇒ true | None ⇒ false ]
                    | _ ⇒ false
                    ]) 
                (λs.eq_instructions … (code … s) (main … prog) ∧ isnilb … (cont … s) ∧ store … s == init_store … p' ∧ io_info … s)
                (λs.match (cont … s) with
                    [ nil ⇒ match (code … s) with
                            [ EMPTY ⇒ true
                            | RETURN _ ⇒ true
                            | _ ⇒ false
                            ]
                    | _ ⇒ false
                    ])
                ???.
@hide_prf
[ #s1 #s2 #l #H #H1 inversion H1 #st #st'
 [ #hd #tl
 | #i #cd #s #opt_l 
 |3,4: #exp #ltrue #i_true #lfalse #i_false #cd #new_m 
 |5,6: #exp #ltrue #i_true #lfalse #ifalse #new_m
 | #lin #io #lout #cd #mem
 | #f #act_p #r_lb #cd #mem #env_it
 | #r_t #mem #tl #rb #cd
 ]
 #EQcode
 [ #EQ1 #EQ2 #EQ3 #EQ4 #x #EQ5 #H2 #EQ' #EQ6 #EQ7 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 #EQ10 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 #EQ10 
 ]
 #_ destruct >EQcode in H; normalize nodelta /2 by ex_intro/
 [ cases(io_info ??) normalize nodelta] #EQ destruct
| #s1 #s2 #l #H #H1 inversion H1 #st #st'
 [ #hd #tl
 | #i #cd #s #opt_l 
 |3,4: #exp #ltrue #i_true #lfalse #i_false #cd #new_m 
 |5,6: #exp #ltrue #i_true #lfalse #ifalse #new_m
 | #lin #io #lout #cd #mem
 | #f #act_p #r_lb #cd #mem #env_it
 | #r_t #mem #tl #rb #cd
 ]
 #EQcode
 [ #EQ1 #EQ2 #EQ3 #EQ4 #x #EQiost' #H2 #EQ' #EQ6 #EQ7 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQiost' #EQ7 #EQ8 #EQ9 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQiost' #EQ7 #EQ8 #EQ9 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQiost' #EQ7 #EQ8 #EQ9
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQiost' #EQ7 #EQ8 #EQ9 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQiost' #EQ7 #EQ8 #EQ9 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQiost' #EQ6 #EQ7 #EQ8 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQiost' #EQ8 #EQ9 #EQ10 
 | #EQ1 #EQ2 #EQ3 #EQiost' #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 #EQ10 
 ]
 #_ destruct
 cases(code ? st') in H; normalize nodelta >EQiost' normalize nodelta
 #eq destruct try (#eq1 destruct) try (#eq2 destruct) try (#eq3 destruct)
 try (#eq4 destruct) try (#eq5 destruct) try (#eq6 destruct) %{lin} %
| #s1 #s2 #l #H #H1 inversion H1 #st #st'
 [ #hd #tl
 | #i #cd #s #opt_l 
 |3,4: #exp #ltrue #i_true #lfalse #i_false #cd #new_m 
 |5,6: #exp #ltrue #i_true #lfalse #ifalse #new_m
 | #lin #io #lout #cd #mem
 | #f #act_p #r_lb #cd #mem #env_it
 | #r_t #mem #tl #rb #cd
 ]
 #EQcode
 [ #EQ1 #EQ2 #EQ3 #EQ4 #x #EQiost' #H2 #EQ' #EQ6 #EQ7 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQiost #EQiost' #EQ7 #EQ8 #EQ9 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQiost #EQiost' #EQ7 #EQ8 #EQ9 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQiost #EQiost' #EQ7 #EQ8 #EQ9
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQiost #EQiost' #EQ7 #EQ8 #EQ9 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQiost #EQiost' #EQ7 #EQ8 #EQ9 
 | #EQ1 #EQ2 #EQ3 #EQiost #EQiost' #EQ6 #EQ7 #EQ8 
 | #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQiost #EQiost' #EQ8 #EQ9 #EQ10 
 | #EQ1 #EQ2 #EQiost #EQiost' #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 #EQ10 
 ]
 #_ destruct >EQcode in H; normalize nodelta [|*: #EQ destruct]
 cases(io_info … st) in x; normalize nodelta [2: #_ #EQ destruct]
 #H3 #_ @H3 %
]
qed.

let rec eqb_list (D : DeqSet) (l1 : list D) on l1 : list D → bool ≝ 
match l1 with
[ nil ⇒ λl2.match l2 with [nil ⇒ true | _ ⇒ false ]
| cons x xs ⇒ λl2.match l2 with [ cons y ys ⇒ x == y ∧ eqb_list … xs ys | _ ⇒ false ]
]. 

definition DeqSet_List : DeqSet → DeqSet ≝ 
λX.mk_DeqSet (list X) (eqb_list …) ?.
#x elim x [ * /2 by refl, conj/ #y #ys normalize % #EQ destruct] -x
#x #xs #IH * [ normalize % #EQ destruct] #y #ys normalize % inversion(x == y)
#EQ normalize nodelta 
[ #H >(proj1 … (IH ys) H) @eq_f2 [2: %] @(proj1 … (eqb_true …)) assumption
| #EQ destruct
| #EQ1 destruct @(proj2 … (IH …)) %
| #EQ1 destruct <EQ @(proj2 … (eqb_true …)) %
]
qed.

unification hint  0 ≔ C; 
    X ≟ DeqSet_List C
(* ---------------------------------------- *) ⊢ 
    list C ≡ carr X.


unification hint  0 ≔ D,p1,p2; 
    X ≟ DeqSet_List D
(* ---------------------------------------- *) ⊢ 
    eqb_list D p1 p2 ≡ eqb X p1 p2.

definition associative_list : DeqSet → Type[0] → Type[0] ≝ 
λA,B.list (A × (list B)).

let rec update_list (A : DeqSet) (B : Type[0]) (l : associative_list A B) 
   on l : A → list B → associative_list A B ≝ 
λa,b.match l with 
    [ nil ⇒ [〈a,b〉]
    | cons x xs ⇒ if (a == (\fst x)) then 〈a,b〉 :: xs
                  else x :: (update_list … xs a b)  
    ].

let rec get_element (A :DeqSet) (B : Type[0]) (l: associative_list A B) on l : A → list B ≝ 
λa.match l with [ nil ⇒ nil ? 
                | cons x xs ⇒ if (a == \fst x) then \snd x else get_element … xs a 
                ].

let rec domain_of_associative_list (A :DeqSet) (B : Type[0]) (l: associative_list A B) on l : list A ≝ 
 match l with
  [ nil ⇒ [] 
  | cons x xs ⇒ \fst x :: domain_of_associative_list … xs 
  ].

lemma get_element_append_l:
 ∀A,B. ∀l1,l2: associative_list A B. ∀x.
  x ∈ domain_of_associative_list … l1 →
   get_element … (l1@l2) x = get_element … l1 x.
#A #B #l1 elim l1 normalize [ #l2 #x * ] #hd #tl #IH #l2 #x cases (dec_eq … x (\fst hd))
#H [ >(\b H) | >(\bf H) ] normalize /2/
qed.

lemma get_element_append_r:
 ∀A,B. ∀l1,l2: associative_list A B. ∀x.
  ¬ (bool_to_Prop (x ∈ domain_of_associative_list … l1)) →
   get_element ?? (l1@l2) x = get_element … l2 x.
#A #B #l1 elim l1 normalize [ #l2 #x // ] #hd #tl #IH #l2 #x cases (dec_eq … x (\fst hd))
#H [ >(\b H) | >(\bf H) ] normalize /2 by/ * #K cases (K I)
qed.

lemma get_element_append_l1 : 
 ∀A,B. ∀l1,l2: associative_list A B. ∀x.
  ¬ (bool_to_Prop (x ∈ domain_of_associative_list … l2)) →
   get_element ?? (l1@l2) x = get_element … l1 x.
#A #B #l1 elim l1 normalize [2: #x #xs #IH #l2 #a #H >IH // ]
#l2 elim l2 // #y #ys #IH #a normalize cases(a == \fst y) normalize
[ * #H @⊥ @H % ] #H @IH assumption
qed.

lemma get_element_append_r1 : 
 ∀A,B. ∀l1,l2: associative_list A B. ∀x.
  ¬ (bool_to_Prop (x ∈ domain_of_associative_list … l1)) →
   get_element ?? (l1@l2) x = get_element … l2 x.
#A #B #l1 elim l1 normalize // #x #xs #IH #l2 #a cases (?==?)
normalize [* #H cases H //] #H >IH normalize //
qed.

definition fresh_nf_label : ℕ → NonFunctionalLabel × ℕ ≝ 
λx.〈a_non_functional_label x,S x〉.

definition fresh_cc_labe : ℕ → CallCostLabel × ℕ ≝ 
λx.〈a_call_label x,S x〉.

definition fresh_rc_label : ℕ → ReturnPostCostLabel × ℕ ≝ 
λx.〈a_return_cost_label (S x),S x〉.

record call_post_info (p : instr_params) : Type[0] ≝ 
{ gen_labels : list CostLabel
; t_code : Instructions p
; fresh : ℕ
; lab_map : associative_list DEQCostLabel CostLabel
; lab_to_keep : list ReturnPostCostLabel
}.

let rec call_post_trans (p : instr_params) (i : Instructions p) (n : ℕ) on i : 
list CostLabel → call_post_info p ≝
λabs.
match i with
[ EMPTY ⇒ mk_call_post_info ? abs (EMPTY …) n (nil ?) (nil ?)
| RETURN x ⇒ mk_call_post_info ? abs (RETURN … x) n (nil ?) (nil ?)
| SEQ x lab instr ⇒
   let ih ≝ call_post_trans … instr n abs in
   match lab with 
   [ None ⇒ mk_call_post_info ? (gen_labels … ih) (SEQ … x (None ?) (t_code … ih))
             (fresh … ih) (lab_map … ih) (lab_to_keep … ih)
   | Some lbl ⇒ 
      mk_call_post_info ? (nil ?) (SEQ … x (Some ? lbl) (t_code …  ih)) (fresh … ih) 
      (〈a_non_functional_label lbl,(a_non_functional_label lbl :: (gen_labels … ih))〉 :: (lab_map … ih))
      (lab_to_keep … ih)
   ]
| COND x ltrue i1 lfalse i2 i3 ⇒ 
   let ih3 ≝ call_post_trans … i3 n abs in
   let ih2 ≝ call_post_trans … i2 (fresh … ih3) (gen_labels … ih3) in
   let ih1 ≝ call_post_trans … i1 (fresh … ih2) (gen_labels … ih3) in
   mk_call_post_info ? (nil ?) (COND … x ltrue (t_code … ih1) lfalse (t_code … ih2) (t_code … ih3))
    (fresh … ih1)  
    (〈a_non_functional_label ltrue,(a_non_functional_label ltrue :: (gen_labels … ih1))〉::
      〈a_non_functional_label lfalse,(a_non_functional_label lfalse :: (gen_labels … ih2))〉::
       ((lab_map … ih1) @ (lab_map …  ih2) @ (lab_map … ih3)))
    ((lab_to_keep … ih1) @ (lab_to_keep … ih2) @ (lab_to_keep … ih3))
| LOOP x ltrue i1 lfalse i2 ⇒ 
   let ih2 ≝ call_post_trans … i2 n abs in
   let ih1 ≝ call_post_trans … i1 (fresh … ih2) (nil ?) in
   mk_call_post_info ? (nil ?) (LOOP … x ltrue (t_code … ih1) lfalse (t_code … ih2)) (fresh … ih1) 
    (〈a_non_functional_label lfalse,(a_non_functional_label lfalse :: (gen_labels … ih2))〉 :: 
     〈a_non_functional_label ltrue,(a_non_functional_label ltrue :: (gen_labels … ih1))〉 :: 
      ((lab_map … ih1) @ (lab_map … ih2)))
    ((lab_to_keep … ih1) @ (lab_to_keep … ih2))
| CALL f act_p r_lb i1 ⇒ 
   let ih ≝ call_post_trans … i1 n abs in
   match r_lb with 
   [ None ⇒ let 〈l',f''〉 ≝ fresh_rc_label (fresh … ih) in
       mk_call_post_info ? ((a_return_post l')::(gen_labels … ih)) 
         (CALL … f act_p (Some ? l') (t_code … ih))  f'' (lab_map … ih) (lab_to_keep … ih)
   | Some lbl ⇒ 
      mk_call_post_info ? (nil ?) (CALL ? f act_p (Some ? lbl) (t_code … ih)) (fresh … ih) 
       (〈a_return_post lbl,(a_return_post lbl :: (gen_labels … ih))〉 :: (lab_map … ih)) 
       (lbl :: lab_to_keep … ih)
   ]
| IO lin io lout i1 ⇒ 
    let ih ≝ call_post_trans … i1 n abs in
    mk_call_post_info ? (nil ?) (IO ? lin io lout (t_code … ih)) (fresh … ih)
     (〈a_non_functional_label lout,(a_non_functional_label lout :: (gen_labels … ih))〉 ::
      〈a_non_functional_label lin,[a_non_functional_label lin]〉 :: (lab_map … ih)) (lab_to_keep … ih)
].


let rec call_post_clean (p : instr_params) (i : Instructions p) on i :
associative_list DEQCostLabel CostLabel → list ReturnPostCostLabel → list CostLabel → 
option ((list CostLabel) × (Instructions p)) ≝ 
λm,keep,abs.
 match i with
[ EMPTY ⇒ Some ? 〈abs,EMPTY …〉
| RETURN x ⇒ Some ? 〈abs,RETURN … x〉
| SEQ x lab instr ⇒
   ! 〈l,i1〉 ← call_post_clean … instr m keep abs;
   match lab with 
   [ None ⇒ return 〈l,SEQ … x (None ?) i1〉
   | Some lbl ⇒ if ((get_element … m lbl) == lbl :: l) 
                then return 〈nil ?,SEQ … x (Some ? lbl) i1〉
                else None ?
   ]
| COND x ltrue i1 lfalse i2 i3 ⇒ 
    ! 〈l3,instr3〉 ← call_post_clean … i3 m keep abs;
    ! 〈l2,instr2〉 ← call_post_clean … i2 m keep l3;
    ! 〈l1,instr1〉 ← call_post_clean … i1 m keep l3;
    if ((get_element … m ltrue) == ltrue :: l1) ∧ 
       ((get_element … m lfalse) == lfalse :: l2)
    then return 〈nil ?,COND … x ltrue instr1 lfalse instr2 instr3〉
    else None ?
| LOOP x ltrue i1 lfalse i2 ⇒ 
   ! 〈l2,instr2〉 ← call_post_clean … i2 m keep abs;
   ! 〈l1,instr1〉 ← call_post_clean … i1 m keep (nil ?);
   if ((get_element … m ltrue) == ltrue :: l1) ∧
      ((get_element … m lfalse) == lfalse :: l2)
   then return 〈nil ?,LOOP … x ltrue instr1 lfalse instr2〉
   else None ?
| CALL f act_p r_lb i1 ⇒ 
  ! 〈l1,instr1〉 ← call_post_clean … i1 m keep abs;
  match r_lb with
  [ None ⇒ None ?
  | Some lbl ⇒ if (lbl ∈ keep)
               then if ((get_element … m lbl) == lbl :: l1)
                    then return 〈nil ?,CALL … f act_p (Some ? lbl) instr1〉
                    else None ?
               else return 〈(a_return_post lbl) :: l1,CALL … f act_p (None ?) instr1〉 
  ]
| IO lin io lout i1 ⇒ 
   ! 〈l1,instr1〉 ← call_post_clean … i1 m keep abs;
   if ((get_element … m lout) == lout :: l1) ∧ ((get_element … m lin) == [lin])
   then return 〈nil ?,IO … lin io lout instr1〉
   else None ?   
].

let rec foldr2 (A : Type[0]) (B : Type[0]) (C : Type[0]) (a : A) (l1 : list B)
(l2 : list C) (f : A → B → C → A) on l1 : option A≝ 
match l1 with
[ nil ⇒ match l2 with [ nil ⇒ return a | cons y ys ⇒ None ? ]
| cons x xs ⇒ 
        match l2 with
        [ nil ⇒ None ?
        | cons y ys ⇒ ! ih ← (foldr2 … a xs ys f);
                      return f ih x y
        ]
].

