source: src/RTLabs/RTLAbstoRTL.ma @ 1280

include "RTLabs/syntax.ma".
include "RTL/RTL.ma".
include "common/AssocList.ma".
include "common/FrontEndOps.ma".
include "common/Graphs.ma".
include "joint/TranslateUtils.ma".

(*
let rec register_freshes (runiverse: universe RegisterTag) (n: nat) on n ≝
  match n with
  [ O ⇒ [ ]
  | S n' ⇒ register_fresh runiverse :: (register_freshes runiverse n')
  ].
*)

definition choose_rest ≝
  λA: Type[0].
  λleft, right: list A.
  λprfl: 0 < |left|.
  λprfr: 0 < |right|.
  match left return λx. 0 < |x| → list A with
  [ nil ⇒ λnil_prf.
    match right return λx. 0 < |x| → list A with
    [ nil ⇒ λnil_nil_absrd. ?
    | _ ⇒ λnil_cons_prf. right
    ] prfr
  | _ ⇒ λcons_prf. right
  ] prfl.
  normalize in nil_prf;
  cases(not_le_Sn_O 0)
  #HYP
  cases(HYP nil_prf)
qed.

definition complete_regs ≝
  λglobals.
  λdef.
  λsrcrs1.
  λsrcrs2.
  if leb (length … srcrs2) (length … srcrs1) then
   let 〈def, added_regs〉 ≝ fresh_regs rtl_params0 globals def (minus (length ? srcrs1) (length ? srcrs2)) in
    〈〈srcrs1, srcrs2 @ added_regs〉, added_regs〉
  else
   let 〈def, added_regs〉 ≝ fresh_regs rtl_params0 globals def (minus (length ? srcrs2) (length ? srcrs1)) in
    〈〈srcrs1 @ added_regs, srcrs2〉, added_regs〉.

lemma complete_regs_length:
  ∀globals,def,left,right.
   |\fst (\fst (complete_regs globals def left right))| = |\snd (\fst (complete_regs globals def left right))|.
 #globals #def #left #right
 whd in match complete_regs normalize nodelta
 @leb_elim normalize nodelta #H
 [ generalize in match (fresh_regs_length rtl_params0 globals def (minus (length … left) (length … right)))
 | generalize in match (fresh_regs_length rtl_params0 globals def (minus (length … right) (length … left)))]
 cases (fresh_regs ????) #def' #fresh normalize >append_length
 generalize in match H -H;
 generalize in match (length … left) generalize in match (length … right) generalize in match (length … fresh)
 [ /2/ | #x #y #z #H generalize in match (not_le_to_lt … H) -H #H #E >E >commutative_plus
         <plus_minus_m_m /2/ ]
qed.

definition size_of_sig_type ≝
  λsig.
  match sig with
  [ ASTint isize sign ⇒
    let isize' ≝ match isize with [ I8 ⇒ 8 | I16 ⇒ 16 | I32 ⇒ 32 ] in
      isize' ÷ (nat_of_bitvector ? int_size)
  | ASTfloat _ ⇒ ? (* dpm: not implemented *)
  | ASTptr rgn ⇒ nat_of_bitvector ? ptr_size
  ].
  cases not_implemented;
qed.

inductive register_type: Type[0] ≝
  | register_int: register → register_type
  | register_ptr: register → register → register_type.

definition local_env ≝ BitVectorTrie (list register) 16.

definition mem_local_env : register → local_env → bool ≝
  λr. member … (word_of_identifier … r).

definition add_local_env : register → list register → local_env → local_env ≝
  λr. insert … (word_of_identifier … r).

definition find_local_env : register → local_env → list register ≝
  λr: register.λbvt. lookup … (word_of_identifier … r) bvt [].

(*
definition initialize_local_env_internal ≝
  λruniverse.
  λlenv.
  λr_sig.
  let 〈r, sig〉 ≝ r_sig in
  let size ≝ size_of_sig_type sig in
  let rs ≝ register_freshes runiverse size in
    add_local_env r rs lenv.

definition initialize_local_env ≝
  λruniverse.
  λregisters.
  λresult.
  let registers ≝ registers @
    match result with
    [ None ⇒ [ ]
    | Some rt ⇒ [ rt ]
    ]
  in
    foldl ? ? (initialize_local_env_internal runiverse) (Stub …) registers.
*)

definition map_list_local_env_internal ≝
  λlenv,res,r. res @ (find_local_env r lenv).
    
definition map_list_local_env ≝
  λlenv,regs. foldl … (map_list_local_env_internal lenv) [ ] regs.

definition make_addr ≝
  λA.
  λlst: list A.
  λprf: 2 ≤ length A lst.
  match lst return λx. 2 ≤ |x| → A × A with
  [ nil ⇒ λlst_nil_prf. ?
  | cons hd tl ⇒ λprf.
    match tl return λx. 1 ≤ |x| → A × A with
    [ nil ⇒ λtl_nil_prf. ?
    | cons hd' tl' ⇒ λtl_cons_prf. 〈hd, hd'〉
    ] ?
  ] prf.
  [1: normalize in lst_nil_prf;
      cases(not_le_Sn_O 1);
      #HYP cases(HYP lst_nil_prf)
  |2: normalize in prf;
      @le_S_S_to_le
      assumption
  |3: normalize in tl_nil_prf;
      cases(not_le_Sn_O 0)
      #HYP cases(HYP tl_nil_prf)]
qed.

definition find_and_addr ≝
  λr,lenv. make_addr ? (find_local_env r lenv).

definition rtl_args ≝
  λregs_list,lenv. flatten … (map … (λr. find_local_env r lenv) regs_list).

definition translate_cst_int_internal ≝
  λglobals,dest_lbl,r,i. joint_st_sequential rtl_params_ globals (joint_instr_int … r i) dest_lbl.

