Theory "Decode"

Signature

Definitions

wf_decoder_def

⊢ ∀p d.
      wf_decoder p d ⇔
      ∀x.
          if p x then ∃a. ∀b c. (d b = SOME (x,c)) ⇔ (b = a ++ c)
          else ∀a b. d a ≠ SOME (x,b)

enc2dec_def

⊢ ∀p e l.
      enc2dec p e l =
      if ∃x t. p x ∧ (l = e x ++ t) then SOME (@(x,t). p x ∧ (l = e x ++ t))
      else NONE

decode_unit_def

⊢ ∀p. decode_unit p = enc2dec p encode_unit

decode_tree_def

⊢ ∀p d. decode_tree p d = enc2dec p (encode_tree (dec2enc d))

decode_sum_def

⊢ ∀p d1 d2.
      decode_sum p d1 d2 = enc2dec p (encode_sum (dec2enc d1) (dec2enc d2))

decode_prod_def

⊢ ∀p d1 d2.
      decode_prod p d1 d2 = enc2dec p (encode_prod (dec2enc d1) (dec2enc d2))

decode_option_def

⊢ ∀p d. decode_option p d = enc2dec p (encode_option (dec2enc d))

decode_num_def

⊢ ∀p. decode_num p = enc2dec p encode_num

decode_list_def

⊢ ∀p d. decode_list p d = enc2dec p (encode_list (dec2enc d))

decode_bool_def

⊢ ∀p. decode_bool p = enc2dec p encode_bool

decode_bnum_def

⊢ ∀m p. decode_bnum m p = enc2dec p (encode_bnum m)

decode_blist_def

⊢ ∀p m d. decode_blist p m d = enc2dec p (encode_blist m (dec2enc d))

dec_bnum_def

⊢ (∀l. dec_bnum 0 l = SOME (0,l)) ∧
  ∀m l.
      dec_bnum (SUC m) l =
      case l of
        [] => NONE
      | h::t =>
        case dec_bnum m t of
          NONE => NONE
        | SOME (n,t') => SOME (2 * n + if h then 1 else 0,t')

dec2enc_def

⊢ ∀d x. dec2enc d x = @l. d l = SOME (x,[])

Theorems

wf_enc2dec

⊢ ∀p e. wf_encoder p e ⇒ wf_decoder p (enc2dec p e)

wf_decode_unit

⊢ wf_decoder p (decode_unit p)

wf_decode_tree

⊢ ∀p d.
      wf_decoder p d ⇒ wf_decoder (lift_tree p) (decode_tree (lift_tree p) d)

wf_decode_sum

⊢ ∀p1 p2 d1 d2.
      wf_decoder p1 d1 ∧ wf_decoder p2 d2 ⇒
      wf_decoder (lift_sum p1 p2) (decode_sum (lift_sum p1 p2) d1 d2)

wf_decode_prod

⊢ ∀p1 p2 d1 d2.
      wf_decoder p1 d1 ∧ wf_decoder p2 d2 ⇒
      wf_decoder (lift_prod p1 p2) (decode_prod (lift_prod p1 p2) d1 d2)

wf_decode_option

⊢ ∀p d.
      wf_decoder p d ⇒
      wf_decoder (lift_option p) (decode_option (lift_option p) d)

wf_decode_num

⊢ ∀p. wf_decoder p (decode_num p)

wf_decode_list

⊢ ∀p d. wf_decoder p d ⇒ wf_decoder (EVERY p) (decode_list (EVERY p) d)

wf_decode_bool

⊢ ∀p. wf_decoder p (decode_bool p)

wf_decode_bnum

⊢ ∀m p. wf_pred_bnum m p ⇒ wf_decoder p (decode_bnum m p)

wf_decode_blist

⊢ ∀m p d.
      wf_decoder p d ⇒
      wf_decoder (lift_blist m p) (decode_blist (lift_blist m p) m d)

wf_dec2enc

⊢ ∀p d. wf_decoder p d ⇒ wf_encoder p (dec2enc d)

encode_then_decode_sum

⊢ ∀p1 p2 e1 e2 l t.
      wf_encoder p1 e1 ∧ wf_encoder p2 e2 ∧ lift_sum p1 p2 l ⇒
      (decode_sum (lift_sum p1 p2) (enc2dec p1 e1) (enc2dec p2 e2)
         (encode_sum e1 e2 l ++ t) = SOME (l,t))