definition is_silent_cost_act_b : ActionLabel → bool ≝ 
λact. match act with
 [ cost_act x ⇒ match x with [None ⇒ true | _ ⇒ false ] | _ ⇒ false].

definition eq_ActionLabel : ActionLabel → ActionLabel → bool ≝ 
λc1,c2.
match c1 with
[ call_act f l ⇒ match c2 with [ call_act f' l' ⇒ f == f' ∧ l == l' | _ ⇒ false]
| ret_act x ⇒ match c2 with [ret_act y ⇒ match x with [ None ⇒ match y with [ None ⇒ true | _ ⇒ false]
                                                      | Some l ⇒ match y with [ Some l' ⇒ l == l' | _ ⇒ false]
                                                      ]
                            | _ ⇒ false
                            ]
| cost_act x ⇒ match c2 with [cost_act y ⇒ match x with [ None ⇒ match y with [ None ⇒ true | _ ⇒ false]
                                                        | Some l ⇒ match y with [ Some l' ⇒ l == l' | _ ⇒ false]
                                                        ]
                             | _ ⇒ false
                             ]
| init_act ⇒ match c2 with [init_act ⇒ true | _ ⇒ false]
].

definition ret_costed_abs : list ReturnPostCostLabel → option ReturnPostCostLabel → option CostLabel ≝ 
λkeep,x.
 match x with 
              [ Some lbl ⇒ if lbl ∈ keep then return (a_return_post lbl)
                           else None ?
              | None ⇒ None ?
              ].


definition check_continuations : ∀p : instr_params.
∀l1,l2 : list (ActionLabel × (Instructions p)). 
associative_list DEQCostLabel CostLabel → 
list ReturnPostCostLabel →  option (Prop × (list CostLabel) × (list CostLabel)) ≝ 
λp,cont1,cont2,m,keep.
foldr2 ??? 〈True,nil ?,nil ?〉 cont1 cont2
 (λx,y,z.
   let 〈cond,abs_top',abs_tail'〉 ≝ x in
   match call_post_clean p (\snd z) m keep abs_top' with
   [ None ⇒ 〈False,nil ?,nil ?〉
   | Some w ⇒
      match \fst z with
       [ ret_act opt_x ⇒ 
           match ret_costed_abs keep opt_x with 
           [ Some lbl ⇒ 〈\fst y = \fst z ∧ cond ∧ \snd w = \snd y ∧ 
                               get_element … m lbl = lbl :: (\fst w),(nil ?),abs_tail'〉 
           | None ⇒ 
              〈\fst y = ret_act (None ?) ∧ cond ∧ \snd w = \snd y,(nil ?),(\fst w) @ abs_tail'〉
           ]
       | cost_act opt_x ⇒
           match opt_x with
           [ None ⇒ 〈\fst y = \fst z ∧ cond ∧ \snd w = \snd y,\fst w,abs_tail'〉
           | Some xx ⇒ 〈\fst y = \fst z ∧ cond ∧ \snd w = \snd y ∧ 
                               get_element … m xx = xx :: (\fst w),(nil ?),abs_tail'〉]
       | _ ⇒ (* dummy *) 〈False,nil ?,nil ?〉]]).

(*
definition check_continuations : ∀p : instr_params.
∀l1,l2 : list (ActionLabel × (Instructions p)). 
associative_list DEQCostLabel CostLabel → list CostLabel → list CostLabel → 
list (list CostLabel) →  option (Prop × (list CostLabel) × (list (list CostLabel))) ≝ 
λp,cont1,cont2,m,keep,abs_top,abs_tail.
foldr2 ??? 〈True,abs_top,abs_tail〉 cont1 cont2
 (λx,y,z.
   let 〈cond,abs_top',abs_tail'〉 ≝ x in
   match call_post_clean p (\snd z) m keep abs_top' with
   [ None ⇒ 〈False,nil ?,nil ?〉
   | Some w ⇒
      match \fst z with
       [ ret_act opt_x ⇒ 
           match ret_costed_abs keep opt_x with 
           [ Some lbl ⇒ 〈\fst y = \fst z ∧ cond ∧ \snd w = \snd y ∧ 
                               get_element … m lbl = lbl :: (\fst w),(nil ?),(nil ?) :: abs_tail'〉 
           | None ⇒ 
              〈\fst y = ret_act (None ?) ∧ cond ∧ \snd w = \snd y,(nil ?),(\fst w) :: abs_tail'〉
           ]
       | cost_act opt_x ⇒
           match opt_x with
           [ None ⇒ 〈\fst y = \fst z ∧ cond ∧ \snd w = \snd y,\fst w,abs_tail'〉
           | Some xx ⇒ 〈\fst y = \fst z ∧ cond ∧ \snd w = \snd y ∧ 
                               get_element … m xx = xx :: (\fst w),(nil ?),abs_tail'〉]
       | _ ⇒ (* dummy *) 〈False,nil ?,nil ?〉]]). *)
(* in input :
     abs_top is the list of labels to be propageted to the deepest level of the call stack
     abs_tail are the lists of labels to be propagated to the levels "below" the deepest level
   in output :
     abs_top is the list of labels to be propageted from the current level of the call stack
     abs_tail are the lists of labels to be propagated from the levels "below" the current level 
  *)       
       

definition state_rel : ∀p.
associative_list DEQCostLabel CostLabel → list ReturnPostCostLabel → list CostLabel →(list CostLabel) → 
relation (state p) ≝ λp,m,keep,abs_top,abs_tail,st1,st2.
match check_continuations ? (cont ? st1) (cont … st2) m keep with
[ Some x ⇒ let 〈prf1,abs_top',abs_tail'〉 ≝ x in
           prf1 ∧ call_post_clean … (code … st2) m keep abs_top' = return 〈abs_top,(code … st1)〉
           ∧ store … st1 = store … st2 ∧ io_info … st1 = io_info … st2 ∧ abs_tail = abs_tail'
| None ⇒ False
].

include "Simulation.ma".

let rec len (s : abstract_status) (st1 : s) (st2 : s) (t : raw_trace s st1 st2)
on t : ℕ ≝ 
match t with
[ t_base s ⇒ O
| t_ind s1 s2 s3 l prf lt ⇒ S (len … lt)
].
(*
lemma associative_max : ∀n1,n2,n3.max (max n1 n2) n3 = max n1 (max n2 n3).
#n1 #n2 #n3 normalize @leb_elim normalize
[ @leb_elim normalize
  [ #H1 #H2 @leb_elim normalize
    [ @leb_elim normalize // * #H @⊥ @H assumption  
    | @leb_elim normalize
      [ #H3 * #H @⊥ @H @(transitive_le … H1) assumption
      | * #H @⊥ @H assumption
      ]
    ]
  | #H1 #H2 @leb_elim normalize
    [ @leb_elim normalize // #_ #H cases H1 #H1 @⊥ @H1 assumption
    | @leb_elim normalize
      [ #_ * #H @⊥ @H assumption
      | * #H @⊥ @H @(transitive_le … H2) 
*)
let rec compute_max_n (p : instr_params) (i : Instructions p) on i : ℕ ≝ 
match i with 
[ EMPTY ⇒ O
| RETURN x ⇒ O
| SEQ x lab instr ⇒ let n ≝ compute_max_n … instr in
                    match lab with
                    [ None ⇒ n
                    | Some l ⇒ 
                        match l with 
                        [ a_non_functional_label n' ⇒ S (max n n') ]
                    ]
| COND x ltrue i1 lfalse i2 i3 ⇒ 
  let n1 ≝ compute_max_n … i1 in
  let n2 ≝ compute_max_n … i2 in
  let n3 ≝ compute_max_n … i3 in
  let mx ≝ max (max n1 n2) n3 in
  match ltrue with 
  [ a_non_functional_label lt ⇒ 
    match lfalse with
    [a_non_functional_label lf ⇒ S (max (max mx lt) lf) ] ]
| LOOP x ltrue i1 lfalse i2 ⇒ 
   let n1 ≝ compute_max_n … i1 in
   let n2 ≝ compute_max_n … i2 in
   let mx ≝ max n1 n2 in
   match ltrue with 
  [ a_non_functional_label lt ⇒ 
    match lfalse with
    [a_non_functional_label lf ⇒ S (max (max mx lt) lf) ] ]
| CALL f act_p r_lb i1 ⇒ 
   let n ≝ compute_max_n … i1 in
   match r_lb with 
   [ None ⇒ n
   | Some lbl ⇒ match lbl with [a_return_cost_label l ⇒ S(max l n) ]
   ]
| IO lin io lout i1 ⇒ 
  let n ≝ compute_max_n … i1 in
  match lin with
  [a_non_functional_label l1 ⇒ 
    match lout with
    [a_non_functional_label l2 ⇒ S(max (max n l1) l2) ] ]
].


definition same_fresh_map_on : list CostLabel → 
relation (associative_list DEQCostLabel CostLabel) ≝ 
λdom,m1,m2.∀x.bool_to_Prop (x ∈ dom) → get_element … m1 x = get_element … m2 x.

definition same_to_keep_on : list CostLabel → relation (list ReturnPostCostLabel) ≝ 
λdom,keep1,keep2.∀x. bool_to_Prop (a_return_post x ∈ dom) → (x ∈ keep1) = (x ∈ keep2).

lemma memb_not_append : ∀D : DeqSet.∀l1,l2 : list D.∀x : D.
x ∈ l1 = false → x ∈ l2 = false → x ∈ (l1 @ l2) = false.
#D #l1 elim l1
[ #l2 #x #_ #H @H ]
#x #xs #IH #l2 #x1 whd in match (memb ???); inversion (x1 == x) normalize nodelta
[ #_ #EQ destruct] #EQx1 #EQxs #EQl2 whd in match (memb ???); >EQx1
normalize nodelta @IH //
qed.

lemma memb_no_duplicates_append : ∀A : DeqSet.∀x.∀l1,l2 : list A .
no_duplicates … (l1 @ l2) → x ∈  l1 → x ∈ l2 → False.
#A #x #l1 elim l1 // #x1 #xs #IH #l2 * #H1 #H2 whd in match (memb ???);
inversion (x == x1) normalize nodelta 
[ #H3 #_ #H4 >memb_append_l2 in H1; [2: <(\P H3) @H4 ] * #H @H %
| #_ @IH //
]
qed.


lemma same_to_keep_on_append : ∀dom1,dom2,dom3 : list CostLabel.
∀l1,l2,l3,l : list ReturnPostCostLabel.
no_duplicates … (dom1@dom2@dom3) → (∀x.x ∈ l1 → a_return_post x ∈ dom1) →
(∀x.x ∈ l3 → a_return_post x ∈ dom3) → 
same_to_keep_on (dom1@dom2@dom3) (l1@l2@l3) l → 
same_to_keep_on dom2 l2 l.
#dom1 #dom2 #dom3 #l1 #l2 #l3 #l #no_dup #sub_set1 #sub_set3 #H2
#x #Hx inversion (x ∈ l2)
 [ #EQkeep <H2 [> memb_append_l2 // >memb_append_l1 // ]
   >memb_append_l2 // >memb_append_l1 // >Hx //
 | #EQno_keep <H2
   [2: >memb_append_l2 // >memb_append_l1 // >Hx //
   | @sym_eq @memb_not_append [2: @memb_not_append //]
   [ <associative_append in no_dup; #no_dup ]
   lapply(memb_no_duplicates_append … (a_return_post x) … no_dup) #H
   inversion(memb ???) // #H1 cases H 
   [1,4: [>memb_append_l2 | >memb_append_l1] // >Hx //
   | @sub_set3 >H1 //
   | @sub_set1 >H1 //
   ]
   ]
 ]
qed.

lemma same_fresh_map_on_append : ∀dom1,dom2,dom3,l1,l2,l3,l.
no_duplicates … (dom1 @dom2 @ dom3) → (∀x.x ∈ domain_of_associative_list … l1 → x ∈ dom1) →
(∀x.x ∈ domain_of_associative_list … l3 → x ∈ dom3) → 
same_fresh_map_on … (dom1 @dom2 @dom3) (l1 @l2 @ l3) l → 
same_fresh_map_on … dom2 l2 l.
#dom1 #dom2 #dom3 #l1 #l2 #l3 #l #no_dup #subset1 #subset3 whd in ⊢ (% → ?); #H1
whd #x #Hx <H1
[2: >memb_append_l2 // >memb_append_l1 // >Hx //]
>get_element_append_r [ >get_element_append_l1 // ] % #K
[ lapply(subset3 … K) | lapply(subset1 … K) ] #ABS
[ <associative_append in no_dup; #no_dup] @(memb_no_duplicates_append … x … no_dup)
// [>memb_append_l2 | >memb_append_l1 ] // >Hx //
qed.


lemma lab_to_keep_in_domain : ∀p.∀i : Instructions p.
∀x,n,l.
x ∈ lab_to_keep … (call_post_trans … i n l) → a_return_post x ∈ get_labels_of_code …  i.
#p #i elim i //
[ #seq #opt_l #instr #IH #x #n #l whd in match (call_post_trans ????);
  cases opt_l -opt_l normalize nodelta [|#lbl] 
  whd in match (get_labels_of_code ??); #H [2: @orb_Prop_r] /2/
| #cond #ltrue #i1 #lfalse #i2 #i3 #IH1 #IH2 #IH3 #x #n #l
  whd in match (call_post_trans ????); whd in match (get_labels_of_code ??);
  #H cases(memb_append … H) -H #H @orb_Prop_r @orb_Prop_r
  [ >memb_append_l1 // @IH1 [3: >H // |*: ]
  | >memb_append_l2 // cases(memb_append … H) -H #H
     [>memb_append_l1 // @IH2 [3: >H // |*: ]
     | >memb_append_l2 // @IH3 [3: >H // |*: ]
     ]
  ]
| #loop #ltrue #i1 #lfalse #i2 #IH1 #IH2 #x #n #l 
  whd in match (call_post_trans ????); whd in match (get_labels_of_code ??);
  #H cases(memb_append … H) -H #H @orb_Prop_r @orb_Prop_r
  [ >memb_append_l1 | >memb_append_l2 ] // [ @IH1 | @IH2 ] [3,6: >H |*: ] //
| #f #act_p * [|#lbl] #i1 #IH #x #n #l whd in match (call_post_trans ????);
  whd in match (get_labels_of_code ??); /2/ whd in match (memb ???);
  inversion(x == lbl) #Hlbl normalize nodelta 
  [*  >(\P Hlbl) @orb_Prop_l @eq_costlabel_elim // * #H @H %
  | #H @orb_Prop_r @IH //
  ]
| #lin #io #lout #i1 #IH #x #n #l whd in match (call_post_trans ????);
  whd in match (get_labels_of_code ??); #H @orb_Prop_r @orb_Prop_r @IH //
]
qed.

lemma domain_of_associative_list_append : ∀A,B.∀l1,l2 : associative_list A B.
domain_of_associative_list ?? (l1 @ l2) = 
 (domain_of_associative_list ?? l1) @ (domain_of_associative_list ?? l2).
#A #B #l1 elim l1 // #x #xs #IH #l2 normalize //
qed.