(*CSC: XXXXX *)
axiom translate_cst:
  ∀globals.
  ∀cst: constant.
  ∀destrs: list register.
  ∀start_lbl: label.
  ∀dest_lbl: label.
  ∀def: rtl_internal_function globals.
    rtl_internal_function globals.

definition translate_move_internal ≝
  λglobals.
  λstart_lbl: label.
  λdestr: register.
  λsrcr: register.
    joint_st_sequential rtl_params_ globals (joint_instr_move … 〈destr,srcr〉) start_lbl.

definition translate_move ≝
  λglobals.
  λdestrs: list register.
  λsrcrs: list register.
  λstart_lbl: label.
    match reduce_strong register register destrs srcrs with
    [ dp crl_crr len_proof ⇒
      let commonl ≝ \fst (\fst crl_crr) in
      let commonr ≝ \fst (\snd crl_crr) in
      let restl ≝ \snd (\fst crl_crr) in
      let restr ≝ \snd (\snd crl_crr) in
      let f_common ≝ translate_move_internal globals start_lbl in
      let translate1 ≝ adds_graph rtl_params1 … (map2 … f_common commonl commonr ?) in
      let translate2 ≝ translate_cst … (Ointconst ? (repr I8 0)) restl in (* should this be 8? *)
        add_translates … [ translate1 ; translate2 ] start_lbl
    ].
    @len_proof
qed.

let rec make
  (A: Type[0]) (elt: A) (n: nat) on n ≝
  match n with
  [ O ⇒ [ ]
  | S n' ⇒ elt :: make A elt n'
  ].
  
lemma make_length:
  ∀A: Type[0].
  ∀elt: A.
  ∀n: nat.
    n = length ? (make A elt n).
  #A #ELT #N
  elim N
  [ normalize %
  | #N #IH
    normalize <IH %
  ]
qed.

definition translate_cast_unsigned ≝
  λglobals.
  λdestrs.
  λstart_lbl.
  λdest_lbl.
  λdef: joint_internal_function … (rtl_params globals).
  let 〈def, tmp_zero〉 ≝ fresh_reg … def in
  let zeros ≝ make … tmp_zero (length … destrs) in
    add_translates … [
      adds_graph rtl_params1 … [
        joint_st_sequential rtl_params_ … (joint_instr_int rtl_params_ ? tmp_zero (bitvector_of_nat ? 0)) start_lbl
        ];
      translate_move globals destrs zeros
    ] start_lbl dest_lbl def.

definition translate_cast_signed ≝
  λglobals.
  λdestrs.
  λsrcr.
  λstart_lbl.
  λdest_lbl.
  λdef.
  let 〈def, tmp_128〉 ≝ fresh_reg … def in
  let 〈def, tmp_255〉 ≝ fresh_reg … def in
  let 〈def, tmpr〉 ≝ fresh_reg … def in
  let 〈def, dummy〉 ≝ fresh_reg … def in
  let insts ≝ [
    rtl_st_int tmp_128 (bitvector_of_nat ? 128) start_lbl;
    rtl_st_op2 And tmpr tmp_128 srcr start_lbl;
    rtl_st_opaccs DivuModu tmpr dummy tmpr tmp_128 start_lbl;
    rtl_st_int tmp_255 (bitvector_of_nat ? 255) start_lbl;
    rtl_st_opaccs Mul tmpr dummy tmpr tmp_255 start_lbl
  ] in
  let srcrs ≝ make … tmpr (length … destrs) in
    add_translates … [
      adds_graph … insts;
      translate_move destrs srcrs
    ] start_lbl dest_lbl def.

definition translate_cast ≝
  λsrc_size.
  λsrc_sign.
  λdest_size.
  λdestrs.
  λsrcrs.
  match |srcrs| return λx. |srcrs| = x → ? with
  [ O ⇒ λzero_prf. adds_graph [ ]
  | S n' ⇒ λsucc_prf.
    if ltb dest_size src_size then
      translate_move destrs srcrs
    else
      match reduce_strong register register destrs srcrs with
      [ dp crl len_proof ⇒
        let commonl ≝ \fst (\fst crl) in
        let commonr ≝ \fst (\snd crl) in
        let restl ≝ \snd (\fst crl) in
        let restr ≝ \snd (\snd crl) in
        let insts_common ≝ translate_move commonl commonr in
        let sign_reg ≝ last_safe ? srcrs ? in
        let insts_sign ≝
          match src_sign with
          [ Unsigned ⇒ translate_cast_unsigned restl
          | Signed ⇒ translate_cast_signed restl sign_reg
          ]
        in
          add_translates [ insts_common; insts_sign ]
      ]
  ] (refl ? (|srcrs|)).
  >succ_prf //
qed.

definition translate_negint ≝
  λdestrs.
  λsrcrs.
  λstart_lbl.
  λdest_lbl.
  λdef.
  λprf: | destrs | = | srcrs |. (* assert in caml code *)
  let 〈def, tmpr〉 ≝ fresh_reg def in
  let f_cmpl ≝ λdestr. λsrcr. rtl_st_op1 Cmpl destr srcr start_lbl in
  let insts_cmpl ≝ map2 ? ? ? f_cmpl destrs srcrs prf in
  let insts_init ≝ [
    rtl_st_set_carry start_lbl;
    rtl_st_int tmpr (zero ?) start_lbl
  ] in
  let f_add ≝ λdestr. rtl_st_op2 Addc destr destr tmpr start_lbl in
  let insts_add ≝ map … f_add destrs in
    adds_graph (insts_cmpl @ insts_init @ insts_add) start_lbl dest_lbl def.

definition translate_notbool ≝
  λdestrs.
  λsrcrs.
  λstart_lbl.
  λdest_lbl.
  λdef.
  match destrs with
  [ nil ⇒ add_graph start_lbl (rtl_st_skip dest_lbl) def
  | cons destr destrs ⇒
    let 〈def, tmpr〉 ≝ fresh_reg def in
    let 〈def, tmp_srcrs〉 ≝ fresh_regs def (length ? srcrs) in
    let save_srcrs ≝ translate_move tmp_srcrs srcrs in
    let init_destr ≝ rtl_st_int destr (bitvector_of_nat ? 1) start_lbl in
    let f ≝ λtmp_srcr. [
      rtl_st_clear_carry start_lbl;
      rtl_st_int tmpr (zero ?) start_lbl;
      rtl_st_op2 Sub tmpr tmpr tmp_srcr start_lbl;
      rtl_st_int tmpr (zero ?) start_lbl;
      rtl_st_op2 Addc tmpr tmpr tmpr start_lbl;
      rtl_st_op2 Xor destr destr tmpr start_lbl
    ] in
    let insts ≝ init_destr :: (flatten ? (map … f tmp_srcrs)) in
    let epilogue ≝ translate_cst (Ointconst I8 (bitvector_of_nat ? 0)) destrs in
      add_translates [
        save_srcrs; adds_graph insts; epilogue
      ] start_lbl dest_lbl def
  ].