encode_then_decode_prod

⊢ ∀p1 p2 e1 e2 l t.
      wf_encoder p1 e1 ∧ wf_encoder p2 e2 ∧ lift_prod p1 p2 l ⇒
      (decode_prod (lift_prod p1 p2) (enc2dec p1 e1) (enc2dec p2 e2)
         (encode_prod e1 e2 l ++ t) = SOME (l,t))

encode_then_decode_option

⊢ ∀p e l t.
      wf_encoder p e ∧ lift_option p l ⇒
      (decode_option (lift_option p) (enc2dec p e) (encode_option e l ++ t) =
       SOME (l,t))

encode_then_decode_list

⊢ ∀p e l t.
      wf_encoder p e ∧ EVERY p l ⇒
      (decode_list (EVERY p) (enc2dec p e) (encode_list e l ++ t) = SOME (l,t))

encode_then_decode_blist

⊢ ∀m p e l t.
      wf_encoder p e ∧ lift_blist m p l ⇒
      (decode_blist (lift_blist m p) m (enc2dec p e) (encode_blist m e l ++ t) =
       SOME (l,t))

enc2dec_some_alt

⊢ ∀p e l x.
      wf_encoder p e ⇒
      ((enc2dec p e l = SOME x) ⇔ p (FST x) ∧ (l = e (FST x) ++ SND x))

enc2dec_some

⊢ ∀p e l x t.
      wf_encoder p e ⇒ ((enc2dec p e l = SOME (x,t)) ⇔ p x ∧ (l = e x ++ t))

enc2dec_none

⊢ ∀p e l. (enc2dec p e l = NONE) ⇔ ∀x t. p x ⇒ l ≠ e x ++ t

enc2dec_dec2enc

⊢ ∀p d. wf_decoder p d ⇒ (enc2dec p (dec2enc d) = d)

decode_unit

⊢ decode_unit p l = if p () then SOME ((),l) else NONE

decode_tree

⊢ wf_decoder p d ⇒
  (decode_tree (lift_tree p) d l =
   case d l of
     NONE => NONE
   | SOME (a,t) =>
     case decode_list (EVERY (lift_tree p)) (decode_tree (lift_tree p) d) t of
       NONE => NONE
     | SOME (ts,t') => SOME (Node a ts,t'))

decode_sum

⊢ wf_decoder p1 d1 ∧ wf_decoder p2 d2 ⇒
  (decode_sum (lift_sum p1 p2) d1 d2 l =
   case l of
     [] => NONE
   | T::t => (case d1 t of NONE => NONE | SOME (x,t') => SOME (INL x,t'))
   | F::t => case d2 t of NONE => NONE | SOME (x,t') => SOME (INR x,t'))

decode_prod

⊢ wf_decoder p1 d1 ∧ wf_decoder p2 d2 ⇒
  (decode_prod (lift_prod p1 p2) d1 d2 l =
   case d1 l of
     NONE => NONE
   | SOME (x,t) => case d2 t of NONE => NONE | SOME (y,t') => SOME ((x,y),t'))

decode_option

⊢ wf_decoder p d ⇒
  (decode_option (lift_option p) d l =
   case l of
     [] => NONE
   | T::t => (case d t of NONE => NONE | SOME (x,t') => SOME (SOME x,t'))
   | F::t => SOME (NONE,t))

decode_num_total

⊢ decode_num (K T) l =
  case l of
    [] => NONE
  | [T] => NONE
  | T::T::t => SOME (0,t)
  | T::F::t =>
    (case decode_num (K T) t of
       NONE => NONE
     | SOME (v,t') => SOME (2 * v + 1,t'))
  | F::t' =>
    case decode_num (K T) t' of
      NONE => NONE
    | SOME (v,t') => SOME (2 * v + 2,t')

decode_num

⊢ decode_num p l =
  case l of
    [] => NONE
  | [T] => NONE
  | T::T::t => if p 0 then SOME (0,t) else NONE
  | T::F::t =>
    (case decode_num (K T) t of
       NONE => NONE
     | SOME (v,t') => if p (2 * v + 1) then SOME (2 * v + 1,t') else NONE)
  | F::t' =>
    case decode_num (K T) t' of
      NONE => NONE
    | SOME (v,t') => if p (2 * v + 2) then SOME (2 * v + 2,t') else NONE

decode_list

⊢ wf_decoder p d ⇒
  (decode_list (EVERY p) d l =
   case l of
     [] => NONE
   | T::t =>
     (case d t of
        NONE => NONE
      | SOME (x,t') =>
        case decode_list (EVERY p) d t' of
          NONE => NONE
        | SOME (xs,t'') => SOME (x::xs,t''))
   | F::t => SOME ([],t))