lemma lab_map_in_domain: ∀p.∀i: Instructions p.
 ∀x,n,l.
  x ∈ domain_of_associative_list … (lab_map p (call_post_trans … i n l)) →
   x ∈ get_labels_of_code … i.
#p #i elim i //
[ #seq * [|#lbl] #i1 #IH #x #n #l whd in match(call_post_trans ????);
  whd in match (get_labels_of_code ??); /2/ whd in match (memb ???);
  inversion(x==lbl) #Hlbl normalize nodelta [#_ whd in match (memb ???); >Hlbl % ]
  #H >memb_cons // @IH //
| #cond #ltrue #i1 #lfalse #i2 #i3 #IH1 #IH2 #IH3 #x #n #l 
  whd in match (call_post_trans ????); whd in match (memb ???);
  whd in match (get_labels_of_code ??); inversion(x == ltrue) #Hlbl normalize nodelta
  [ #_ whd in match (memb ???); >Hlbl % ] whd in match (memb ???);
  inversion(x == lfalse) #Hlbl1 normalize nodelta 
  [ #_ whd in match (memb ???); >Hlbl normalize nodelta whd in match (memb ???);
    >Hlbl1 % ] #H >memb_cons // >memb_cons // >domain_of_associative_list_append in H;
    #H cases(memb_append … H) [ #H1 >memb_append_l1 // @IH1 [3: >H1 // |*:] ]
    >domain_of_associative_list_append #H1 cases(memb_append … H1)
    #H2 >memb_append_l2 // [ >memb_append_l1 | >memb_append_l2 ] //
    [ @IH2 | @IH3] /2 by eq_true_to_b/
| #loop #ltrue #i1 #lfalse #i2 #IH1 #IH2 #x #n #l whd in match (call_post_trans ????);
  whd in match (get_labels_of_code ??); whd in match (memb ???); inversion(x == lfalse)
  #Hlfalse normalize nodelta [ #_ >memb_cons // whd in match (memb ???); >Hlfalse // ]
  whd in match (memb ???); inversion(x==ltrue) normalize nodelta #Hltrue
  [ #_ whd in match (memb ???); >Hltrue %] >domain_of_associative_list_append #H
  cases(memb_append … H) #H1 >memb_cons // >memb_cons // [ >memb_append_l1 | >memb_append_l2 ]
  // [ @IH1 | @IH2] /2/
| #f #act_p * [|#lbl] #i1 #IH #x #n #l whd in match (call_post_trans ????); 
  whd in match (get_labels_of_code ??); /2/ whd in match (memb ???); inversion (x == lbl)
  #Hlbl normalize nodelta [ #_ whd in match (memb ???); >Hlbl % ] #H >memb_cons // @IH /2/
| #lin #io #lout #i1 #IH #x #n #l whd in match (memb ???); inversion(x == lout) #Hlout
  normalize nodelta [ #_ >memb_cons // whd in match (memb ???); >Hlout % ]
  whd in match (memb ???); inversion(x==lin) #Hlin normalize nodelta
  [ #_ whd in match (memb ???); >Hlin % ] #H >memb_cons // >memb_cons // @IH /2/
] 
qed.

lemma eq_a_non_functional_label : 
∀c1,c2.c1 == c2 → a_non_functional_label c1 == a_non_functional_label c2.
#c1 #c2 #EQ cases(eqb_true DEQNonFunctionalLabel c1 c2) #H1 #_ lapply(H1 (eq_true_to_b … EQ)) -EQ
#EQ destruct >(\b (refl …)) @I
qed.

let rec is_fresh_for_return (keep : list CostLabel) (n : ℕ) on keep : Prop ≝
match keep with
[ nil ⇒ True
| cons x xs ⇒ let ih ≝ is_fresh_for_return xs n in
              match x with 
              [ a_return_post y ⇒ match y with [a_return_cost_label m ⇒ m ≤ n ∧ ih ]
              | _ ⇒ ih
              ]
].

lemma fresh_ok_call_post_trans : ∀p : instr_params.∀i1 : Instructions p.∀n : ℕ.∀l.
n ≤ fresh … (call_post_trans p i1 n l).
#p #i1 elim i1 normalize /2 by transitive_le, le_n/
[ #seq * [|#lbl] #i2 #IH #n #l normalize /2 by /
| #cond #ltrue #i_true #lfalse #i_false #i2 #IH1 #IH2 #IH3 #n #l
  @(transitive_le … (IH3 …)) [2: @(transitive_le … (IH2 …)) [2: @(transitive_le … (IH1 …)) ]] //
| #f #act_p * [|#lbl] normalize #i2 #IH /2 by le_S/
]
qed.

lemma fresh_keep_n_ok : ∀n,m,l.
is_fresh_for_return l n → n ≤ m → is_fresh_for_return l m.
#n #m #l lapply n -n lapply m -m elim l // ** #x #xs #IH #n #m
normalize [2: * #H1] #H2 #H3 [ %] /2 by transitive_le/
qed.

definition cast_return_to_cost_labels ≝ map … (a_return_post …).
coercion cast_return_to_cost_labels.

lemma fresh_false : ∀n.∀l: list ReturnPostCostLabel.is_fresh_for_return l n → 
a_return_cost_label (S n) ∈ l = false.
#n #l lapply n -n elim l // * #x #xs #IH #n whd in ⊢ (% → ??%?); * #H1 
#H2 @eq_return_cost_lab_elim
[ #EQ destruct @⊥ /2 by absurd/
| #_ >IH //
]
qed.

lemma inverse_call_post_trans : ∀p : instr_params.∀i1 : Instructions p.∀n : ℕ.
let dom ≝ get_labels_of_code … i1 in
no_duplicates … dom → 
∀m,keep.
∀info,l.call_post_trans p i1 n l = info → 
same_fresh_map_on dom (lab_map … info) m → 
same_to_keep_on dom (lab_to_keep … info) keep → 
is_fresh_for_return keep n → 
call_post_clean ? (t_code ? info) m keep l 
 = return 〈gen_labels … info,i1〉.
#p #i1 elim i1
[ #n #no_dup #m #keep * #gen_lab #t_code #fresh #lab_map #lab_keep #l whd in ⊢ (??%? → ?);
  #EQ destruct(EQ) //
| #x #n #no_dup #m #keep * #gen_lab #t_code #fresh #lab_map #lab_keep #l whd in ⊢ (??%? → ?);
  #EQ destruct(EQ) //
| #seq * [|#lbl] #instr #IH #n #no_dup #m #keep * #gen_lab #t_code #fresh #lab_map #lab_keep
  #l whd in ⊢ (??%? → ?); #EQ destruct(EQ) #H1 #H2 #H3 whd in ⊢ (??%%); normalize nodelta
  >IH //
  [1,4: whd whd in H2; #x #Hx @H2 whd in match (get_labels_of_code ??); //
  |2,5: whd whd in H1; #x #Hx [ @H1 // ] cases no_dup #H3 #_ <H1 
       [2: whd in match (get_labels_of_code ??); @orb_Prop_r // ]
       whd in ⊢ (???%); cases(eqb_true … x lbl) inversion(x == lbl) normalize nodelta
       [2: //] #_ #H4 >H4 in Hx; // #H5 >H5 in H3; * #ABS @⊥ @ABS %
  |6: cases no_dup //
  ]
  normalize nodelta <(H1 lbl) 
  [2: whd in match (get_labels_of_code ??); @orb_Prop_l cases(eqb_true … (a_non_functional_label lbl) lbl)
      #H3 #H4 >H4 % ]
  whd in match (get_element ????); >(\b (refl …)) normalize nodelta 
  >(\b (refl …)) %
| #cond #ltrue #i1 #lfalse #i2 #i3 #IH1 #IH2 #IH3 #n normalize nodelta 
  #no_dup #m #keep * #gen_lab #t_code #fresh #lab_map #lab_keep #l whd in ⊢ (??%? → ?);
  #EQ destruct(EQ) #H1 #H2 #H3 whd in ⊢ (??%?); >IH3 //
  [2: whd  in match (get_labels_of_code ??) in H2;
      change with ([?;?]@?) in match (?::?) in H2;
      <associative_append in H2; <associative_append
      <(append_nil … (?@?)) <associative_append in ⊢ (??%? → ?);
      <(append_nil … (?@?)) in ⊢ (??%? → ?); >associative_append
      >associative_append in ⊢ (??%? → ?); #H2
      @(same_to_keep_on_append … H2) // [ >append_nil
      whd in ⊢ (??%); whd in no_dup:(??%); >associative_append // ]
      #x #Hx cases (memb_append … Hx) -Hx #Hx @orb_Prop_r @orb_Prop_r
      [ >memb_append_l1 | >memb_append_l2 ] //
      @(lab_to_keep_in_domain … (eq_true_to_b … Hx))
  |3: -H2 whd in match (get_labels_of_code ??) in H1;
      change with ([?;?]@?) in match (?::?) in H1;
      <associative_append in H1; <associative_append      
      <(append_nil … (?@?)) >associative_append
      change with ([?;?]@?) in match (?::?::?) in ⊢ (??%? → ?);
      <associative_append in ⊢ (??%? → ?);
      <associative_append in ⊢ (??%? → ?);
      <(append_nil … (?@?)) in ⊢ (??%? → ?);
      >associative_append in ⊢ (??%? → ?); #H1
      @(same_fresh_map_on_append … H1) //
      [ >append_nil >associative_append // ]
      #x whd in match (memb ???); inversion(x == ltrue) 
      [ #Hltrue normalize nodelta #_ whd in match (memb ???); >Hltrue %
      | #Hltrue normalize nodelta whd in match (memb ???); inversion(x == lfalse)
         [ #Hlfalse #_ @orb_Prop_r @orb_Prop_l >Hlfalse %
         | #Hlfalse normalize nodelta #Hx @orb_Prop_r @orb_Prop_r
           >domain_of_associative_list_append in Hx; #H
           cases(memb_append … H) #H2 [ >memb_append_l1 | >memb_append_l2 ]
           // @(lab_map_in_domain … (eq_true_to_b … H2))
         ]
      ]
  |4: cases no_dup #_ * #_ #H @no_duplicates_append_r [2: @(no_duplicates_append_r … H) |]
  ]
  normalize nodelta >IH2 
  [5: %
  |2: /2 by fresh_keep_n_ok/
  |3: whd  in match (get_labels_of_code ??) in H2;
   change with ([?;?]@?) in match (?::?) in H2;
   <associative_append in H2; #H2
   @(same_to_keep_on_append … H2) // #x #Hx [ @orb_Prop_r @orb_Prop_r ]
   @(lab_to_keep_in_domain … (eq_true_to_b … Hx))
  |4: whd  in match (get_labels_of_code ??) in H1;
   change with ([?;?]@?) in match (?::?) in H1;
   change with ([?;?]@?) in match (?::?::?) in H1 : (??%?);
   <associative_append in H1; <associative_append in ⊢ (??%? → ?); #H1
   @(same_fresh_map_on_append … H1) // [2: /2 by lab_map_in_domain/ ]
   #x >domain_of_associative_list_append #H cases(memb_append … H)
   [ whd in ⊢ (??%? → ?%); cases(x == ltrue) // normalize nodelta
     whd in ⊢ (??%? → ?%); cases(x == lfalse) // normalize nodelta
     normalize #EQ destruct
   | #H1 @orb_Prop_r @orb_Prop_r
     @(lab_map_in_domain … (eq_true_to_b … H1))
   ]
  |6: cases no_dup #_ * #_ #K lapply (no_duplicates_append_r … K) @no_duplicates_append_l
  |*:
  ]
  >m_return_bind >IH1 
  [5: %
  |2: /3 by fresh_keep_n_ok/
  |3:  whd  in match (get_labels_of_code ??) in H2;
   change with ([?;?]@?) in match (?::?) in H2;
   change with ([ ] @ ?) in match (lab_to_keep ??) in H2;
   >associative_append in H2 : (??%?); #H2
   @(same_to_keep_on_append … H2) //  #x #Hx cases(memb_append … Hx)
   -Hx #Hx [ >memb_append_l1 | >memb_append_l2] //
   @(lab_to_keep_in_domain … (eq_true_to_b … Hx))
  |4:  whd  in match (get_labels_of_code ??) in H1;
   change with ([?;?]@?) in match (?::?) in H1;
   change with ([?;?]@?) in match (?::?::?) in H1 : (??%?);
    @(same_fresh_map_on_append … H1) // #x >domain_of_associative_list_append
    #Hx cases(memb_append … Hx) -Hx #Hx [ >memb_append_l1 | >memb_append_l2 ]
    // @(lab_map_in_domain … (eq_true_to_b … Hx))
  |6: cases no_dup #_ * #_ @no_duplicates_append_l
  |*:
  ]
  >m_return_bind normalize nodelta whd in H1; <H1 
  [2: whd in match (get_labels_of_code ??); whd in match (memb ???);
      >(\b (refl …)) % ] whd in match (get_element ????); >(\b (refl …))
      normalize nodelta >(\b (refl …)) <H1 
      [2: whd in match (get_labels_of_code ??); >memb_cons // 
      whd in match (memb ???); >(\b (refl …)) % ]
      whd in match (get_element ????); @eq_costlabel_elim normalize nodelta
      [ #ABS @⊥ cases no_dup >ABS * #H #_ @H @orb_Prop_l  
      >(\b (refl ? (a_non_functional_label ltrue))) % ] #_
      whd in match (get_element ????); >(\b (refl …)) normalize nodelta
      >(\b (refl …)) %
| #loop #ltrue #i1 #lfalse #i2 #IH1 #IH2 #n #no_dup #m #keep #info #l whd in ⊢ (??%? → ?);
  #EQ destruct(EQ) whd in match (get_labels_of_code ??); #fresh_map #keep_on #f_k
  whd in ⊢ (??%?); >(IH2 … (refl …)) 
  [ normalize nodelta >(IH1 … (refl …))
    [ >m_return_bind <fresh_map [2: @orb_Prop_l >(\b (refl …)) % ]
      whd in match (get_element ????);
      inversion(a_non_functional_label ltrue == a_non_functional_label lfalse) 
      #Hltrue normalize nodelta
      [ cases no_dup whd in match (memb ???); 
        cases(eqb_true … (a_non_functional_label ltrue) (a_non_functional_label lfalse))
        #H1 #_ lapply(H1 Hltrue) #EQ destruct(EQ) >(\b (refl …)) * #ABS @⊥ @ABS % ]
      whd in match (get_element ????); >(\b (refl …)) normalize nodelta >(\b (refl …))
      <fresh_map [2: @orb_Prop_r @orb_Prop_l >(\b (refl …)) % ] 
      whd in match (get_element ????); >(\b (refl …)) normalize nodelta
      >(\b (refl …)) %
    | /2 by fresh_keep_n_ok/
    | change with ([?;?]@?@?) in keep_on : (?%??); change with ((nil ?) @ ? @ ?) in keep_on : (??%?);
      @(same_to_keep_on_append … keep_on) // #x /2 by lab_to_keep_in_domain/ 
    | change with ([?;?]@?@?) in fresh_map : (?%%?); @(same_fresh_map_on_append … fresh_map)
      /2 by lab_map_in_domain/ #x whd in match (memb ???); inversion(x==lfalse) #Hlfalse
      normalize nodelta
      [ #_ @orb_Prop_r whd in match (memb ???); >Hlfalse %
      | whd in match (memb ???); inversion(x==ltrue) #Hltrue normalize nodelta [2: *] #_
        @orb_Prop_l >Hltrue %
      ]
    | cases no_dup #_ * #_ /2/
    ]
  | //
  | change with ([?;?]@?@?) in keep_on : (?%??); <associative_append in keep_on; 
    <(append_nil … (?@?)) <(append_nil … (?@?)) in ⊢ (??%? → ?); 
    >associative_append in ⊢ (?%%? → ?); >associative_append in ⊢ (??%? → ?); 
    #keep_on @(same_to_keep_on_append … keep_on) //
    [ >associative_append >append_nil //
    | #x #Hx @orb_Prop_r @orb_Prop_r /2 by lab_to_keep_in_domain/
    ]
  | change with ([?;?]@?@?) in fresh_map : (?%??); <associative_append in fresh_map; 
    <(append_nil … (?@?)) change with ([?;?]@?@?) in ⊢ (??%? → ?);
    <associative_append in ⊢ (??%? → ?); <(append_nil … (?@?)) in ⊢ (??%? → ?); 
    >associative_append in ⊢ (?%%? → ?); >associative_append in ⊢ (??%? → ?);
    #fresh_map @(same_fresh_map_on_append … fresh_map) //
    [ >append_nil //
    | #x >domain_of_associative_list_append #Hx cases(memb_append … Hx)
      [2: #Hx1 @orb_Prop_r @orb_Prop_r @(lab_map_in_domain … (eq_true_to_b … Hx1)) ]
      whd in match (memb ???); inversion(x == lfalse) normalize nodelta #Hlfalse
      [ #_ @orb_Prop_r @orb_Prop_l >Hlfalse %
      | whd in match (memb ???); inversion (x==ltrue) normalize nodelta #Hltrue
        [ #_ @orb_Prop_l >Hltrue %
        | whd in match (memb ???); #EQ destruct
        ]
      ]
    ]
  | cases no_dup #_ * #_ /2 by no_duplicates_append_r/
  ]
| #f #act_p * [|#r_lb] #i #IH #n #no_dup #m #keep #info #l whd in ⊢ (??%? → ?); 
  #EQ destruct(EQ) #fresh_map #same_keep #f_k whd in ⊢ (??%?);
  >(IH … (refl …))
  [1,6: normalize nodelta
     [ >fresh_false [2: /2 by fresh_keep_n_ok/] %
     | <same_keep 
       [ whd in match (memb ???); >(\b (refl …)) normalize nodelta
         <fresh_map
         [ whd in match (get_element ????); >(\b (refl …)) normalize nodelta
           >(\b (refl …)) %
         | whd in match (memb ???); >(\b (refl …)) %
         ]
       | whd in match (memb ???); >(\b (refl …)) %
       ]
    ]
  |2,7: //
  |3,8: whd in match (get_labels_of_code ??) in same_keep; // #x #Hx <same_keep
        [2: >memb_cons // >Hx // ] cases no_dup * #ABS #_ whd in ⊢ (???%);
        inversion(x==?) [2: #_ //] #ABS1 @⊥ @ABS <(\P ABS1) >Hx //
  |4,9: whd in match (get_labels_of_code ??) in fresh_map; // #x #Hx <fresh_map
        [2: >memb_cons // >Hx //] cases no_dup * #ABS #_ whd in ⊢ (???%);
        inversion(x==?) [2: #_ //] #ABS1 @⊥ @ABS <(\P ABS1) //
  |5,10: [ @no_dup | cases no_dup // ]
  ]
| #lin #io #lout #i #IH #n whd in match (get_labels_of_code ??); #no_dup
  #m #keep #info #l whd in ⊢ (??%? → ?); #EQ destruct(EQ) #fresh_map #same_keep
  #f_k whd in ⊢ (??%?); >(IH … (refl …))
  [ normalize nodelta <fresh_map [2: >memb_cons // >memb_hd // ]
    whd in match (get_element ????); >(\b (refl …)) normalize nodelta
    >(\b (refl …)) <fresh_map [2: >memb_hd //] whd in match (get_element ????);
    inversion(lin==lout)
    [ #ABS @⊥ cases no_dup * #ABS1 #_ @ABS1 whd in match (memb ???); >(\P ABS)
      >(\b (refl …)) //
    | #H inversion (a_non_functional_label lin== ? lout) 
      [ #ABS lapply(\P ABS) #EQ destruct >(\b (refl …)) in H; #EQ destruct
      | #_ normalize nodelta whd in match (get_element ????); >(\b (refl …))
        normalize nodelta >(\b (refl …)) %
      ]
    ]
  | //
  | #x #Hx >same_keep [2: >memb_cons // >memb_cons // >Hx % ] %
  | #x #Hx <fresh_map [2: >memb_cons // >memb_cons // >Hx %]
    cases no_dup * #ABS1 ** #ABS2 #_ whd in ⊢ (???%); inversion(x == lout) 
    normalize nodelta
    [2: #_ whd in ⊢ (???%); inversion(x==lin) normalize nodelta //
        #H @⊥ @ABS1 >memb_cons // <(\P H) >Hx //
    | #H @⊥ @ABS2 <(\P H) >Hx //
    ]
  | cases no_dup #_ * #_ //
  ]
]
qed.