(* TODO: examine this properly.  This is a change from the O'caml code due
   to us dropping the explicit use of a cast destination size field.  We
   instead infer the size of the cast's destination from the context.  Is
   this correct?
*)
definition translate_op1 ≝
  λop1: unary_operation.
  λdestrs: list register.
  λsrcrs: list register.
  λprf: |destrs| = |srcrs|.
  λstart_lbl: label.
  λdest_lbl: label.
  λdef: rtl_internal_function.
  match op1 with
  [ Ocastint src_sign src_size ⇒
    let dest_size ≝ |destrs| * 8 in
    let src_size ≝ bitsize_of_intsize src_size in
      translate_cast src_size src_sign dest_size destrs srcrs start_lbl dest_lbl def
  | Onegint ⇒
      translate_negint destrs srcrs start_lbl dest_lbl def prf
  | Onotbool ⇒
      translate_notbool destrs srcrs start_lbl dest_lbl def
  | Onotint ⇒
    let f ≝ λdestr. λsrcr. rtl_st_op1 Cmpl destr srcr start_lbl in
    let l ≝ map2 … f destrs srcrs prf in
      adds_graph l start_lbl dest_lbl def
  | Optrofint r ⇒
      translate_move destrs srcrs start_lbl dest_lbl def
  | Ointofptr r ⇒
      translate_move destrs srcrs start_lbl dest_lbl def
  | Oid ⇒
      translate_move destrs srcrs start_lbl dest_lbl def
  | _ ⇒ ? (* float operations implemented in runtime *)
  ].
  cases not_implemented
qed.
  
definition translate_op ≝
  λop.
  λdestrs.
  λsrcrs1.
  λsrcrs2.
  λstart_lbl.
  λdest_lbl.
  λdef.
  match reduce_strong register register srcrs1 srcrs2 with
  [ dp reduced first_reduced_proof ⇒
    let srcrsl_common ≝ \fst (\fst reduced) in
    let srcrsr_common ≝ \fst (\snd reduced) in
    let srcrsl_rest ≝ \snd (\fst reduced) in
    let srcrsr_rest ≝ \snd (\snd reduced) in
    let srcrs_rest ≝ choose_rest ? srcrsl_rest srcrsr_rest ? ? in
    match reduce_strong register register destrs srcrsl_common with
    [ dp reduced second_reduced_proof ⇒
      let destrs_common ≝ \fst (\fst reduced) in
      let destrs_rest ≝ \snd (\fst reduced) in
      match reduce_strong register register destrs_rest srcrs_rest with
      [ dp reduced third_reduced_proof ⇒
        let destrs_cted ≝ \fst (\fst reduced) in
        let destrs_rest ≝ \snd (\fst reduced) in
        let srcrs_cted ≝ \fst (\snd reduced) in
        let 〈def, tmpr〉 ≝ fresh_reg def in
        let insts_init ≝ [
          rtl_st_clear_carry start_lbl;
          rtl_st_int tmpr (zero ?) start_lbl
        ] in
        let f_add ≝ λdestr. λsrcr1. λsrcr2. rtl_st_op2 op destr srcr1 srcr2 start_lbl in
        let insts_add ≝ map3 … f_add destrs_common srcrsl_common srcrsr_common ? ? in
        let f_add_cted ≝ λdestr. λsrcr. rtl_st_op2 op destr srcr tmpr start_lbl in
        let insts_add_cted ≝ map2 ? ? ? f_add_cted destrs_cted srcrs_cted ? in
        let f_rest ≝ λdestr. rtl_st_op2 op destr tmpr tmpr start_lbl in
        let insts_rest ≝ map … f_rest destrs_rest in
          adds_graph (insts_init @ insts_add @ insts_add_cted @ insts_rest) start_lbl dest_lbl def
      ]
    ]
  ].
  [1: @third_reduced_proof
  |3: @first_reduced_proof
  |*: cases daemon (* TODO *)
  ]
qed.

let rec translate_mul1
  (dummy: register) (tmpr: register) (destrs: list register)
  (srcrs1: list register) (srcr2: register) (start_lbl: label)
    on srcrs1 ≝
  match destrs with
  [ nil ⇒ adds_graph [ rtl_st_skip start_lbl ] start_lbl
  | cons destr tl ⇒
    match tl with
    [ nil ⇒
      match srcrs1 with
      [ nil ⇒
        adds_graph [
          rtl_st_int tmpr (zero ?) start_lbl;
          rtl_st_op2 Addc destr destr tmpr start_lbl
        ] start_lbl
      | cons srcr1 tl' ⇒
        adds_graph [
          rtl_st_opaccs Mul tmpr dummy srcr2 srcr1 start_lbl;
          rtl_st_op2 Addc destr destr tmpr start_lbl
        ] start_lbl
      ]
    | cons destr2 destrs ⇒
      match srcrs1 with
      [ nil ⇒
        add_translates [
          adds_graph [
            rtl_st_int tmpr (zero ?) start_lbl;
            rtl_st_op2 Addc destr destr tmpr start_lbl;
            rtl_st_op2 Addc destr2 tmpr tmpr start_lbl
          ];
          translate_cst (Ointconst I8 (zero ?)) destrs
        ] start_lbl
      | cons srcr1 srcrs1 ⇒
        match destrs with
        [ nil ⇒
          add_translates [
            adds_graph [
              rtl_st_int tmpr (zero ?) start_lbl;
              rtl_st_op2 Addc destr destr tmpr start_lbl;
              rtl_st_op2 Addc destr2 tmpr tmpr start_lbl
            ];
            translate_cst (Ointconst I8 (zero ?)) destrs
          ] start_lbl
        | cons destr2 destrs ⇒
          add_translates [
            adds_graph [
              rtl_st_opaccs Mul tmpr destr2 srcr2 srcr1 start_lbl;
              rtl_st_op2 Addc destr destr tmpr start_lbl
            ];
            translate_mul1 dummy tmpr (destr2 :: destrs) srcrs1 srcr2
          ] start_lbl
        ]
      ]
    ]
  ].