decode_dec2enc_append

⊢ ∀p d x t. wf_decoder p d ∧ p x ⇒ (d (dec2enc d x ++ t) = SOME (x,t))

decode_dec2enc

⊢ ∀p d x. wf_decoder p d ∧ p x ⇒ (d (dec2enc d x) = SOME (x,[]))

decode_bool

⊢ decode_bool p l =
  case l of [] => NONE | h::t => if p h then SOME (h,t) else NONE

decode_bnum

⊢ wf_pred_bnum m p ⇒
  (decode_bnum m p l =
   case dec_bnum m l of
     NONE => NONE
   | SOME (n,t) => if p n then SOME (n,t) else NONE)

decode_blist

⊢ wf_decoder p d ⇒
  (decode_blist (lift_blist m p) m d l =
   case m of
     0 => SOME ([],l)
   | SUC n =>
     case d l of
       NONE => NONE
     | SOME (x,t) =>
       case decode_blist (lift_blist n p) n d t of
         NONE => NONE
       | SOME (xs,t') => SOME (x::xs,t'))

dec_bnum_lt

⊢ ∀m l n t. (dec_bnum m l = SOME (n,t)) ⇒ n < 2 ** m

dec_bnum_inj

⊢ ∀m l n t. (dec_bnum m l = SOME (n,t)) ⇒ (l = encode_bnum m n ++ t)

dec_bnum_compute

⊢ (∀l. dec_bnum 0 l = SOME (0,l)) ∧
  (∀m l.
       dec_bnum (NUMERAL (BIT1 m)) l =
       case l of
         [] => NONE
       | h::t =>
         case dec_bnum (NUMERAL (BIT1 m) − 1) t of
           NONE => NONE
         | SOME (n,t') => SOME (2 * n + if h then 1 else 0,t')) ∧
  ∀m l.
      dec_bnum (NUMERAL (BIT2 m)) l =
      case l of
        [] => NONE
      | h::t =>
        case dec_bnum (NUMERAL (BIT1 m)) t of
          NONE => NONE
        | SOME (n,t') => SOME (2 * n + if h then 1 else 0,t')

dec2enc_some

⊢ ∀p d x l. wf_decoder p d ⇒ ((dec2enc d x = l) ∧ p x ⇔ (d l = SOME (x,[])))

dec2enc_enc2dec

⊢ ∀p e x. wf_encoder p e ∧ p x ⇒ (dec2enc (enc2dec p e) x = e x)

dec2enc_decode_unit

⊢ ∀p x. p x ⇒ (dec2enc (decode_unit p) x = encode_unit x)

dec2enc_decode_sum

⊢ ∀p1 p2 d1 d2 x.
      wf_decoder p1 d1 ∧ wf_decoder p2 d2 ∧ lift_sum p1 p2 x ⇒
      (dec2enc (decode_sum (lift_sum p1 p2) d1 d2) x =
       encode_sum (dec2enc d1) (dec2enc d2) x)

dec2enc_decode_prod

⊢ ∀p1 p2 d1 d2 x.
      wf_decoder p1 d1 ∧ wf_decoder p2 d2 ∧ lift_prod p1 p2 x ⇒
      (dec2enc (decode_prod (lift_prod p1 p2) d1 d2) x =
       encode_prod (dec2enc d1) (dec2enc d2) x)

dec2enc_decode_option

⊢ ∀p d x.
      wf_decoder p d ∧ lift_option p x ⇒
      (dec2enc (decode_option (lift_option p) d) x =
       encode_option (dec2enc d) x)

dec2enc_decode_num

⊢ ∀p x. p x ⇒ (dec2enc (decode_num p) x = encode_num x)

dec2enc_decode_list

⊢ ∀p d x.
      wf_decoder p d ∧ EVERY p x ⇒
      (dec2enc (decode_list (EVERY p) d) x = encode_list (dec2enc d) x)

dec2enc_decode_bool

⊢ ∀p x. p x ⇒ (dec2enc (decode_bool p) x = encode_bool x)

dec2enc_decode_bnum

⊢ ∀m p x.
      wf_pred_bnum m p ∧ p x ⇒ (dec2enc (decode_bnum m p) x = encode_bnum m x)

dec2enc_decode_blist

⊢ ∀m p d l.
      wf_decoder p d ∧ lift_blist m p l ⇒
      (dec2enc (decode_blist (lift_blist m p) m d) l =
       encode_blist m (dec2enc d) l)