definition fresh_for_prog_aux : ∀p,p'.Program p p' → ℕ → ℕ ≝ 
λp,p',prog,n.foldl … (λn,i.max n (compute_max_n … (f_body … i))) n (env … prog).

include alias "arithmetics/nat.ma".

lemma max_1 : ∀n,m,k.k ≤ m → k ≤ max m n.
#m #n #k #H normalize @leb_elim // normalize nodelta #H1 
/2 by transitive_le/
qed.

lemma max_2 : ∀n,m,k.k≤ n → k ≤ max m n.
#m #n #k #H normalize @leb_elim // #H1 normalize nodelta
@(transitive_le … H) @(transitive_le … (not_le_to_lt … H1)) //
qed.

lemma fresh_aux_ok : ∀p,p'.∀prog : Program p p'.∀n,m.n ≤ m → 
fresh_for_prog_aux … prog n ≤ fresh_for_prog_aux … prog m.
#p #p' * #env #main elim env // #hd #tl #IH #n #m #H whd in ⊢ (?%%);
@IH whd in ⊢ (?%?); @leb_elim normalize nodelta
[ #H1 @max_2 // | #_ @max_1 // ]
qed.

definition fresh_for_prog : ∀p,p'.Program p p' → ℕ ≝ 
λp,p',prog.fresh_for_prog_aux … prog 
(compute_max_n … (main … prog)).

definition translate_env ≝ 
λp,p'.λenv : list (env_item p p').λmax_all.(foldr ??
           (λi,x.let 〈t_env,n,m,keep〉 ≝ x in
           let info ≝ call_post_trans … (f_body … i) n (nil ?) in
                   〈(mk_env_item ?? 
                       (mk_signature ??(f_name ?? i) (f_pars … i) (f_ret … i))
                       (f_lab … i) (t_code … info)) :: t_env,
                     fresh … info, 〈a_call (f_lab … i),(a_call (f_lab … i)) :: (gen_labels ? info)〉 :: 
                                     ((lab_map … info) @ m),(lab_to_keep … info) @ keep〉) 
          (〈nil ?,max_all,nil ?,nil ?〉) env).

definition trans_prog : ∀p,p'.Program p p' → 
((Program p p') × (associative_list DEQCostLabel CostLabel)) × ((list ReturnPostCostLabel))≝ 
λp,p',prog.
let max_all ≝ fresh_for_prog … prog in
let info_main ≝ (call_post_trans … (main … prog) max_all (nil ?)) in
let 〈t_env,n,m,keep〉 ≝ translate_env … (env … prog) (fresh … info_main) in
〈mk_Program ?? t_env (t_code … info_main),m @ (lab_map … info_main),keep @ (lab_to_keep … info_main)〉.

definition map_labels_on_trace : 
(associative_list DEQCostLabel CostLabel) → list CostLabel → list CostLabel ≝ 
λm,l.foldr … (λlab,t.(get_element … m lab) @ t) (nil ?) l.

lemma map_labels_on_trace_append:
 ∀m,l1,l2. map_labels_on_trace m (l1@l2) =
  map_labels_on_trace m l1 @ map_labels_on_trace m l2.
 #m #l1 elim l1 // #hd #tl #IH #l2 >associative_append <IH //
qed.

include "../src/common/Errors.ma".

axiom is_permutation: ∀A.list A → list A → Prop.
axiom is_permutation_eq : ∀A.∀l : list A.is_permutation … l l.
axiom is_permutation_cons : ∀A.∀l1,l2,x.is_permutation A l1 l2 → 
                                       is_permutation A (x :: l1) (x :: l2).

(*
inductive is_permutation (A : Type[0]) : list A → list A → Prop ≝ 
| p_empty : is_permutation A (nil ?) (nil ?)
| p_append : ∀x,x1,x2,y,y1,y2.
               x = y → is_permutation A (x1 @ x2) (y1 @ y2) → 
                 is_permutation A (x1 @ [x] @ x2) (y1 @ [y] @ y2).

lemma is_permutation_eq : ∀A.∀l : list A.is_permutation … l l.
#A #l elim l // #x #xs #IH 
change with ((nil ?) @ (x :: xs)) in ⊢ (??%%);
>append_cons >associative_append
@(p_append ? x (nil ?) xs x (nil ?) xs (refl …)) @IH
qed.

lemma is_permutation_append : ∀A.∀l1,l2,l3,l4 : list A.
is_permutation A l1 l3 → is_permutation A l2 l4 → 
is_permutation A (l1 @ l2) (l3 @ l4).
#A #l1 inversion (|l1|)  [2: #n lapply l1 elim n
[ #l2 #l3 #l4 #H inversion H // #x #x1 #x2 #y #y1 #y2 #EQ #H1 #_ 
 #ABS cases(nil_to_nil … (sym_eq ??? ABS)) -ABS #_ #ABS
 cases(nil_to_nil … ABS) #EQ1 destruct(EQ1) ]
#x #xs #IH #l2 #l3 #l4 #H inversion H 
[#EQ lapply(jmeq_to_eq ??? EQ) -EQ #EQ destruct(EQ) ]
#y #y1 #y2 #z #z1 #z2 #EQ destruct(EQ) #H1 #_ #EQx_xs #EQ destruct(EQ) #_
*)

lemma memb_append_l22 : ∀A : DeqSet.∀x : A.∀l1,l2 : list A. 
¬ (x ∈ l1) → x∈ l1 @ l2 = (x ∈ l2).
#A #x #l1 elim l1 normalize // #y #ys #IH #l2 cases(x==y) 
normalize [*] @IH
qed.

lemma memb_append_l12 : ∀A : DeqSet.∀x : A.∀l1,l2 : list A. 
¬ (x ∈ l2) → x∈ l1 @ l2 = (x ∈ l1).
#A #x #l1 elim l1 
[ #l2 #H whd in match (append ???); @not_b_to_eq_false @Prop_notb >H % ]
#y #ys #IH #l2 #H1 whd in match (memb ???); >IH //
qed.


lemma lookup_ok_append : ∀p,p',l,f,env_it.
lookup p p' l f = return env_it → ∃l1,l2. l = l1 @ [env_it] @ l2 ∧
f_name … env_it = f.
#p #p' #l elim l [ #f #env_it normalize #EQ destruct]
#x #xs #IH #f #env_it whd in ⊢ (??%? → ?); @eq_function_name_elim
[ #EQ destruct(EQ) normalize nodelta whd in ⊢ (???% → ?); #EQ destruct
  %{(nil ?)} %{xs} /2/
| #Hno_f normalize nodelta #EQ_env_it cases(IH … EQ_env_it)
  #l1 * #l2 * #EQ1 #EQ2 %{(x :: l1)} %{l2} >EQ1 /2/
]
qed.
(*
lemma foldr_append :
  ∀A,B:Type[0]. ∀l1, l2 : list A. ∀f:A → B → B. ∀seed. foldr ?? f seed (l1 @ l2) = foldr ?? f (foldr ?? f seed l2) l1.
#A #B #l1 elim l1 //
#hd #tl #Hind #l2 #f #seed normalize >Hind @refl
qed.
*)

lemma foldr_map_append :
  ∀A,B:Type[0]. ∀l1, l2 : list A.
   ∀f:A → list B. ∀seed.
    foldr ?? (λx,acc. (f x) @ acc) seed (l1 @ l2) =
     append ? (foldr ?? (λx,acc. (f x) @ acc) (nil ?) l1) 
       (foldr  ?? (λx,acc. (f x) @ acc) seed l2).
#A #B #l1 elim l1 normalize // /3 by eq_f, trans_eq/
qed.

lemma cons_append : ∀A.∀x : A.∀l.x::l = ([x]@l).
//
qed.

lemma eq_a_call_label : 
∀c1,c2.c1 == c2 → a_call c1 == a_call c2.
#c1 #c2 #EQ cases(eqb_true ? c1 c2) #H1 #_ lapply(H1 (eq_true_to_b … EQ)) -EQ
#EQ destruct >(\b (refl …)) @I
qed.


lemma no_duplicates_append_commute : ∀ A : DeqSet.∀l1,l2 : list A.
no_duplicates … (l1 @ l2) → 
no_duplicates … (l2 @ l1).
#A #l1 elim l1
[ #l2 >append_nil //]
#x #xs #IH #l2 * #H1 #H2 lapply(IH … H2) lapply H1 -H1 -IH -H2
elim l2 -l1
[ >append_nil #H1 #H2 % // ]
#y #ys #IH * #H1 * #H2 #H3 %
[2: @IH
   [ % #H4 @H1 cases(memb_append … H4)
     [ #H5 >memb_append_l1 //
     | #H5 >memb_append_l2 // @orb_Prop_r >H5 //
     ]
   | //
   ]
| % #H4 cases(memb_append … H4)
  [ #H5 @(absurd ?? H2) >memb_append_l1 //
  | whd in match (memb ???); inversion(y==x)
    [ #H5 #_ <(\P H5) in H1; #H1 @H1 >memb_append_l2 //
    | #_ normalize nodelta #H5 @(absurd ?? H2) >memb_append_l2 //
    ]
  ]
]
qed.

(* aggiungere fresh_to_keep al lemma seguente??*)

let rec subset (A : Type[0]) (l1,l2 : list A) on l1 : Prop ≝ 
match l1 with
[ nil ⇒ True
| cons x xs ⇒ mem … x l2 ∧ subset … xs l2
].

interpretation "subset" 'subseteq a b = (subset ? a b).

lemma subset_append_l2 : ∀A,l2,l1.subset A l2 (l1 @ l2).
#A #l2 elim l2 // normalize #x #xs #IH #l1 % // @mem_append_l2 whd /2/
qed.

lemma refl_subset : ∀A.reflexive … (subset A).
#A #l1 elim l1 // #x #xs #IH normalize % /2/
qed.

lemma fresh_for_subset : ∀l1,l2,n.l1 ⊆ l2 → is_fresh_for_return … l2 n → 
is_fresh_for_return … l1 n.
#l1 elim l1 // ** #x #xs #IH #l2 #n * #H1 #H2 #H3 whd
try(@IH) // % [2: @IH //] elim l2 in H1 H3; normalize [*]
** #y #ys #IH normalize 
[2,3: * [2,4: #H3 [2: @IH //] * #H4 #H5 @IH //
|*: #EQ destruct * //
]]
*
[ #EQ destruct ] #H3 #H4 @IH //
qed.

lemma fresh_append : ∀n,l1,l2.is_fresh_for_return l1 n → is_fresh_for_return l2 n → 
is_fresh_for_return (l1@l2) n.
#n #l1 lapply n -n elim l1 // ** #x #xs #IH #n #l2 [2: * #H1 ] #H2 #H3 
[ % // @IH //] @IH //
qed.

definition labels_of_prog : ∀p,p'.Program p p' → ? ≝ 
λp,p',prog.foldr … (λx,l.l @ (get_labels_of_code … (f_body … x))) 
 (get_labels_of_code … (main … prog)) (env … prog).

lemma cast_return_append : ∀l1,l2.cast_return_to_cost_labels … (l1 @ l2) = 
(cast_return_to_cost_labels … l1) @ (cast_return_to_cost_labels … l2).
#l1 #l2 @(sym_eq … (map_append …)) qed.

include alias "arithmetics/nat.ma".


lemma is_fresh_code : ∀p.∀i : Instructions p.
is_fresh_for_return (get_labels_of_code … i) (compute_max_n … i).
#p #main  elim main //
[ #seq * [| * #lbl] #i #IH normalize // @(fresh_keep_n_ok … IH)
  inversion(leb ? lbl) // @leb_elim #Hlbl #EQ destruct normalize nodelta 
  /2 by le_S/
| #cond * #ltrue #i1 * #lfalse #i2 #i3 #IH1 #IH2 #IH3 whd in ⊢ (?%%); 
  @fresh_append
  [ @(fresh_keep_n_ok … IH1) @le_S @max_1 @max_1 @max_1 @max_1 //
  | @fresh_append
    [ @(fresh_keep_n_ok … IH2) @le_S @max_1 @max_1 @max_1 @max_2 //
    | @(fresh_keep_n_ok … IH3) @le_S @max_1 @max_1 @max_2 //
    ]
  ]
| #cond * #ltrue #i1 * #lfalse #i2 #IH1 #IH2 whd in ⊢ (?%%); @fresh_append
  [ @(fresh_keep_n_ok … IH1) @le_S @max_1 @max_1 @max_1 //
  | @(fresh_keep_n_ok … IH2) @le_S @max_1 @max_1 @max_2 //
  ]
| #f #act_p * [| * #lbl] #i #IH whd in ⊢ (?%%); // 
  change with ([?]@?) in ⊢ (?%?); @fresh_append
  [ whd % // @le_S @max_1 //
  | @(fresh_keep_n_ok … IH) @le_S @max_2 //
  ]
| * #lin #io * #lout #i #IH whd in ⊢ (?%%); @(fresh_keep_n_ok … IH)
  @le_S @max_1 @max_1 //
]
qed.