definition translate_muli ≝
  λdummy: register.
  λtmpr: register.
  λdestrs: list register.
  λtmp_destrs: list register.
  λsrcrs1: list register.
  λdummy_lbl: label.
  λi: nat.
  λi_prf.
  λtranslates: list ?.
  λsrcr2i: register.
  let 〈tmp_destrs1, tmp_destrs2〉 ≝ split ? tmp_destrs i i_prf in
  let tmp_destrs2' ≝
    match tmp_destrs2 with
    [ nil ⇒ [ ]
    | cons tmp_destr2 tmp_destrs2 ⇒ [
        adds_graph [
          rtl_st_clear_carry dummy_lbl;
          rtl_st_int tmp_destr2 (zero ?) dummy_lbl
        ];
        translate_mul1 dummy tmpr tmp_destrs2 srcrs1 srcr2i;
        translate_cst (Ointconst I8 (zero ?)) tmp_destrs1;
        adds_graph [
          rtl_st_clear_carry dummy_lbl
        ];
        translate_op Addc destrs destrs tmp_destrs
      ]
    ]
  in
    translates @ tmp_destrs2'.

axiom translate_mul:
  ∀destrs: list register.
  ∀srcrs1: list register.
  ∀srcrs2: list register.
  ∀start_lbl: label.
  ∀dest_lbl: label.
  ∀def: rtl_internal_function.
    rtl_internal_function.

(*
definition translate_mul ≝
  λdestrs: list register.
  λsrcrs1: list register.
  λsrcrs2: list register.
  λstart_lbl: label.
  λdest_lbl: label.
  λdef: rtl_internal_function.
  let 〈def, dummy〉 ≝ fresh_reg def in
  let 〈def, tmpr〉 ≝ fresh_reg def in
  let 〈def, tmp_destrs〉 ≝ fresh_regs def (|destrs|) in
  let 〈def, fresh_srcrs1〉 ≝ fresh_regs def (|srcrs1|) in
  let 〈def, fresh_srcrs2〉 ≝ fresh_regs def (|srcrs2|) in
  let insts_init ≝ [
    translate_move fresh_srcrs1 srcrs1;
    translate_move fresh_srcrs2 srcrs2;
    translate_cst (Ointconst I8 (zero ?)) destrs
  ]
  in
  let f ≝ λi. translate_muli dummy tmpr destrs tmp_destrs ? fresh_srcrs1 start_lbl i ? in
  let insts_mul ≝ foldi ? ? [ ] srcrs2 in ?. [5: check insts_init.
    add_translates (insts_init @ insts_mul) start_lbl dest_lbl def.
*)

definition translate_divumodu8 ≝
  λorder: bool.
  λdestrs: list register.
  λsrcr1: register.
  λsrcr2: register.
  λstart_lbl: label.
  λdest_lbl: label.
  λdef: rtl_internal_function.
  match destrs with
  [ nil ⇒ add_graph start_lbl (rtl_st_skip dest_lbl) def
  | cons destr destrs ⇒
    let 〈def, dummy〉 ≝ fresh_reg def in
    let 〈destr1, destr2〉 ≝ match order with [ true ⇒ 〈destr, dummy〉 | _ ⇒ 〈dummy, destr〉 ] in
    let inst_div ≝ adds_graph [
      rtl_st_opaccs DivuModu destr1 destr2 srcr1 srcr2 start_lbl
    ]
    in
    let insts_rest ≝ translate_cst (Ointconst I8 (zero ?)) destrs in
      add_translates [ inst_div; insts_rest ] start_lbl dest_lbl def
  ].

definition translate_ne: ? → ? → ? → ? → ? → ? → rtl_internal_function ≝
  λdestrs: list register.
  λsrcrs1: list register.
  λsrcrs2: list register.
  λstart_lbl: label.
  λdest_lbl: label.
  λdef: rtl_internal_function.
  match destrs with
  [ nil ⇒ add_graph start_lbl (rtl_st_skip dest_lbl) def
  | cons destr destrs ⇒
    let 〈def, tmpr〉 ≝ fresh_reg def in
    let 〈def, tmp_zero〉 ≝ fresh_reg def in
    let 〈def, tmp_srcrs1〉 ≝ fresh_regs def (|srcrs1|) in
    let save_srcrs1 ≝ translate_move tmp_srcrs1 srcrs1 in
    let 〈def, tmp_srcrs2〉 ≝ fresh_regs def (|srcrs2|) in
    let save_srcrs2 ≝ translate_move tmp_srcrs2 srcrs2 in
    match reduce_strong register register tmp_srcrs1 tmp_srcrs2 with
    [ dp crl their_proof ⇒
      let commonl ≝ \fst (\fst crl) in
      let commonr ≝ \fst (\snd crl) in
      let restl ≝ \snd (\snd crl) in
      let restr ≝ \snd (\snd crl) in
      let rest ≝ choose_rest ? restl restr ? ? in
      let inits ≝ [
        rtl_st_int destr (zero ?) start_lbl;
        rtl_st_int tmp_zero (zero ?) start_lbl
      ]
      in
      let f_common ≝ λtmp_srcr1. λtmp_srcr2. [
        rtl_st_clear_carry start_lbl;
        rtl_st_op2 Sub tmpr tmp_srcr1 tmp_srcr2 start_lbl;
        rtl_st_op2 Or destr destr tmpr start_lbl
      ]
      in
      let insts_common ≝ flatten ? (map2 … f_common commonl commonr ?) in
      let f_rest ≝ λtmp_srcr. [
        rtl_st_clear_carry start_lbl;
        rtl_st_op2 Sub tmpr tmp_zero tmp_srcr start_lbl;
        rtl_st_op2 Or destr destr tmpr start_lbl
      ]
      in
      let insts_rest ≝ flatten ? (map ? ? f_rest rest) in
      let insts ≝ inits @ insts_common @ insts_rest in
      let epilogue ≝ translate_cst (Ointconst I8 (zero ?)) destrs in
        add_translates [
          save_srcrs1; save_srcrs2; adds_graph insts; epilogue
        ] start_lbl dest_lbl def
    ]
  ].
  [1: @their_proof
  |*: cases not_implemented (* choose rest requires non-emptiness proofs but
                               these appear impossible to obtain, similar proof
                               to above *)
  ]
qed.

definition translate_eq_reg ≝
  λtmp_zero: register.
  λtmp_one: register.
  λtmpr1: register.
  λtmpr2: register.
  λdestr: register.
  λdummy_lbl: label.
  λsrcr12: register × register.
  let 〈srcr1, srcr2〉 ≝ srcr12 in
  [ rtl_st_clear_carry dummy_lbl;
    rtl_st_op2 Sub tmpr1 srcr1 srcr2 dummy_lbl;
    rtl_st_op2 Addc tmpr1 tmp_zero tmp_zero dummy_lbl;
    rtl_st_op2 Sub tmpr2 srcr2 srcr1 dummy_lbl;
    rtl_st_op2 Addc tmpr2 tmp_zero tmp_zero dummy_lbl;
    rtl_st_op2 Or tmpr1 tmpr1 tmpr2 dummy_lbl;
    rtl_st_op2 Xor tmpr1 tmpr1 tmp_one dummy_lbl;
    rtl_st_op2 And destr destr tmpr1 dummy_lbl ].

definition translate_eq_list ≝
  λtmp_zero: register.
  λtmp_one: register.
  λtmpr1: register.
  λtmpr2: register.
  λdestr: register.
  λleq: list (register × register).
  λdummy_lbl: label.
  let f ≝ translate_eq_reg tmp_zero tmp_one tmpr1 tmpr2 destr dummy_lbl in
    (rtl_st_int destr (bitvector_of_nat ? 1) dummy_lbl) ::
      flatten ? (map ? ? f leq).

definition translate_atom ≝
  λtmp_zero: register.
  λtmp_one: register.
  λtmpr1: register.
  λtmpr2: register.
  λtmpr3: register.
  λdestr: register.
  λdummy_lbl: label.
  λleq: list (register × register).
  λsrcr1: register.
  λsrcr2: register.
    translate_eq_list tmp_zero tmp_one tmpr1 tmpr2 tmpr3 leq dummy_lbl @
    [ rtl_st_clear_carry dummy_lbl;
      rtl_st_op2 Sub tmpr1 srcr1 srcr2 dummy_lbl;
      rtl_st_op2 Addc tmpr1 tmp_zero tmp_zero dummy_lbl;
      rtl_st_op2 And tmpr3 tmpr3 tmpr1 dummy_lbl;
      rtl_st_op2 Or destr destr tmpr3 dummy_lbl ].

definition translate_lt_main ≝
  λtmp_zero: register.
  λtmp_one: register.
  λtmpr1: register.
  λtmpr2: register.
  λtmpr3: register.
  λdestr: register.
  λdummy_lbl: label.
  λsrcrs1: list register.
  λsrcrs2: list register.
  λproof: |srcrs1| = |srcrs2|.
  let f ≝ λinsts_leq. λsrcr1. λsrcr2.
    let 〈insts, leq〉 ≝ insts_leq in
    let added_insts ≝ translate_atom tmp_zero tmp_one tmpr1 tmpr2 tmpr3 destr dummy_lbl leq srcr1 srcr2 in
      〈insts @ added_insts, leq @ [〈srcr1, srcr2〉]〉
  in
    \fst (fold_left2 ? ? ? f 〈[ ], [ ]〉 srcrs1 srcrs2 proof).

definition fresh_regs_strong:
  rtl_internal_function → ∀n: nat. Σfresh: rtl_internal_function × (list register). |\snd fresh| = n ≝
  λdef.
  λn.
    fresh_regs def n.
  @fresh_regs_length
qed.

definition complete_regs_strong:
  rtl_internal_function → list register → list register → Σcomplete: (list register) × (list register) × (list register). |\fst (\fst complete)| = |\snd (\fst complete)| ≝
  λdef.
  λleft.
  λright.
    complete_regs def left right.
  @complete_regs_length
qed.

definition translate_lt ≝
  λdestrs: list register.
  λprf_destrs: lt 0 (|destrs|).
  λsrcrs1: list register.
  λsrcrs2: list register.
  λstart_lbl: label.
  λdest_lbl: label.
  λdef: rtl_internal_function.
  match destrs with
  [ nil ⇒ add_graph start_lbl (rtl_st_skip dest_lbl) def
  | _ ⇒
    match fresh_regs_strong def (|destrs|) with
    [ dp def_tmp_destrs tmp_destrs_proof ⇒
      let def ≝ \fst def_tmp_destrs in
      let tmp_destrs ≝ \snd def_tmp_destrs in
      let tmp_destr ≝ hd_safe ? tmp_destrs ? in
      let 〈def, tmp_zero〉 ≝ fresh_reg def in
      let 〈def, tmp_one〉 ≝ fresh_reg def in
      let 〈def, tmpr1〉 ≝ fresh_reg def in
      let 〈def, tmpr2〉 ≝ fresh_reg def in
      let 〈def, tmpr3〉 ≝ fresh_reg def in
      match complete_regs_strong def srcrs1 srcrs2 with
      [ dp srcrs12_added srcrs12_proof ⇒
        let srcrs1 ≝ \fst (\fst srcrs12_added) in
        let srcrs2 ≝ \snd (\fst srcrs12_added) in
        let added ≝ \snd srcrs12_added in
        let srcrs1' ≝ rev ? srcrs1 in
        let srcrs2' ≝ rev ? srcrs2 in
        let insts_init ≝ [
          translate_cst (Ointconst I8 (zero ?)) tmp_destrs;
          translate_cst (Ointconst I8 (zero ?)) added;
          adds_graph [
            rtl_st_int tmp_zero (zero ?) start_lbl;
            rtl_st_int tmp_one (bitvector_of_nat ? 1) start_lbl
          ]
        ]
        in
        let insts_main ≝
          translate_lt_main tmp_zero tmp_one tmpr1 tmpr2 tmpr3 tmp_destr start_lbl srcrs1' srcrs2' ? in
          let insts_main ≝ [ adds_graph insts_main ] in
          let insts_exit ≝ [ translate_move destrs tmp_destrs ] in
            add_translates (insts_init @ insts_main @ insts_exit) start_lbl dest_lbl def
      ]
    ]
  ].
  [2: >tmp_destrs_proof @prf_destrs
  |1: normalize nodelta
      generalize in match srcrs12_proof
      #HYP >rev_length >rev_length @HYP
  ]
qed.