lemma is_fresh_fresh_for_prog : ∀p,p'.∀prog : Program p p'.
is_fresh_for_return (labels_of_prog … prog) (fresh_for_prog … prog).
#p #p' * #env #main whd in match fresh_for_prog; normalize nodelta whd in ⊢ (?%?);
elim env // * #sig #cost #i #tail #IH  whd in ⊢ (?%?); @fresh_append
[ @(fresh_keep_n_ok … IH) @fresh_aux_ok @max_1 //
| @(fresh_keep_n_ok … (is_fresh_code … i)) whd in match fresh_for_prog_aux; normalize nodelta
  whd in ⊢ (??%); elim tail [ @max_2 // ] #hd1 #tl1 #IH1 @(transitive_le …  IH1)
  whd in ⊢ (??%); change with (fresh_for_prog_aux ?? (mk_Program ?? tl1 main) ?) in ⊢ (?%%);
  @fresh_aux_ok @max_1 //
]
qed.

lemma subset_def : ∀A : DeqSet.∀l1,l2 : list A.(∀x.x ∈l1 → x ∈ l2) → l1 ⊆ l2.
#A #l1 elim l1 // #x #xs #IH #l2 #H % 
[ @memb_to_mem >H // >memb_hd // 
| @IH #y #H1 @H >memb_cons // >H1 //
]
qed.

lemma memb_cast_return : ∀keep,x.x ∈ cast_return_to_cost_labels keep → 
∃ y.x = a_return_post y ∧ bool_to_Prop (y ∈ keep).
#keep elim keep
[ #x *] #x #xs #IH #y whd in match cast_return_to_cost_labels;
whd in match (map ????); whd in match (memb ???); inversion(y==x)
[ #Hx #_ %{x} >(\P Hx) %{(refl …)} >memb_hd //
| #Hx normalize nodelta #H cases(IH … H) #z * #H1 #H2 %{z} %{H1} >memb_cons // >H2 //
]
qed.

lemma subset_append : ∀A.∀l1,l2,l3 : list A.l1 ⊆ l3 → l2 ⊆ l3 → (l1 @ l2) ⊆ l3.
#A #l1 elim l1 // #x #xs #IH #l2 #l3 * #H1 #H2 #H3 % /2/
qed.

lemma subset_def_inv : ∀A.∀l1,l2 : list A. l1 ⊆ l2 → ∀x.mem … x l1 → mem … x l2.
#A #l1 elim l1 [ #l2 * #x * ] #x #xs #IH #l2 * #H1 #H2 #y *
[ #EQ destruct // | #H3 @IH // ]
qed.

lemma transitive_subset : ∀A.transitive … (subset A).
#A #l1 elim l1 // #x #xs #IH #l2 #l3 * #H1 #H2 #H3 %
[ @(subset_def_inv … H3) // | @IH // ]
qed.

lemma subset_append_h1 : ∀A.∀l1,l2,l3 : list A.l1 ⊆ l3 → l1 ⊆ (l3 @ l2).
#A #l1 elim l1 // #x #x2 #IH #l2 #l3 * #H1 #H2 % [ @mem_append_l1 // | @IH // ]
qed.

lemma subset_append_h2 : ∀A.∀l1,l2,l3 : list A.l2 ⊆ l3 → l2 ⊆ (l1 @ l3).
#A #l1 elim l1 // #x #xs #IH #l2 elim l2 // #y #ys #IH #l3 * #H1 #H2 %
[ @mem_append_l2 // | @IH // ]
qed.

lemma lab_to_keep_in_prog : ∀p,p'.∀prog : Program p p'.
∀t_prog,m,keep.trans_prog … prog = 〈t_prog,m,keep〉 → 
(cast_return_to_cost_labels keep) ⊆ (labels_of_prog p p' prog).
#p #p' * #env #main #t_prog #m #keep whd in match trans_prog; normalize nodelta
@pair_elim * #env1 #fresh * #m1 #keep1 #EQenv1 normalize nodelta #EQ destruct
lapply EQenv1 -EQenv1 lapply keep1 -keep1 lapply m1 -m1 lapply fresh -fresh
lapply env1 -env1 generalize in match (fresh_for_prog ???); elim env
[ #n #t_env #n1 #m #keep whd in ⊢ (??%? → ?); #EQ destruct whd in match (append ???);
  @subset_def #x #H whd in match (labels_of_prog); normalize nodelta
  whd in match (foldr ?????); cases(memb_cast_return … H) -H #x1 * #EQ1 #H destruct
  @(lab_to_keep_in_domain … H)
| #x #xs #IH #n #t_env #n1 #m #keep whd in ⊢ (??%? → ?); @pair_elim
  * #t_env_tail #fresh_tail * #t_m_tail #t_keep_tail 
  change with (translate_env ????) in match (foldr ?????); #EQt_env_tail
  normalize nodelta #EQ1 destruct >cast_return_append @subset_append
  [ >cast_return_append @subset_append
    [ whd in match labels_of_prog; normalize nodelta whd in match (foldr ?????);
      @subset_def #y #H cases(memb_cast_return … H) -H #y1 * #EQ destruct #H
      >memb_append_l2 // @(lab_to_keep_in_domain … H)
    | whd in match labels_of_prog; normalize nodelta whd in match (foldr ?????);
      change with (labels_of_prog ?? (mk_Program ?? xs ?)) in match (foldr ?????);
      @subset_append_h1 @(transitive_subset … (IH … EQt_env_tail))
      >cast_return_append @subset_append_h1 //
    ]
  | whd in match labels_of_prog; normalize nodelta whd in match (foldr ?????);
    change with (labels_of_prog ?? (mk_Program ?? xs ?)) in match (foldr ?????);
    @subset_append_h1 @(transitive_subset … (IH … EQt_env_tail))
     >cast_return_append @subset_append_h2 //
  ]
]
qed.

lemma fresh_call_post_trans_ok : ∀p.∀i : Instructions p.∀n,l.
n ≤ fresh … (call_post_trans … i n l).
#p #i elim i //
qed.

lemma fresh_translate_env_ok : ∀p,p'.∀env,t_env : list (env_item p p').∀n,n1,m,keep.
translate_env … env n = 〈t_env,n1,m,keep〉 → n ≤ n1.
#p #p' #env elim env
[ #t_env #n #n1 #m #keep whd in ⊢ (??%? → ?); #EQ destruct // ]
#x #xs #IH #t_env #n #n1 #m #keep whd in ⊢ (??%? → ?);
change with (translate_env ????) in match (foldr ?????); @pair_elim
* #t_env_tail #fresh_tail * #t_m_tail #t_keep_tail #EQt_env_tail normalize nodelta
#EQ destruct @(transitive_le … (IH … EQt_env_tail)) @fresh_call_post_trans_ok
qed.
 

lemma trans_env_ok : ∀p : state_params.∀ prog.
no_duplicates_labels … prog → 
let 〈t_prog,m,keep〉 ≝ trans_prog … prog in 
∀f,env_it.lookup p p (env … prog) f = return env_it → 
let dom ≝ get_labels_of_code … (f_body … env_it) in 
∃env_it',n.is_fresh_for_return keep n ∧lookup p p (env … t_prog) f = return env_it' ∧ 
let info ≝ call_post_trans … (f_body … env_it) n (nil ?) in
t_code … info = f_body … env_it' ∧
get_element … m (a_call (f_lab … env_it')) = (a_call (f_lab … env_it')) :: gen_labels … info ∧
f_sig … env_it = f_sig … env_it' ∧ f_lab … env_it = f_lab … env_it' ∧
same_fresh_map_on dom m (lab_map … info) ∧ same_to_keep_on dom keep (lab_to_keep … info).
#p #prog inversion(trans_prog … prog) * #t_prog0 #m0 #keep0 #EQt_prog
lapply EQt_prog normalize nodelta
generalize in match keep0 in ⊢ (% → ? → ? → ? → ? → ??(λ_.??(λ_.?%?)));
#keep1 #EQkeep1 inversion prog in EQt_prog; #env #main #EQprog
whd in match trans_prog; normalize nodelta 
@pair_elim
cut(fresh_for_prog ?? prog ≤ fresh_for_prog ?? (mk_Program … env main)) [ >EQprog //]
generalize in match (fresh_for_prog ???) in ⊢ (??% → %);
lapply t_prog0 lapply m0 lapply keep0
elim env in ⊢ (?→ ? → ? → ? → ? → %);
[ #keep #m #t_prog #n #_ * #env' #fresh * #x #y #_ #_ #_ #f #env_it normalize in ⊢ (% → ?);  #ABS destruct] 
* #hd_sig #hd_lab #hd_code #tail #IH #keep #m #t_prog #fresh1 #Hfresh1 * #env' #fresh * #m' #keep' 
normalize in ⊢ (% → ?); normalize nodelta @pair_elim * #env_tail #fresh_tail
* #m_tail #keep_tail change with (translate_env ????) in ⊢ (??%? → ?); #EQtail normalize nodelta #EQ1 destruct(EQ1) #EQ2 destruct(EQ2)
whd in ⊢ (% → ?); whd in match (foldr ?????); * #Hhd_lab #H lapply(no_duplicates_append_r … H) 
change with (no_duplicates_labels p p (mk_Program p p tail main)) in match
(no_duplicates_labels p p (mk_Program p p tail main)); #no_dup_tail
lapply(no_duplicates_append_l … H) #no_dup_head normalize nodelta 
#f #env_it whd in ⊢ (??%? → ?); @eq_function_name_elim normalize nodelta
[ #EQ destruct(EQ) whd in ⊢ (???% → ?); #EQ destruct(EQ)
  inversion (call_post_trans … hd_code fresh_tail [])
  #gen_labs #t_hd_code #t_fresh #t_lab_map #t_lab_to_keep #EQ_trans_code
  %{(mk_env_item … hd_sig hd_lab t_hd_code)} %{fresh_tail} %
  [ %
    [ @(fresh_keep_n_ok … fresh1)
      [ @(fresh_keep_n_ok … Hfresh1)
        @(fresh_for_subset … (labels_of_prog … prog))
        [ @(lab_to_keep_in_prog … EQkeep1) | @is_fresh_fresh_for_prog ]
       | @(transitive_le … (fresh_translate_env_ok … EQtail)) // 
      ]
    | whd in ⊢ (??%?); @eq_function_name_elim [2: * #H @⊥ @H %] #_ normalize nodelta
      @eq_f cases hd_sig // ]] >EQ_trans_code % [% [ % [ % [% // whd in ⊢ (??%?); >(\b (refl …)) %] % ] % | whd 
    #x #Hx whd in ⊢ (??%?); >(? : (x == hd_lab) = false) 
    [2: inversion(x==hd_lab) // #EQx_hdlab cases Hhd_lab -Hhd_lab #Hhd_lab cases Hhd_lab
        >memb_append_l1 // <(\P EQx_hdlab) >Hx // ]
    normalize nodelta >get_element_append_l1
    [2: cases daemon (*noduplicates*)] @get_element_append_l1 
    % #H1  
    (* subproof with no nice statement *)
    lapply H1 -H1 lapply H -H lapply Hhd_lab -Hhd_lab lapply EQtail -EQtail
    generalize in match fresh1; lapply env_tail -env_tail lapply fresh_tail 
    -fresh_tail lapply m_tail -m_tail lapply keep_tail -keep_tail elim tail 
    normalize nodelta
    [ #keep_tail #m_tail #fresh_tail #env_tail #n whd in ⊢ (??%? → ?);
      #EQ destruct(EQ) whd in match (foldr ?????);
      #H1 #H2 * ]
    #head #tail #IH #keep_tail #m_tail #fresh_tail #env_tail #n whd in ⊢ (??%? → ?);
    whd in match (foldr ?????); 
    @pair_elim * #env_tail_res #fresh_tail_res * #lab_map_res #keep_res #EQres
    normalize nodelta #EQ destruct(EQ) whd in match (foldr ?????);
    #H1 #H2 whd in match (memb ???); inversion(x == ?)
    [ #H3 #_ <(\P H3) in H2; change with ([?]@?) in match (?::?); #H2
      lapply(no_duplicates_append_commute … H2) -H2 ** #ABS #_ @ABS
      >memb_append_l2 // >Hx %
    | #H3 normalize nodelta #H4 @(IH … EQres) 
      [3: >domain_of_associative_list_append in H4; #H4 cases(memb_append … H4) [2: #EQ >EQ %]
          #ABS @⊥ @(memb_no_duplicates_append … x … H2) // @orb_Prop_r >memb_append_l1 //
          @(lab_map_in_domain … (eq_true_to_b … ABS))
      | % #ABS elim H1 -H1 #H1 @H1 cases(memb_append … ABS)
        [ #H5 >memb_append_l1 //
        | #H5 >memb_append_l2 // @orb_Prop_r >memb_append_l2 //
        ]
      | lapply(no_duplicates_append_commute … H2) * #_ >associative_append
        #h @no_duplicates_append_commute @(no_duplicates_append_r … h)
      ]
    ]
    ]
  | whd #x #Hx >memb_append_l12 [2: cases daemon (*TODO*) ] 
    >memb_append_l12 // inversion(memb ???) // #ABS @(memb_no_duplicates_append … (a_return_post x) … H)
    // @⊥
    (* subproof with no nice statement *)
    lapply ABS -ABS lapply H -H lapply Hhd_lab -Hhd_lab lapply EQtail -EQtail
    generalize in match fresh1; lapply env_tail -env_tail lapply fresh_tail 
    -fresh_tail lapply m_tail -m_tail lapply keep_tail -keep_tail elim tail 
    normalize nodelta
    [ #keep_tail #m_tail #fresh_tail #env_tail #n whd in ⊢ (??%? → ?);
      #EQ destruct(EQ) whd in match (foldr ?????);
      #H1 #H2 whd in ⊢ (??%? → ?); #EQ destruct ]
    #head #tail #IH #keep_tail #m_tail #fresh_tail #env_tail #n whd in ⊢ (??%? → ?);
    whd in match (foldr ?????); 
    @pair_elim * #env_tail_res #fresh_tail_res * #lab_map_res #keep_res #EQres
    normalize nodelta #EQ destruct(EQ) whd in match (foldr ?????);
    #H1 #H2 #H3 cases(memb_append … H3) -H3
    [ #H3 change with ([?]@?) in match (?::?) in H2;
      lapply(no_duplicates_append_commute … H2) -H2 * #_ #H4 @(memb_no_duplicates_append … (a_return_post x) … H4)
      [ whd in match (append ???); >memb_append_l1 // >(lab_to_keep_in_domain … (eq_true_to_b … H3)) % 
      | //
      ]
    | #H3 normalize nodelta @(IH … EQres) 
      [3: //
      |  % #ABS elim H1 -H1 #H1 @H1 cases(memb_append … ABS)
        [ #H5 >memb_append_l1 //
        | #H5 >memb_append_l2 // @orb_Prop_r >memb_append_l2 //
        ]
      | lapply(no_duplicates_append_commute … H2) * #_ >associative_append
        #h @no_duplicates_append_commute @(no_duplicates_append_r … h)
      ]
    ] 
  ] 
| #Hf #Henv_it cases(IH … no_dup_tail … Henv_it)
  [9: >EQtail in ⊢ (??%?); %
  |13: %
  |6: assumption
  |10: %
  |*:
  ]
  #new_env_it * #new_fresh ** #is_fresh_newfresh #EQlook_new_env_it ***** #EQt_code #EQ_get_el 
  #EQsign_env_it #EQ_f_lab #same_fresh_map #same_to_keep %{new_env_it} %{new_fresh}
  %
  [ %
     [ assumption
     | whd in ⊢ (??%?); @eq_function_name_elim [ #ABS >ABS in Hf; * #H @⊥ @H %]
        #_ normalize nodelta assumption ]]
   % [2: #x #Hx <same_to_keep // >associative_append @memb_append_l22 
        inversion(memb ???) // #ABS lapply(lab_to_keep_in_domain … (eq_true_to_b … ABS))
        #ABS1 @(memb_no_duplicates_append … (a_return_post x) … H) // 
        cases(lookup_ok_append … Henv_it) #l1 * #l2 * #EQ1 #EQ2 destruct(EQ1 EQ2)
        >foldr_map_append >memb_append_l2 // >foldr_map_append >memb_append_l1 //
        whd in match (foldr ?????); @orb_Prop_r >memb_append_l1 // >Hx % ]
   % [2: #x #Hx <same_fresh_map // >cons_append <associative_append
         <associative_append in ⊢ (??(???(??%?)?)?); >associative_append
         @(get_element_append_r1)
         % >domain_of_associative_list_append #ABS cases(memb_append … ABS)
         [ whd in match (memb ???); inversion(x==hd_lab) normalize nodelta
           [2: #_ whd in match (memb ???); #EQ destruct ] #EQx_hdlab #_
               <(\P EQx_hdlab) in Hhd_lab; cases(lookup_ok_append … Henv_it)
               #tail1 * #tail2 * #EQ1 #EQ2 destruct >foldr_map_append >foldr_map_append
               * #ABS1 @ABS1 >memb_append_l2 // >memb_append_l2 //
               >memb_append_l1 // whd in ⊢ (??%?); cases(x==?) //
               normalize nodelta >memb_append_l1 // >Hx %
         | #ABS1 @(memb_no_duplicates_append … x … H) 
           [ @(lab_map_in_domain … (eq_true_to_b … ABS1))
           | cases(lookup_ok_append … Henv_it)
             #tail1 * #tail2 * #EQ1 #EQ2 destruct >foldr_map_append >foldr_map_append
             >memb_append_l2 // >memb_append_l1 //
             whd in ⊢ (??%?); cases(x==?) //
             normalize nodelta >memb_append_l1 // >Hx %
           ]
         ]
     ]  
   % // % // % // <EQ_get_el >cons_append <associative_append  <associative_append in ⊢ (??(???(??%?)?)?); 
   >associative_append
   @get_element_append_r1 % >domain_of_associative_list_append #ABS cases(memb_append … ABS)
         [ whd in match (memb ???); inversion(a_call (f_lab … new_env_it)== a_call hd_lab) 
           #EQ_hdlab normalize nodelta
           [2: whd in ⊢ (??%? → ?); #EQ destruct ]  
           #_ <(\P EQ_hdlab) in Hhd_lab; cases(lookup_ok_append … Henv_it)
           #tail1 * #tail2 * #EQ1 #EQ2 destruct >foldr_map_append >foldr_map_append
           * #ABS1 @ABS1 >memb_append_l2 // >memb_append_l2 //
           >memb_append_l1 // whd in ⊢ (??%?); >EQ_f_lab >(\b (refl …)) // 
         | #ABS1 @(memb_no_duplicates_append … (a_call (f_lab … new_env_it)) … H) 
           [ @(lab_map_in_domain … (eq_true_to_b … ABS1))
           | cases(lookup_ok_append … Henv_it)
             #tail1 * #tail2 * #EQ1 #EQ2 destruct >foldr_map_append >foldr_map_append
             >memb_append_l2 // >memb_append_l1 //
             whd in ⊢ (??%?); >EQ_f_lab >(\b (refl …)) //
           ]
         ]
]
qed.