definition add_128_to_last ≝
  λtmp_128: register.
  λrs.
  λprf: 0 < |rs|.
  λstart_lbl: label.
  match rs with
  [ nil ⇒ adds_graph [ ] start_lbl
  | _ ⇒
    let r ≝ last_safe ? rs prf in
      adds_graph [
        rtl_st_op2 Add r r tmp_128 start_lbl
      ] start_lbl
  ].

definition translate_lts ≝
  λdestrs: list register.
  λdestrs_prf: lt 0 (|destrs|).
  λsrcrs1: list register.
  λsrcrs2: list register.
  λsrcrs1_lt_prf: 0 < |srcrs1|.
  λsrcrs2_lt_prf: 0 < |srcrs2|.
  λstart_lbl: label.
  λdest_lbl: label.
  λdef: rtl_internal_function.
  match fresh_regs_strong def (|srcrs1|) with
  [ dp def_tmp_srcrs1 srcrs1_prf ⇒
    let def ≝ \fst def_tmp_srcrs1 in
    let tmp_srcrs1 ≝ \snd def_tmp_srcrs1 in
    match fresh_regs_strong def (|srcrs2|) with
    [ dp def_tmp_srcrs2 srcrs2_prf ⇒
      let def ≝ \fst def_tmp_srcrs2 in
      let tmp_srcrs2 ≝ \snd def_tmp_srcrs2 in
      let 〈def, tmp_128〉 ≝ fresh_reg def in
        add_translates [
          translate_move tmp_srcrs1 srcrs1;
          translate_move tmp_srcrs2 srcrs2;
          adds_graph [
            rtl_st_int tmp_128 (bitvector_of_nat ? 128) start_lbl
          ];
          add_128_to_last tmp_128 tmp_srcrs1 ?;
          add_128_to_last tmp_128 tmp_srcrs2 ?;
          translate_lt destrs destrs_prf tmp_srcrs1 tmp_srcrs2
        ] start_lbl dest_lbl def
    ]
  ].
  [1: >srcrs1_prf @srcrs1_lt_prf
  |2: >srcrs2_prf @srcrs2_lt_prf
  ]
qed.

definition translate_op2 ≝
  λop2.
  λdestrs: list register.
  λdestrs_prf: lt 0 (|destrs|).
  λsrcrs1: list register.
  λsrcrs2: list register.
  λsrcrs1_prf: lt 0 (|srcrs1|). (* can this be relaxed? i think it can
                                   but we need more dep. typ. in modu/divu
                                   cases *)
  λsrcrs2_prf: lt 0 (|srcrs2|).
  λstart_lbl: label.
  λdest_lbl: label.
  λdef: rtl_internal_function.
  match op2 with
  [ Oadd ⇒
    translate_op Addc destrs srcrs1 srcrs2 start_lbl dest_lbl def
  | Oaddp ⇒
    translate_op Addc destrs srcrs1 srcrs2 start_lbl dest_lbl def
  | Osub ⇒
    translate_op Sub destrs srcrs1 srcrs2 start_lbl dest_lbl def
  | Osubpi ⇒
    translate_op Sub destrs srcrs1 srcrs2 start_lbl dest_lbl def
  | Osubpp sz ⇒
    translate_op Sub destrs srcrs1 srcrs2 start_lbl dest_lbl def
  | Omul ⇒
    translate_mul destrs srcrs1 srcrs2 start_lbl dest_lbl def
  | Odivu ⇒
    match srcrs1 return λx. 0 < |x| → rtl_internal_function with
    [ cons hd tl ⇒ λcons_prf.
      match tl with
      [ nil ⇒ translate_divumodu8 true destrs hd (hd_safe register srcrs2 ?) start_lbl dest_lbl def
      | _ ⇒ ? (* not implemented *)
      ] 
    | nil ⇒ λnil_absrd. ?
    ] srcrs1_prf
  | Omodu ⇒
    match srcrs1 return λx. 0 < |x| → rtl_internal_function with
    [ cons hd tl ⇒ λcons_prf.
      match tl with
      [ nil ⇒ translate_divumodu8 false destrs hd (hd_safe register srcrs2 ?) start_lbl dest_lbl def
      | _ ⇒ ? (* not implemented *)
      ] 
    | nil ⇒ λnil_absrd. ?
    ] srcrs1_prf
  | Oand ⇒
    translate_op And destrs srcrs1 srcrs2 start_lbl dest_lbl def
  | Oor ⇒
    translate_op Or destrs srcrs1 srcrs2 start_lbl dest_lbl def
  | Oxor ⇒
    translate_op Xor destrs srcrs1 srcrs2 start_lbl dest_lbl def
  | Ocmp c ⇒
    match c with
    [ Ceq ⇒
      add_translates [
        translate_ne destrs srcrs1 srcrs2;
        translate_op1 Onotbool destrs destrs (refl ? (|destrs|))
      ] start_lbl dest_lbl def
    | Cne ⇒ translate_ne destrs srcrs1 srcrs2 start_lbl dest_lbl def
    | Clt ⇒ translate_lts destrs destrs_prf srcrs1 srcrs2 ? ? start_lbl dest_lbl def
    | Cgt ⇒ translate_lts destrs destrs_prf srcrs2 srcrs1 ? ? start_lbl dest_lbl def
    | Cle ⇒
      add_translates [
        translate_lts destrs destrs_prf srcrs2 srcrs1 ? ?;
        translate_op1 Onotbool destrs destrs (refl ? (|destrs|))
      ] start_lbl dest_lbl def
    | Cge ⇒
      add_translates [
        translate_lts destrs destrs_prf srcrs1 srcrs2 ? ?;
        translate_op1 Onotbool destrs destrs (refl ? (|destrs|))
      ] start_lbl dest_lbl def
    ]
  | Ocmpu c ⇒
    match c with
    [ Ceq ⇒
      add_translates [
        translate_ne destrs srcrs1 srcrs2;
        translate_op1 Onotbool destrs destrs (refl ? (|destrs|))
      ] start_lbl dest_lbl def
    | Cne ⇒ translate_ne destrs srcrs1 srcrs2 start_lbl dest_lbl def
    | Clt ⇒ translate_lt destrs destrs_prf srcrs1 srcrs2 start_lbl dest_lbl def
    | Cgt ⇒ translate_lt destrs destrs_prf srcrs2 srcrs1 start_lbl dest_lbl def
    | Cle ⇒
      add_translates [
        translate_lt destrs destrs_prf srcrs2 srcrs1;
        translate_op1 Onotbool destrs destrs (refl ? (|destrs|))
      ] start_lbl dest_lbl def
    | Cge ⇒
      add_translates [
        translate_lt destrs destrs_prf srcrs1 srcrs2;
        translate_op1 Onotbool destrs destrs (refl ? (|destrs|))
      ] start_lbl dest_lbl def
    ]
  | Ocmpp c ⇒
    match c with
    [ Ceq ⇒
      add_translates [
        translate_ne destrs srcrs1 srcrs2;
        translate_op1 Onotbool destrs destrs (refl ? (|destrs|))
      ] start_lbl dest_lbl def
    | Cne ⇒ translate_ne destrs srcrs1 srcrs2 start_lbl dest_lbl def
    | Clt ⇒ translate_lt destrs destrs_prf srcrs1 srcrs2 start_lbl dest_lbl def
    | Cgt ⇒ translate_lt destrs destrs_prf srcrs2 srcrs1 start_lbl dest_lbl def
    | Cle ⇒
      add_translates [
        translate_lt destrs destrs_prf srcrs2 srcrs1;
        translate_op1 Onotbool destrs destrs (refl ? (|destrs|))
      ] start_lbl dest_lbl def
    | Cge ⇒
      add_translates [
        translate_lt destrs destrs_prf srcrs1 srcrs2;
        translate_op1 Onotbool destrs destrs (refl ? (|destrs|))
      ] start_lbl dest_lbl def
    ]
  | _ ⇒ ? (* assert false, implemented in run time or float op *)
  ].
  [ 2,6: normalize in nil_absrd;
         cases(not_le_Sn_O 0)
         #HYP cases(HYP nil_absrd)
  | 3,7,12,17,13,15,18,19,20,21,14,16:
         assumption
  | *:   cases not_implemented (* yes, really *)
  ]
qed.