lemma len_append : ∀S : abstract_status.∀st1,st2,st3 : S.
∀t1 : raw_trace … st1 st2.∀t2 : raw_trace … st2 st3.
len … (t1 @ t2) = len … t1 + len …  t2.
#S #st1 #st2 #st3 #t1 elim t1 normalize //
qed.

axiom permute_ok : ∀A.∀l1,l2,l3,l4,l5,l6,l7,l8,l9,x,y.
  (is_permutation A ((l5 @l1) @l6@l7) ((l4 @[]) @l6@l7)
 →is_permutation A ((l6 @l2) @l7) ((l3 @l8) @l9)
 →is_permutation A
   (y ::((l6 @((x ::l5) @(l1 @l2))) @l7))
   (((x ::l4 @y ::l3) @l8) @l9)).
    

lemma correctness : ∀p,p',prog.
no_duplicates_labels … prog → 
let 〈t_prog,m,keep〉 ≝ trans_prog … prog in
∀s1,s2,s1' : state p.∀abs_top,abs_tail.
∀t : raw_trace (operational_semantics … p' prog) s1 s2.
pre_measurable_trace … t → 
state_rel … m keep abs_top abs_tail s1 s1' → 
∃abs_top',abs_tail'.∃s2'.∃t' : raw_trace (operational_semantics … p' t_prog) s1' s2'.
state_rel  … m keep abs_top' abs_tail' s2 s2' ∧
is_permutation ? ((abs_top @ (map_labels_on_trace m (get_costlabels_of_trace … t))) @ abs_tail)  
                 (((get_costlabels_of_trace … t') @ abs_top' ) @ abs_tail') ∧
 len … t = len … t' ∧ 
 ∀k.∀i.option_hd … (cont … s2) = Some ? 〈ret_act (None ?),i〉 → 
 cont … s1 = k @ cont … s2 → 
All … (λx.let 〈lab,i〉 ≝ x in ¬ is_unlabelled_ret_act lab) k → 
∃k'.cont … s1' = k' @ cont … s2' ∧ |k| = |k'|.
#p #p' #prog #no_dup @pair_elim * #t_prog #m #keep #EQtrans
#s1 #s2 #s1' #abs_top #abs_tail #t #Hpre lapply abs_top -abs_top lapply abs_tail
-abs_tail lapply s1' -s1' elim Hpre
[ #st #_ #s1' #abs_tail #abs_top #H %{abs_top} %{abs_tail} %{s1'} %{(t_base …)} 
  % 
  [ % [2: %] %{H} whd in match (get_costlabels_of_trace ????); >append_nil
    @is_permutation_eq
  | #k #i #_ #EQcont_st #wf %{[ ]} % // @sym_eq @eq_f @append_l1_injective_r //
  ]
| #st1 #st2 #st3 #lab whd in ⊢ (% → ?); #H inversion H in ⊢ ?; #st11 #st12
  [ * #lab1 #new_code #new_cont #EQcode11 #EQcont11 #EQcode12 #EQcont12 #EQstore
    #Hio11 #EQio12 #Hl1 #EQ1 #EQ2 #EQ3 #EQ4 destruct #tl #_ * #opt_l #EQ destruct(EQ) 
    #pre_tl #IH #s1' #abs_tail #abs_top whd in match state_rel in ⊢ (% → ?); normalize nodelta >EQcont11 in ⊢ (% → ?);
    whd in match check_continuations; normalize nodelta whd in match (foldr2 ???????);
    inversion (cont ? s1') [ #_ *] * #l1' #new_code' #new_cont' #_ #EQconts1'
    normalize nodelta change with (check_continuations ?????) in match (foldr2 ???????); 
    inversion(check_continuations ?????) [ #_ *] ** #H1 #l2 #ll2 #EQHl2 
    >m_return_bind normalize nodelta inversion (call_post_clean ?????) [ #_ ***** ]
    * #l3 #code_c_st12 #EQcode_c_st12 normalize nodelta
    cases l1' in EQconts1'; (*in Hio11 Hl1 EQcont11 H;*) normalize nodelta
    [1,4: [ #x #y ] #_ (*#_ #_ #_ #H*) *****
    | #x #_ cases (ret_costed_abs ??) normalize nodelta [|#c] ******[|*] #EQ @⊥ >EQ in Hl1;
      normalize * /2/ ] *
    [ #EQconts1' normalize nodelta ****** #EQ destruct(EQ)
      #HH1 #EQ destruct(EQ) >EQcode11 in ⊢ (% → ?); inversion(code … s1')
      [
      | #x 
      | #seq #lbl #i #_ 
      | #cond #ltrue #i1 #lfalse #i2 #i3 #_ #_ #_
      | #cond #ltrue #i1 #lfalse #i2 #_ #_
      | #f #act_p #ret_post #i #_
      | #l_in #io #l_out #i #_
      ]
      [|*: #_ whd in ⊢ (??%% → ?); [ #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct(EQ)]
           cases(call_post_clean ?????) normalize nodelta
           [1,3,5,7,9: #EQ destruct(EQ)] cases daemon (* da fare assurdi!!!*) ]
      #EQcodes1' whd in ⊢ (??%% → ?); #EQ destruct(EQ) #EQstore11 #EQio11 #EQ destruct(EQ)
      cases(IH (mk_state ? new_code' new_cont' (store … s1') false) ll2 abs_top …)
      [2: whd whd in match check_continuations; normalize nodelta 
         change with (check_continuations ?????) in match (foldr2 ???????); >EQHl2
         normalize nodelta % // % // % // % // @EQcode_c_st12 ]
      #abs_top' * #abs_tail' * #st3' * #t' *** #Hst3st3' #EQcosts #EQlen #stack_safety 
      %{abs_top'} %{abs_tail'} %{st3'} %{(t_ind … (cost_act (None ?)) …  t')}
      [ @hide_prf whd @(empty ????? 〈(cost_act (None ?)),?〉)
        [3: assumption |4: assumption |*:] /3 by nmk/ ]
      % [%   [2: whd in ⊢ (??%%); >EQlen % ] %{Hst3st3'} @EQcosts]
      * [| #hd #tl] #i #EQcont_st3 >EQcont11 #EQ whd in EQ : (???%);
      [ <EQ in EQcont_st3; whd in ⊢ (??%% → ?); #EQ1 destruct ]
      destruct * #_ #H cases(stack_safety … EQcont_st3 … e0 H) #k1 *
      #EQk1 #EQlength %{(〈cost_act (None ?),new_code'〉::k1)} 
      % [ >EQk1 % | whd in ⊢ (??%%); @eq_f // ]
    | #lbl #EQconts1' normalize nodelta ******* #EQ destruct(EQ)
      #HH1 #EQ destruct(EQ) #EQget_el >EQcode11 in ⊢ (% → ?); 
      inversion(code … s1')
      [
      | #x 
      | #seq #lbl #i #_ 
      | #cond #ltrue #i1 #lfalse #i2 #i3 #_ #_ #_
      | #cond #ltrue #i1 #lfalse #i2 #_ #_
      | #f #act_p #ret_post #i #_
      | #l_in #io #l_out #i #_
      ]
      [|*: #_ whd in ⊢ (??%% → ?); #EQ cases daemon (* da fare assurdi !!!*) ]
      #EQcodes1' whd in ⊢ (??%% → ?); #EQ destruct(EQ) #EQstore #EQio #EQ destruct(EQ)
      cases(IH (mk_state ? new_code' new_cont' (store … s1') false) ll2 l3 …)
      [2: whd whd in match check_continuations; normalize nodelta
         change with (check_continuations ?????) in match (foldr2 ???????); 
         >EQHl2 in ⊢ (match % with [ _⇒ ? | _ ⇒ ?]);
         normalize nodelta % // % // % // % // @EQcode_c_st12 ]
      #abs_top' * #abs_tail' * #st3' * #t' *** #Hst3st3' #EQcost #EQlen #stack_safety 
      %{abs_top'} %{abs_tail'} %{st3'}
      %{(t_ind … (cost_act (Some ? lbl)) … t')}
      [ @hide_prf whd @(empty ????? 〈(cost_act (Some ? lbl)),?〉)
        [3: assumption |4: assumption |*:] /3 by nmk/ ]
      % [% [2: whd in ⊢ (??%%); >EQlen % ] %{Hst3st3'} >map_labels_on_trace_append
          whd in match (map_labels_on_trace ? [lbl]); >append_nil >EQget_el
          @is_permutation_cons assumption
        | * [| #hd #tl] #i #EQcont_st3 >EQcont11 #EQ whd in EQ : (???%);
          [ <EQ in EQcont_st3; whd in ⊢ (??%% → ?); #EQ1 destruct ]
          destruct * #_ #H cases(stack_safety … EQcont_st3 … e0 H) #k1 *
          #EQk1 #EQlength %{(〈cost_act (Some ? lbl),new_code'〉::k1)} 
          % [ >EQk1 % | whd in ⊢ (??%%); @eq_f // ]
        ]
    ]
  | #seq #i #mem * [|#lbl] #EQcode_st11 #EQeval_seq #EQ destruct(EQ) #EQcont
    #EQ destruct(EQ) #EQio11 #EQio12 #EQ1 #EQ2 #EQ3 #EQ4 destruct #t #Hst11
    #_ #Hpre_t #IH #s1' #abs_tail #abs_top whd in ⊢ (% → ?);
    inversion(check_continuations ?????) normalize nodelta [1,3: #_ *]
    ** #H1 #abs_top_cont #abs_tail_cont #EQcheck normalize nodelta ****
    #HH1 [ >EQcode_st11 in ⊢ (% → ?); | >EQcode_st11 in ⊢ (% → ?); ]
    inversion(code ??) [1,2,4,5,6,7,8,9,11,12,13,14: cases daemon (*assurdi*)]
    #seq1 #opt_l #i' #_ #EQcode_s1' change with (m_bind ?????) in ⊢ (??%? → ?);
    inversion(call_post_clean ????) [1,3: #_ whd in ⊢ (??%% → ?); #EQ destruct]
    * #gen_labs #i'' #EQclean >m_return_bind inversion(opt_l) normalize nodelta
    [2,4: #lab1 ] #EQ destruct(EQ)
    [1,2: inversion(?==?) normalize nodelta #EQget_el ]
    whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct(EQ)
    #EQstore #EQinfo #EQ destruct(EQ)
    cases(IH … (mk_state ? i' (cont … s1') (store … st12) (io_info … s1')) abs_tail_cont ?)
    [3,6: whd 
       [ <EQcont in ⊢ (match % with [_ ⇒ ? | _⇒ ?]); >EQcheck in ⊢ (match % with [_ ⇒ ? | _⇒ ?]); 
       | <EQcont in ⊢ (match % with [_ ⇒ ? | _⇒ ?]); >EQcheck in ⊢ (match % with [_ ⇒ ? | _⇒ ?]);
       ]
       normalize nodelta % // % // % // % // assumption
    |2,5:
    ]
    #abs_top1 * #abs_tail1 * #s2' * #t' *** #Hst3_s2' #EQcosts #EQlen #stack_safety 
    %{abs_top1}
    %{abs_tail1} %{s2'} %{(t_ind … t')}
    [1,4: @hide_prf @(seq_sil … EQcode_s1') //
    |2,5:
    ]
    % [1,3: % [2,4: whd in ⊢ (??%%); @eq_f // ] %{Hst3_s2'} [2: assumption]
        whd in ⊢ (??%%); whd in match (get_costlabels_of_trace ????);
        whd in match (map_labels_on_trace ??); >(\P EQget_el) @is_permutation_cons
        assumption
      |*: #k #i #EQcont_st3 >EQcont #EQ #H 
          cases(stack_safety … EQcont_st3 … EQ H) #k1 * #EQk1 #EQlength %{k1} % //
      ]
  | #cond #ltrue #i_true #lfalse #i_false #i #mem #EQcode_st11 #EQcond #EQcont_st12
    #EQ1 #EQ2 destruct(EQ1 EQ2) #EQio_st11 #EQio_st12 #EQ1 #EQ2 #EQ3 #EQ4 destruct #t
    #Hclass #_ #Hpre #IH #s1' #abs_tail #abs_top whd in ⊢ (% → ?); inversion(check_continuations ?????)
    [ #_ *] ** #H1 #abs_top_cont #abs_tail_cont #EQcheck normalize nodelta **** #HH1
    >EQcode_st11 in ⊢ (% → ?); inversion(code ??)
    [1,2,3,5,6,7: cases daemon (*ASSURDI*)] #cond1 #ltrue1 #i_true1 #lfalse1 #ifalse1 #i' #_ #_ #_
    #EQcode_s1' whd in ⊢ (??%% → ?); inversion(call_post_clean ?????) normalize nodelta
    [ #_ #EQ destruct ] * #gen_lab_i' #i'' #EQi'' inversion(call_post_clean ????)
    [ #_ whd in ⊢ (??%% → ?); #EQ destruct] * #gen_lab_ifalse1 #i_false' #EQi_false'
    >m_return_bind inversion(call_post_clean ????) [ #_ whd in ⊢ (??%% → ?); #EQ destruct]
    * #gen_lab_i_true' #i_true' #EQi_true' >m_return_bind inversion(?==?) normalize nodelta
    [2: #_ #EQ destruct] #EQget_ltrue1 inversion(?==?) normalize nodelta #EQget_lfalse1
    whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct(EQ) #EQstore11 #EQio11
    #EQ destruct(EQ) 
    cases(IH … 
     (mk_state ? i_true1 (〈cost_act (None ?),i'〉 :: cont … s1') (store … st12) (io_info … st12)) 
     abs_tail_cont gen_lab_i_true')
    [2: whd whd in match (check_continuations ?????); 
        >EQcont_st12 in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?]); whd in match (foldr2 ???????);
        change with (check_continuations ?????) in match (foldr2 ???????); 
        >EQcheck in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?]); normalize nodelta
        >EQi'' normalize nodelta % // % // % // % [2: assumption ] % // % // 
    ] #abs_top1 * #abs_tail1 * #s2' * #t' *** #Hst3_s2' #EQcost #EQlen 
      #stack_safety %{abs_top1} %{abs_tail1}
    %{s2'} %{(t_ind … t')}
    [ @hide_prf @(cond_true … EQcode_s1') //
    |
    ]
    % 
    [ % [2: whd in ⊢ (??%%); @eq_f // ] %{Hst3_s2'} 
      whd in match (get_costlabels_of_trace ????); whd in match(map_labels_on_trace ??);
      >(\P EQget_ltrue1) whd in match (get_costlabels_of_trace ????) in ⊢ (???%); @is_permutation_cons
      assumption
    |  #k #i1 #EQcont_st3 #EQcont_st11 #H 
       cases(stack_safety … (〈cost_act (None NonFunctionalLabel),i〉::k) … EQcont_st3 …)
       [2: >EQcont_st12 >EQcont_st11 % |3: % // % whd in ⊢ (% → ?); #EQ destruct] 
       * [|#hd1 #tl1] * #EQk1 whd in ⊢ (??%% → ?); #EQlength destruct 
       %{tl1} % // whd in EQk1 : (??%%); destruct //
    ]
  | #cond #ltrue #i_true #lfalse #i_false #i #mem #EQcode_st11 #EQcond #EQcont_st12
    #EQ1 #EQ2 destruct(EQ1 EQ2) #EQio_st11 #EQio_st12 #EQ1 #EQ2 #EQ3 #EQ4 destruct #t
    #Hclass #_ #Hpre #IH #s1' #abs_tail #abs_top whd in ⊢ (% → ?); inversion(check_continuations ?????)
    [ #_ *] ** #H1 #abs_top_cont #abs_tail_cont #EQcheck normalize nodelta **** #HH1
    >EQcode_st11 in ⊢ (% → ?); inversion(code ??)
    [1,2,3,5,6,7: cases daemon (*ASSURDI*)] #cond1 #ltrue1 #i_true1 #lfalse1 #ifalse1 #i' #_ #_ #_