definition translate_cond ≝
  λsrcrs: list register.
  λstart_lbl: label.
  λlbl_true: label.
  λlbl_false: label.
  λdef: rtl_internal_function.
  let 〈def, tmpr〉 ≝ fresh_reg def in
  let 〈def, tmp_lbl〉 ≝ fresh_label def in
  let init ≝ rtl_st_int tmpr (zero ?) start_lbl in
  let f ≝ λsrcr. rtl_st_op2 Or tmpr tmpr srcr start_lbl in
  let def ≝ adds_graph (init :: (map … f srcrs)) start_lbl tmp_lbl def in
    add_graph tmp_lbl (rtl_st_cond tmpr lbl_true lbl_false) def.  

definition translate_load ≝
  λaddr.
  λaddr_prf: 2 ≤ |addr|.
  λdestrs: list register.
  λstart_lbl: label.
  λdest_lbl: label.
  λdef: rtl_internal_function.
  match fresh_regs_strong def (|addr|) with
  [ dp def_save_addr save_addr_prf ⇒
    let def ≝ \fst def_save_addr in
    let save_addr ≝ \snd def_save_addr in
    match fresh_regs_strong def (|addr|) with
    [ dp def_tmp_addr tmp_addr_prf ⇒
      let def ≝ \fst def_tmp_addr in
      let tmp_addr ≝ \snd def_tmp_addr in
      let 〈tmp_addr1, tmp_addr2〉 ≝ make_addr ? tmp_addr ? in
      let 〈def, tmpr〉 ≝ fresh_reg def in
      let insts_save_addr ≝ translate_move save_addr addr in
      let f ≝ λtranslates_off. λr.
        let 〈translates, off〉 ≝ translates_off in
        let translates ≝ translates @ [
          adds_graph [
            rtl_st_int tmpr off start_lbl
          ];
          translate_op2 Oaddp tmp_addr ? save_addr [tmpr] ? ?;
          adds_graph [
            rtl_st_load r tmp_addr1 tmp_addr2 dest_lbl
          ]
        ]
        in
        let 〈carry, result〉 ≝ half_add ? off int_size in
          〈translates, result〉
      in
      let 〈translates, ignore〉 ≝ foldl ? ? f 〈[ ], (zero ?)〉 destrs in
        add_translates (insts_save_addr :: translates) start_lbl dest_lbl def
    ]
  ].
  [1: normalize //
  |2: >save_addr_prf normalize
      @(transitive_le 1 2 (|addr|)) //
  |3: >tmp_addr_prf normalize
      @(transitive_le 1 2 (|addr|)) //
  |*: >tmp_addr_prf assumption
  ]
qed.