    #EQcode_s1' whd in ⊢ (??%% → ?); inversion(call_post_clean ?????) normalize nodelta
    [ #_ #EQ destruct ] * #gen_lab_i' #i'' #EQi'' inversion(call_post_clean ????)
    [ #_ whd in ⊢ (??%% → ?); #EQ destruct] * #gen_lab_ifalse1 #i_false' #EQi_false'
    >m_return_bind inversion(call_post_clean ????) [ #_ whd in ⊢ (??%% → ?); #EQ destruct]
    * #gen_lab_i_true' #i_true' #EQi_true' >m_return_bind inversion(?==?) normalize nodelta
    [2: #_ #EQ destruct] #EQget_ltrue1 inversion(?==?) normalize nodelta #EQget_lfalse1
    whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct(EQ) #EQstore11 #EQio11
    #EQ destruct(EQ) 
    cases(IH … 
     (mk_state ? ifalse1 (〈cost_act (None ?),i'〉 :: cont … s1') (store … st12) (io_info … st12)) 
     abs_tail_cont gen_lab_ifalse1)
    [2: whd whd in match (check_continuations ?????); 
        >EQcont_st12 in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?]); whd in match (foldr2 ???????);
        change with (check_continuations ?????) in match (foldr2 ???????); 
        >EQcheck in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?]); normalize nodelta
        >EQi'' normalize nodelta % // % // % // % [2: assumption ] % // % // 
    ] #abs_top1 * #abs_tail1 * #s2' * #t' *** #Hst3_s2' #EQcost #EQlen #stack_safety 
    %{abs_top1} %{abs_tail1} %{s2'} %{(t_ind … t')}
    [ @hide_prf @(cond_false … EQcode_s1') //
    |
    ]
    % 
    [ % [2: whd in ⊢ (??%%); @eq_f // ] %{Hst3_s2'} 
      whd in match (get_costlabels_of_trace ????); whd in match(map_labels_on_trace ??);
      >(\P EQget_lfalse1) whd in match (get_costlabels_of_trace ????) in ⊢ (???%); @is_permutation_cons
      assumption
    | #k #i1 #EQcont_st3 #EQcont_st11 #H 
      cases(stack_safety … (〈cost_act (None NonFunctionalLabel),i〉::k) … EQcont_st3 …)
      [2: >EQcont_st12 >EQcont_st11 % |3: % // % whd in ⊢ (% → ?); #EQ destruct] 
      * [|#hd1 #tl1] * #EQk1 whd in ⊢ (??%% → ?); #EQlength destruct 
      %{tl1} % // whd in EQk1 : (??%%); destruct //
    ]
 | #cond #ltrue #i_true #lfalse #i_false #store #EQcode_st11 #EQev_loop #EQcont_st12 #EQ1 #EQ2 destruct
   #EQio_11 #EQio_12 #EQ1 #EQ2 #EQ3 #EQ4 destruct #t #Hclass11 #_ #Hpre #IH #s1' #abs_top #abs_tail
   whd in ⊢ (% → ?); inversion(check_continuations ?????) [ #_ * ] ** #H1 #abs_top_cont #abs_tail_cont
   #EQcheck normalize nodelta **** #HH1 >EQcode_st11 in ⊢ (% → ?); inversion(code ??)
   [1,2,3,4,6,7: cases daemon (*assurdi*) ] #cond1 #ltrue1 #i_true1 #lfalse1 #i_false1 #_ #_
   #EQcode_s1' whd in ⊢ (??%% → ?); inversion(call_post_clean ?????) normalize nodelta
   [ #_ #EQ destruct] * #gen_ifalse1 #i_false2 #EQi_false2 inversion(call_post_clean ?????)
   [ #_ whd in ⊢ (??%% → ?); #EQ destruct ] * #gen_itrue1 #i_true2 #EQi_true2 >m_return_bind
   inversion (?==?) normalize nodelta [2: #_ #EQ destruct] #EQget_ltrue1
   inversion(?==?) #EQget_lfalse1 whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) 
   -EQ #EQ destruct(EQ) #EQstore_11 #EQ_info11 #EQ destruct
   cases(IH … 
    (mk_state ? i_true1 
     (〈cost_act (None ?),LOOP … cond ltrue i_true1 lfalse i_false1〉 ::
        cont … s1') (store … st12) (io_info … st12)) abs_tail_cont gen_itrue1)
   [2: whd >EQcont_st12 whd in match (check_continuations ?????);
       whd in match (foldr2 ???????);
       change with (check_continuations ?????) in match (foldr2 ???????);
       >EQcheck normalize nodelta whd in match (call_post_clean ?????);
       >EQi_false2 normalize nodelta >EQi_true2 >m_return_bind >EQget_ltrue1
       >EQget_lfalse1 normalize nodelta % // % // % // % [2: assumption] % //
       % //
   ]
   #abs_cont1 * #abs_tail1 * #s2' * #t' *** #Hst3_s2' #EQcosts #EQlen #stack_safety
   %{abs_cont1} %{abs_tail1} %{s2'} %{(t_ind … t')}
   [ @hide_prf @(loop_true … EQcode_s1') // |]
   % 
   [ % [2: whd in ⊢ (??%%); @eq_f //] %{Hst3_s2'}
     whd in match (get_costlabels_of_trace ????); whd in match(map_labels_on_trace ??);
      >(\P EQget_ltrue1) whd in match (get_costlabels_of_trace ????) in ⊢ (???%); 
     @is_permutation_cons assumption
   | #k #i1 #EQcont_st3 #EQcont_st11 #H 
      cases(stack_safety … (〈cost_act (None NonFunctionalLabel), LOOP p cond ltrue (code p st12) lfalse i_false〉::k) … EQcont_st3 …)
      [2: >EQcont_st12 >EQcont_st11 % |3: % // % whd in ⊢ (% → ?); #EQ destruct] 
      * [|#hd1 #tl1] * #EQk1 whd in ⊢ (??%% → ?); #EQlength destruct 
      %{tl1} % // whd in EQk1 : (??%%); destruct //
   ]
 | #cond #ltrue #i_true #lfalse #i_false #mem #EQcode_st11 #EQev_loop #EQcont_st12 #EQ1 #EQstore11 destruct
   #EQio_11 #EQio_12 #EQ1 #EQ2 #EQ3 #EQ4 destruct #t #Hclass11 #_ #Hpre #IH #s1' #abs_top #abs_tail
   whd in ⊢ (% → ?); inversion(check_continuations ?????) [ #_ * ] ** #H1 #abs_top_cont #abs_tail_cont
   #EQcheck normalize nodelta **** #HH1 >EQcode_st11 in ⊢ (% → ?); inversion(code ??)
   [1,2,3,4,6,7: cases daemon (*assurdi*) ] #cond1 #ltrue1 #i_true1 #lfalse1 #i_false1 #_ #_
   #EQcode_s1' whd in ⊢ (??%% → ?); inversion(call_post_clean ?????) normalize nodelta
   [ #_ #EQ destruct] * #gen_ifalse1 #i_false2 #EQi_false2 inversion(call_post_clean ?????)
   [ #_ whd in ⊢ (??%% → ?); #EQ destruct ] * #gen_itrue1 #i_true2 #EQi_true2 >m_return_bind
   inversion (?==?) normalize nodelta [2: #_ #EQ destruct] #EQget_ltrue1
   inversion(?==?) #EQget_lfalse1 whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) 
   -EQ #EQ destruct(EQ) #EQstore_11 #EQ_info11 #EQ destruct
   cases(IH … 
    (mk_state ? i_false1 
     (cont … s1') (store … st12) (io_info … st12)) abs_tail_cont gen_ifalse1)
   [2: whd >EQcont_st12 whd in match (check_continuations ?????);
       whd in match (foldr2 ???????);
       change with (check_continuations ?????) in match (foldr2 ???????);
       >EQcheck normalize nodelta whd in match (call_post_clean ?????);
       >EQi_false2 normalize nodelta >EQi_true2 
        % // % // % // % [2: %] //
   ]
   #abs_cont1 * #abs_tail1 * #s2' * #t' *** #Hst3_s2' #EQcosts #EQlen #stack_safety 
   %{abs_cont1} %{abs_tail1} %{s2'} %{(t_ind … t')}
   [ @hide_prf @(loop_false … EQcode_s1') // |]
   %
   [ % [2: whd in ⊢ (??%%); @eq_f //] %{Hst3_s2'}
     whd in match (get_costlabels_of_trace ????); whd in match(map_labels_on_trace ??);
     >(\P EQget_lfalse1) whd in match (get_costlabels_of_trace ????) in ⊢ (???%); 
     @is_permutation_cons assumption
   | #k #i #EQcont_st3 <EQcont_st12 #EQ #H 
     cases(stack_safety … EQcont_st3 … EQ H) #k1 * #EQk1 #EQlength %{k1} % //
   ]
 | #lin #io #lout #i #mem #EQcode_st11 #EQev_io #EQcost_st12 #EQcont_st12
   #EQ destruct #EQio_12 #EQ1 #EQ2 #EQ3 #EQ4 destruct #t #Hclass_11 #_ #Hpre
   #IH #s1' #abs_top #abs_tail whd in ⊢ (% → ?); inversion(check_continuations ?????)
   [#_ *] ** #H1 #abs_top_cont #abs_tail_cont #EQcheck normalize nodelta ****
   #HH1 >EQcode_st11 in ⊢ (% → ?); inversion(code ??)
   [1,2,3,4,5,6: cases daemon (*ASSURDI*)] #lin1 #io1 #lout1 #i1 #_ #EQcode_s1'
   whd in ⊢ (??%% → ?); inversion(call_post_clean ?????) normalize nodelta
   [ #_ #EQ destruct] * #gen_lab #i2 #EQ_i2 inversion(?==?) normalize nodelta
   [2: #_ #EQ destruct] #EQget_lout1 inversion(?==?) #EQget_lin1 normalize nodelta
   whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct(EQ) 
   #EQstore11 #EQinfo11 #EQ destruct
   cases(IH …
    (mk_state ? (EMPTY p) (〈cost_act (Some ? lout),i1〉::cont … s1') (store … st12) true)
    abs_tail_cont [ ])
   [2: whd >EQcont_st12 whd in match (check_continuations ?????);
       whd in match (foldr2 ???????);
       change with (check_continuations ?????) in match (foldr2 ???????);
       >EQcheck normalize nodelta >EQ_i2 normalize nodelta % // % // % // % 
       [2: >EQcost_st12 % ] % [ % // % //] >(\P EQget_lout1) %
   ]
   #abs_cont1 * #abs_tail1 * #s2' * #t' *** #Hst3_s2' #EQcosts #EQlen #stack_safety 
   %{abs_cont1} %{abs_tail1} %{s2'} %{(t_ind … t')}
   [ @hide_prf @(io_in … EQcode_s1') // |]
   % 
   [ % [2: whd in ⊢ (??%%); @eq_f //] %{Hst3_s2'}
     whd in match (get_costlabels_of_trace ????); whd in match(map_labels_on_trace ??);
     >(\P EQget_lin1) whd in match (get_costlabels_of_trace ????) in ⊢ (???%); 
     @is_permutation_cons assumption
   | #k #i1 #EQcont_st3 #EQcont_st11 #H 
      cases(stack_safety … (〈cost_act (Some ? lout),i〉::k) … EQcont_st3 …)
      [2: >EQcont_st12 >EQcont_st11 % |3: % // % whd in ⊢ (% → ?); #EQ destruct] 
      * [|#hd1 #tl1] * #EQk1 whd in ⊢ (??%% → ?); #EQlength destruct 
      %{tl1} % // whd in EQk1 : (??%%); destruct //
   ]
  | #f #act_p #r_lb #cd #mem #env_it #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 #EQ10
    #EQ11 #EQ12 destruct #tl #_ * #x #EQ destruct(EQ)
  | #r_t #mem #con #rb #i #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 #EQ7 #EQ8 #EQ9 #EQ10 #EQ11 #EQ12
    destruct #t #_ * #nf #EQ destruct
  ]
| #st1 #st2 #st3 #lab #st1_noio #H inversion H in ⊢ ?; [1,2,3,4,5,6,7,8: cases daemon (*assurdi*)]
  #st11 #st12 #r_t #mem #cont * [|#x] #i #EQcode_st11 #EQcont_st11 #EQ destruct(EQ) 
  #EQio_11 #EQio_12 #EQev_ret #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 destruct(EQ1 EQ2 EQ3 EQ4 EQ5 EQ6)
  #t * #lbl #EQ destruct(EQ) #pre_t #IH #s1' #abs_tail #abs_top whd in ⊢ (% → ?);
  inversion(check_continuations …) [#_ *] ** #H1 #abs_top_cont #abs_tail_cont 
  >EQcont_st11 in ⊢ (% → ?); inversion(cont … s1') [ #_ whd in ⊢ (??%% → ?); #EQ destruct(EQ)]
  * #act_lab #i #cont_s1'' #_ #EQcont_s1' whd in ⊢ (??%% → ?); 
  change with (check_continuations ?????) in match (foldr2 ???????);
  inversion(check_continuations ?????) normalize nodelta [ #_ whd in ⊢ (??%% → ?); #EQ destruct]
  ** #H2 #abs_top_cont' #abs_tail_cont' #EQcheck normalize nodelta
  inversion(call_post_clean ?????) normalize nodelta [ #_ whd in ⊢ (??%% → ?); #EQ destruct *****]
  * #gen_labs #i' #EQi' inversion act_lab normalize nodelta
  [ #f #lab | * [| #lbl1 ]| * [| #nf_l] |] #EQ destruct(EQ) 
  [1,6: whd in ⊢ (??%% → ?); #EQ destruct ***** 
  |4,5: normalize nodelta whd in ⊢ (??%% → ?); #EQ destruct(EQ) ****** [2: *] #EQ destruct
  ] 
  whd in match ret_costed_abs; normalize nodelta 
  [ whd in ⊢ (??%% → ?); #EQ destruct(EQ) ****** #EQ destruct(EQ) ]
  inversion (memb ???) normalize nodelta #Hlbl1 whd in ⊢ (??%% → ?); #EQ destruct(EQ)
  ****** [ * ] #EQ destruct(EQ) #HH2 #EQ destruct #EQget_el >EQcode_st11 in ⊢ (% → ?);
  inversion(code … s1') [1,3,4,5,6,7: cases daemon (*TODO*) ] #r_t' #EQcode_s1' 
  whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct(EQ) 
  #EQstore11 #EQio_11 #EQ destruct
  cases(IH … (mk_state ? i cont_s1'' (store … st12) (io_info … st12)) abs_tail_cont gen_labs) 
  [2: whd >EQcheck normalize nodelta % // % // % // % // @EQi' ] 
  #abs_top1 * #abs_tail1 * #s2' * #t' *** #Hst3_s2' #EQcosts #EQlen #stack_safety
  %{abs_top1} %{abs_tail1} %{s2'} %{(t_ind … t')}
  [ @hide_prf @(ret_instr … EQcode_s1' … EQcont_s1') //
  | ]
  %
  [ % [2: whd in ⊢ (??%%); @eq_f // ] %{Hst3_s2'} 
    whd in match (get_costlabels_of_trace ????); 
    whd in match (get_costlabels_of_trace ????) in ⊢ (???%);
    whd in match (map_labels_on_trace ??); >EQget_el @is_permutation_cons