definition translate_store ≝
  λaddr.
  λaddr_prf: 2 ≤ |addr|.
  λsrcrs: list register.
  λstart_lbl: label.
  λdest_lbl: label.
  λdef: rtl_internal_function.
  match fresh_regs_strong def (|addr|) with
  [ dp def_tmp_addr tmp_addr_prf ⇒
    let def ≝ \fst def_tmp_addr in
    let tmp_addr ≝ \snd def_tmp_addr in
    let 〈tmp_addr1, tmp_addr2〉 ≝ make_addr ? tmp_addr ? in
    let 〈def, tmpr〉 ≝ fresh_reg def in
    let f ≝ λtranslate_off. λsrcr.
      let 〈translates, off〉 ≝ translate_off in
      let translates ≝ translates @ [
        adds_graph [
          rtl_st_int tmpr off start_lbl
        ];
        translate_op2 Oaddp tmp_addr ? addr [tmpr] ? ?;
        adds_graph [
          rtl_st_store tmp_addr1 tmp_addr2 srcr dest_lbl
        ]
      ]
      in
        let 〈carry, result〉 ≝ half_add ? off int_size in
          〈translates, result〉
    in
    let 〈translates, ignore〉 ≝ foldl ? ? f 〈[ ], zero ?〉 srcrs in
      add_translates translates start_lbl dest_lbl def
  ].
  [1: normalize //
  |2: >tmp_addr_prf normalize
      @(transitive_le 1 2 (|addr|)) //
  |3: >tmp_addr_prf normalize
      @(transitive_le 1 2 (|addr|)) //
  |*: >tmp_addr_prf assumption
  ]
qed.

definition translate_stmt ≝
  λlenv.
  λlbl: label.
  λstmt.
  λdef: rtl_internal_function.
  match stmt with
  [ St_skip lbl' ⇒ add_graph lbl (rtl_st_skip lbl') def
  | St_cost cost_lbl lbl' ⇒ add_graph lbl (rtl_st_cost cost_lbl lbl') def
  | St_const destr cst lbl' ⇒
    translate_cst cst (find_local_env destr lenv) lbl lbl' def
  | St_op1 op1 destr srcr lbl' ⇒
    translate_op1 op1 (find_local_env destr lenv) (find_local_env srcr lenv) ? lbl lbl' def
  | St_op2 op2 destr srcr1 srcr2 lbl' ⇒
    translate_op2 op2 (find_local_env destr lenv) ? (find_local_env srcr1 lenv) (find_local_env srcr2 lenv) ? ? lbl lbl' def
  | St_load ignore addr destr lbl' ⇒
    translate_load (find_local_env addr lenv) ? (find_local_env destr lenv) lbl lbl' def
  | St_store ignore addr srcr lbl' ⇒
    translate_store (find_local_env addr lenv) ? (find_local_env srcr lenv) lbl lbl' def
  | St_call_id f args retr lbl' ⇒
    match retr with
    [ Some retr ⇒
      add_graph lbl (rtl_st_call_id f (rtl_args args lenv) (find_local_env retr lenv) lbl') def
    | None ⇒ add_graph lbl (rtl_st_call_id f (rtl_args args lenv) [ ] lbl') def
    ]
  | St_call_ptr f args retr lbl' ⇒
    match retr with
    [ None ⇒
      let 〈f1, f2〉 ≝ find_and_addr f lenv ? in
        add_graph lbl (rtl_st_call_ptr f1 f2 (rtl_args args lenv) [ ] lbl') def
    | Some retr ⇒
      let 〈f1, f2〉 ≝ find_and_addr f lenv ? in
        add_graph lbl (rtl_st_call_ptr f1 f2 (rtl_args args lenv) (find_local_env retr lenv) lbl') def
    ]
  | St_tailcall_id f args ⇒
    add_graph lbl (rtl_st_tailcall_id f (rtl_args args lenv)) def
  | St_tailcall_ptr f args ⇒
    let 〈f1, f2〉 ≝ find_and_addr f lenv ? in
      add_graph lbl (rtl_st_tailcall_ptr f1 f2 (rtl_args args lenv)) def
  | St_cond r lbl_true lbl_false ⇒
    translate_cond (find_local_env r lenv) lbl lbl_true lbl_false def
  | St_jumptable r l ⇒ ? (* assert false: not implemented yet *)
  | St_return ⇒ add_graph lbl rtl_st_return def
  ].
  [10: cases not_implemented (* jtable case *)
  |*: cases daemon (* TODO: proofs regarding lengths of lists, examine! *)
  ]
qed.

(* XXX: we use a "hack" here to circumvent the proof obligations required to
   create res, an empty internal_function record.  we insert into our graph
   the start and end nodes to ensure that they are in, and place dummy
   skip instructions at these nodes. *)
definition translate_internal ≝
  λdef.
  let runiverse ≝ f_reggen def in
  let lenv ≝ initialize_local_env runiverse
    (f_params def @ f_locals def) (f_result def) in
  let params ≝ map_list_local_env lenv (map ? ? \fst (f_params def)) in
  let locals ≝ map_list_local_env lenv (map ? ? \fst (f_locals def)) in
  let result ≝
    match (f_result def) with
    [ None ⇒ [ ]
    | Some r_typ ⇒
      let 〈r, typ〉 ≝ r_typ in
        find_local_env r lenv
    ]
  in
  let luniverse' ≝ f_labgen def in
  let runiverse' ≝ runiverse in
  let result' ≝ result in
  let params' ≝ params in
  let locals' ≝ locals in
  let stack_size' ≝ f_stacksize def in
  let entry' ≝ f_entry def in
  let exit' ≝ f_exit def in
  let graph' ≝ add ? ? (empty_map ? ?) entry' (rtl_st_skip entry') in
  let graph' ≝ add ? ? graph' exit' (rtl_st_skip exit') in
  let res ≝
    mk_rtl_internal_function luniverse' runiverse' result' params'
                         locals' stack_size' graph' ? ? in
    foldi ? ? ? (translate_stmt lenv) (f_graph def) res.
  [1: %
      [1: @entry'
      |2: normalize nodelta;
          @graph_add_lookup
          @graph_add
      ]
  |2: %
      [1: @exit'
      |2: normalize nodelta;
          @graph_add
      ]
  ]
qed.

(*CSC: here we are not doing any alignment on variables, like it is done in OCaml
  because of CompCert heritage *)
definition rtlabs_to_rtl : RTLabs_program → rtl_program ≝
 λp. transform_program … p (transf_fundef … translate_internal).