    assumption 
  | * [| #hd #tl] #i1 #EQcont_st3 >EQcont_st11 #EQ whd in EQ : (???%);
    [ <EQ in EQcont_st3; whd in ⊢ (??%% → ?); #EQ1 destruct ]
    destruct * #_ #H cases(stack_safety … EQcont_st3 … e0 H) #k1 *
    #EQk1 #EQlength %{(〈ret_act (Some ? lbl1),i〉::k1)} 
    % [ >EQk1 % | whd in ⊢ (??%%); @eq_f // ]
  ]
| #st1 #st2 #st3 #lab #H inversion H in ⊢ ?; #st11 #st12
  [1,2,3,4,5,6,7,9: cases daemon (*absurd!*)
  | #f #act_p #opt_l #i #mem #env_it #EQcode11 #EQenv_it #EQmem
    #EQ destruct(EQ) #EQcode12 #EQcont12 #EQio11 #EQio12 #EQ1 #EQ2 #EQ3 
    destruct(EQ1 EQ2 EQ3) #EQ destruct(EQ) #tl #_ #_ whd in ⊢ (?% → ?); >EQcode11 in ⊢ (% → ?);
    normalize nodelta cases opt_l in EQcode11 EQcont12; normalize nodelta [ #x #y *]
    #lbl #EQcode11 #EQcont12 * #pre_tl #IH #st1' #abs_top #abs_tail
    whd in ⊢ (% → ?); inversion(check_continuations ????) [ #_ *] ** #H1 
    #abs_top_cont #abs_tail_cont #EQcheck
    normalize nodelta **** #HH1 >EQcode11 in ⊢ (% → ?); inversion(code … st1')
    [1,2,3,4,5,7: (*assurdi da fare*) cases daemon] #f' #act_p' #opt_l' #i' #_
    #EQcode_st1' #EQclean #EQstore #EQio #EQ destruct(EQ)
    lapply(trans_env_ok … no_dup) >EQtrans normalize nodelta #H
    cases(H … EQenv_it) -H #env_it' * #fresh' ** #is_fresh_fresh' 
    #EQenv_it' ***** #EQtrans 
    #EQgen_labels #EQsignature #EQlab_env_it #same_map #same_keep
    change with (m_bind ?????) in EQclean : (??%?); inversion(call_post_clean ?????) in EQclean;
    [ #_ whd in ⊢ (??%% → ?); #EQ destruct] * #abs_top'' #i'' #EQi' >m_return_bind
    inversion opt_l' in EQcode_st1'; [| #lbl'] #EQopt_l' #EQcode_st1' normalize nodelta
    [2: inversion(memb ???) normalize nodelta #Hlbl_keep
        [  inversion (get_element ????) normalize nodelta [ #_ whd in ⊢ (??%% → ?); #EQ destruct]
           #lbl'' #l3' #_ #EQget_el whd in match (eqb ???); inversion (eqb ???) normalize nodelta
          [2: #_ whd in ⊢ (??%% → ?); #EQ destruct] #H cases(eqb_true ? lbl'' lbl') #H1 #_
          lapply(H1 H) -H -H1 #EQ destruct(EQ) inversion(eqb_list ???) normalize nodelta
          [2: #_ whd in ⊢ (??%% → ?); #EQ destruct] #H cases(eqb_true (DeqSet_List ?) l3' abs_top'')
          #H1 #_ lapply(H1 H) -H -H1 #EQ destruct(EQ) whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) -EQ
          #EQ destruct(EQ)
          cases(IH (mk_state ? (f_body … env_it') (〈ret_act opt_l',i'〉 :: (cont … st1')) (store ? st12) false) (abs_tail_cont) (gen_labels … (call_post_trans … (f_body … env_it) fresh' (nil ?))))
          [2: whd >EQcont12
            change with (m_bind ??? (check_continuations ?????) ?) in match (check_continuations ?????);
            >EQcheck >m_return_bind normalize nodelta >EQi' normalize nodelta >EQopt_l'
            whd in match ret_costed_abs; normalize nodelta >Hlbl_keep normalize nodelta
            % // % // % // % [/5 by conj/] >EQgen_labels >EQcode12 <EQtrans
            @(inverse_call_post_trans … is_fresh_fresh') 
            [2: % |*: [2,3: /2 by / ] 
                cases(lookup_ok_append … EQenv_it) #env1 * #env2 * #EQ1 #EQ2
                whd in no_dup; destruct >EQ1 in no_dup; >foldr_map_append >foldr_map_append
                #no_dup lapply(no_duplicates_append_r … no_dup) #H1 
                lapply(no_duplicates_append_l … H1) whd in match (foldr ?????); -H1
                change with ([?]@?) in match (?::?); #H1 
                lapply(no_duplicates_append_r … H1) >append_nil //
            ]
          ]
          #abs_top''' * #abs_tail''' * #st3' * #t' *** #Hst3st3' #EQcosts #EQlen #stack_safety 
          %{abs_top'''}
          %{abs_tail'''} %{st3'} %{(t_ind … (call_act f (f_lab … env_it')) …  t')}
          [ @hide_prf @call /2 width=10 by jmeq_to_eq/ ] 
          % 
          [ % [2: whd in ⊢ (??%%); >EQlen %]
            %{Hst3st3'} >map_labels_on_trace_append whd in match (get_costlabels_of_trace ????) in ⊢ (???%);
            >EQlab_env_it >associative_append whd in match (append ???); >associative_append
            >associative_append in ⊢ (???%); >(associative_append … [?]) in ⊢ (???%);
            whd in match (map_labels_on_trace ??); >EQgen_labels @is_permutation_cons
            >append_nil whd in match (append ???) in ⊢ (???%); //
          | #k #i1 #EQcont_st3 #EQcont_st11 #H 
            cases(stack_safety … (〈ret_act (Some ? lbl),i〉::k) … EQcont_st3 …)
            [2: >EQcont12 >EQcont_st11 % |3: % // % whd in ⊢ (% → ?); #EQ destruct] 
            * [|#hd1 #tl1] * #EQk1 whd in ⊢ (??%% → ?); #EQlength destruct 
            %{tl1} % // whd in EQk1 : (??%%); destruct //
          ]
        | whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct(EQ)
        ]
    | whd in ⊢ (??%% → ?); #EQ destruct(EQ)
    ]
  ]
| #st1 #st2 #st3 #st4 #st5 #l1 #l2 #H inversion H in ⊢ ?;
  [1,2,3,4,5,6,7,9: cases daemon (*assurdi*) ]
  #st11 #st12 #f #act_p #opt_l #i #mem #env_it #EQcode11 #EQenv_it #EQmem
  #EQ destruct(EQ) #EQcode12 #EQcont12 #EQio11 #EQio12 #EQ1 #EQ2 #EQ3 
  destruct(EQ1 EQ2 EQ3) #EQ destruct(EQ) #t1 #t2 #H inversion H in ⊢ ?;
  [1,2,3,4,5,6,7,8: cases daemon (*assurdi*) ]
   #st41 #st42 #r_t #mem1 #new_cont #opt_l1 #cd #EQcode41 #EQcont41 #EQ destruct(EQ)
   #EQio41 #EQio42 #EQmem1 #EQ1 #EQ2 #EQ3 #EQ4 #EQ5 #EQ6 destruct #st11_noio 
   #st41_noio #_ whd in ⊢ (?(?%) → ?); >EQcode11 in ⊢ (% → ?); normalize nodelta
   inversion(opt_l) in ⊢(% → ?); normalize nodelta [2: #lbl #_ * ] #EQopt_l destruct(EQopt_l) 
   #_ #Hpre_t1 #Hpre_t2 whd in ⊢ (% → ?); * #EQcont11_42 >EQcode11 in ⊢ (% → ?);
   normalize nodelta #EQ destruct(EQ) whd in ⊢ (% → ?); 
   #EQ destruct #IH1 #IH2 #st1' #abs_top #abs_tail
    whd in ⊢ (% → ?); inversion(check_continuations ????) [ #_ *] ** #H1 
    #abs_top_cont #abs_tail_cont #EQcheck
    normalize nodelta **** #HH1 >EQcode11 in ⊢ (% → ?); inversion(code … st1')
    [1,2,3,4,5,7: (*assurdi da fare*) cases daemon] #f' #act_p' #opt_l' #i' #_
    #EQcode_st1' #EQclean #EQstore #EQio #EQ destruct(EQ)
    lapply(trans_env_ok … no_dup) >EQtrans normalize nodelta #H
    cases(H … EQenv_it) -H #env_it' * #fresh' ** #is_fresh_fresh' #EQenv_it' ***** #EQtrans 
    #EQgen_labels #EQsignature #EQlab_env_it #same_map #same_keep
    change with (m_bind ?????) in EQclean : (??%?); inversion(call_post_clean ?????) in EQclean;
    [ #_ whd in ⊢ (??%% → ?); #EQ destruct] * #abs_top'' #i'' #EQi' >m_return_bind
    inversion opt_l' in EQcode_st1'; [| #lbl'] #EQopt_l' #EQcode_st1' normalize nodelta
    [ whd in ⊢ (??%% → ?); #EQ destruct(EQ) ] destruct(EQopt_l') 
    inversion(memb ???) normalize nodelta #Hlbl_keep'
    [ cases(?==?) normalize nodelta ]
     whd in ⊢ (??%% → ?); #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct
     cases(IH1 
            (mk_state ? 
              (f_body … env_it') 
              (〈ret_act (Some ? lbl'),i'〉 :: (cont … st1')) 
              (store ? st12) false) 
            (abs_top''@abs_tail_cont) 
            (gen_labels … (call_post_trans … (f_body … env_it) fresh' (nil ?))))
     [2: whd >EQcont12
         change with (m_bind ??? (check_continuations ?????) ?) in match (check_continuations ?????);
          >EQcheck >m_return_bind normalize nodelta >EQi' normalize nodelta
          whd in match ret_costed_abs; normalize nodelta >Hlbl_keep' normalize nodelta
          % // % // % // % [/5 by conj/] >EQcode12 <EQtrans
          @(inverse_call_post_trans … fresh') 
          [2: % |*: [2,3: /2 by / ] 
              cases(lookup_ok_append … EQenv_it) #env1 * #env2 * #EQ1 #EQ2
              whd in no_dup; destruct >EQ1 in no_dup; >foldr_map_append >foldr_map_append
              #no_dup lapply(no_duplicates_append_r … no_dup) #H1 
              lapply(no_duplicates_append_l … H1) whd in match (foldr ?????); -H1
              change with ([?]@?) in match (?::?); #H1 
              lapply(no_duplicates_append_r … H1) >append_nil //
          ]
     ]
     #abs_top''' * #abs_tail''' * #st41' * #t1' *** #Hst41st41' #EQcosts #EQlen #stack_safety1
     whd in Hst41st41'; inversion(check_continuations …) in Hst41st41'; [ #_ * ]
     ** #H2 #abs_top_cont' #abs_tail_cont' >EQcont41 in ⊢ (% → ?); 
     whd in ⊢ (??%? → ?); inversion(cont … st41') normalize nodelta 
     [ #_ #EQ destruct(EQ) ] * #act_lbl #i'' #cont_st42' #_ #EQcont_st41'
     change with (check_continuations ?????) in match (foldr2 ???????);
     inversion(check_continuations ?????) [ #_ whd in ⊢ (??%% → ?); #EQ destruct]
     ** #H3 #abs_top_cont'' #abs_tail_cont'' #EQ_contst42 >m_return_bind
     normalize nodelta inversion(call_post_clean ?????) normalize nodelta
     [ #_ whd in ⊢ (??%% → ?); #EQ destruct ***** ] * #abs_top'''' #i'''
     #EQ_clean_i'' inversion(act_lbl) normalize nodelta [1,3,4: cases daemon (*assurdi*)]
     cut(act_lbl = ret_act (Some ? lbl') ∧ cont_st42' = cont … st1' ∧ i'' = i') 
     [ cases(stack_safety1 [ ] …)
       [3: >EQcont41 in ⊢ (??%?); % |4: normalize // |5: % |2:] * [|#x #xs] *
       whd in ⊢ (??%% → ??%% → ?); #EQ1 #EQ2 destruct > EQcont_st41' in EQ1; 
       #EQ destruct(EQ) /3 by conj/
     ]
     ** #EQ1 #EQ2 #EQ3 destruct #x #EQ destruct whd in match ret_costed_abs; normalize nodelta
     >Hlbl_keep' normalize nodelta
     whd in ⊢ (??%% → ?); #EQ destruct ****** #_ #HH3 #EQ destruct(EQ)
     >EQcode41 in ⊢ (% → ?); inversion(code … st41') 
     [1,3,4,5,6,7: cases daemon (*ASSURDI*) ] #r_t' #EQcode_st41' whd in ⊢ (??%% → ?);
     #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct(EQ) #EQstore_st41' #EQinfo_st41'
     #EQ destruct(EQ) >EQcont11_42 in EQcheck; >EQ_contst42 in ⊢ (% → ?);
     #EQ destruct(EQ) >EQi' in EQ_clean_i''; #EQ destruct(EQ)
     cases(IH2 
             (mk_state ? i' (cont … st1') (store … st42) (io_info … st42))
             abs_tail_cont abs_top'''')
     [2: whd >EQ_contst42 normalize nodelta % // % // % // % // @EQi' ]
     #abs_top_1 * #abs_tail_1 * #st5' * #t2' *** #Hst5_st5' #EQcosts' 
     #EQlen'  #stack_safety2 %{abs_top_1} %{abs_tail_1} %{st5'} %{(t_ind … (t1' @ (t_ind … t2')))}
     [3: @hide_prf @(call …  EQcode_st1' …  EQenv_it') //
     |1: @hide_prf @(ret_instr … EQcode_st41' … EQcont_st41') //
     |2,4: skip
     ] 
     % 
     [ % [2: whd in ⊢ (??%%); @eq_f >len_append >len_append @eq_f2 //
           whd in ⊢ (??%%); @eq_f // ]
       %{Hst5_st5'} whd in match (map_labels_on_trace ??);
       change with (map_labels_on_trace ??) in match (foldr ?????);
       >map_labels_on_trace_append >get_cost_label_append
       whd in match (get_costlabels_of_trace ??? (t_ind …));
       whd in match (get_costlabels_of_trace ??? (t_ind …));
       >get_cost_label_append
       whd in match (get_costlabels_of_trace ??? (t_ind …));
       >map_labels_on_trace_append >EQlab_env_it >EQgen_labels
       whd in match (map_labels_on_trace ? [ ]);
       whd in match (append ? (nil ?) ?);
       cut (∀A.∀l: list A. [ ] @ l = l) [//] #nil_append
       >nil_append >nil_append >nil_append @(permute_ok … EQcosts EQcosts')
     | #k #i #EQcont_st3 #EQ #H
       cases(stack_safety2 … EQcont_st3 … EQ H) #k1 * #EQk1 #EQlength %{k1} % //
     ]
]
qed.