Theory "Encode"

Signature

Type	Arity
tree	1
Constant	Type
Node	:α -> α tree list -> α tree
biprefix	:α list reln
collision_free	:num -> (num -> bool) -> bool
encode_blist	:num -> (β -> α list) -> β list -> α list
encode_bnum	:num -> num -> bitstring
encode_bool	:bool -> bitstring
encode_list	:(α -> bitstring) -> α list -> bitstring
encode_num	:num -> bitstring
encode_option	:(α -> bitstring) -> α option -> bitstring
encode_prod	:(α -> bitstring) -> (β -> bitstring) -> α # β -> bitstring
encode_sum	:(α -> bitstring) -> (β -> bitstring) -> α + β -> bitstring
encode_tree	:(α -> bitstring) -> α tree -> bitstring
encode_unit	:unit -> bitstring
lift_blist	:num -> (α -> bool) -> α list -> bool
lift_option	:(α -> bool) -> α option -> bool
lift_prod	:(α -> bool) -> (β -> bool) -> α # β -> bool
lift_sum	:(α -> bool) -> (β -> bool) -> α + β -> bool
lift_tree	:(α -> bool) -> α tree -> bool
tree1_size	:(α -> num) -> α tree list -> num
tree_CASE	:α tree -> (α -> α tree list -> β) -> β
tree_size	:(α -> num) -> α tree -> num
wf_encoder	:(α -> bool) -> (α -> bitstring) -> bool
wf_pred	:(α -> bool) -> bool
wf_pred_bnum	:num -> (num -> bool) -> bool

Definitions

biprefix_def

⊢ ∀a b. biprefix a b ⇔ b ≼ a ∨ a ≼ b

wf_pred_def

⊢ ∀p. wf_pred p ⇔ ∃x. p x

wf_encoder_def

⊢ ∀p e. wf_encoder p e ⇔ ∀x y. p x ∧ p y ∧ e y ≼ e x ⇒ x = y

encode_unit_def

⊢ ∀v0. encode_unit v0 = []

encode_bool_def

⊢ ∀x. encode_bool x = [x]

encode_prod_def

⊢ ∀xb yb x y. encode_prod xb yb (x,y) = xb x ++ yb y

lift_prod_def

⊢ ∀p1 p2 x. lift_prod p1 p2 x ⇔ p1 (FST x) ∧ p2 (SND x)

encode_sum_def

⊢ (∀xb yb x. encode_sum xb yb (INL x) = T::xb x) ∧
  ∀xb yb y. encode_sum xb yb (INR y) = F::yb y

lift_sum_def

⊢ ∀p1 p2 x. lift_sum p1 p2 x ⇔ case x of INL x1 => p1 x1 | INR x2 => p2 x2

encode_option_def

⊢ (∀xb. encode_option xb NONE = [F]) ∧
  ∀xb x. encode_option xb (SOME x) = T::xb x

lift_option_def

⊢ ∀p x. lift_option p x ⇔ case x of NONE => T | SOME y => p y

encode_list_def

⊢ (∀xb. encode_list xb [] = [F]) ∧
  ∀xb x xs. encode_list xb (x::xs) = T::(xb x ++ encode_list xb xs)

encode_blist_def

⊢ (∀e l. encode_blist 0 e l = []) ∧
  ∀m e l. encode_blist (SUC m) e l = e (HD l) ++ encode_blist m e (TL l)

lift_blist_def

⊢ ∀m p x. lift_blist m p x ⇔ EVERY p x ∧ LENGTH x = m

encode_num_primitive_def

⊢ encode_num =
  WFREC
    (@R.
         WF R ∧ (∀n. n ≠ 0 ∧ EVEN n ⇒ R ((n − 2) DIV 2) n) ∧
         ∀n. n ≠ 0 ∧ ¬EVEN n ⇒ R ((n − 1) DIV 2) n)
    (λencode_num a.
         I
           (if a = 0 then [T; T]
            else if EVEN a then F::encode_num ((a − 2) DIV 2)
            else T::F::encode_num ((a − 1) DIV 2)))

encode_bnum_def

⊢ (∀n. encode_bnum 0 n = []) ∧
  ∀m n. encode_bnum (SUC m) n = ¬EVEN n::encode_bnum m (n DIV 2)

collision_free_def

⊢ ∀m p.
      collision_free m p ⇔
      ∀x y. p x ∧ p y ∧ x MOD 2 ** m = y MOD 2 ** m ⇒ x = y

wf_pred_bnum_def

⊢ ∀m p. wf_pred_bnum m p ⇔ wf_pred p ∧ ∀x. p x ⇒ x < 2 ** m

tree_TY_DEF

⊢ ∃rep.
      TYPE_DEFINITION
        (λa0'.
             ∀'tree' '@temp @ind_typeEncode0list' .
                 (∀a0'.
                      (∃a0 a1.
                           a0' =
                           (λa0 a1.
                                ind_type$CONSTR 0 a0
                                  (ind_type$FCONS a1 (λn. ind_type$BOTTOM)))
                             a0 a1 ∧ '@temp @ind_typeEncode0list' a1) ⇒
                      'tree' a0') ∧
                 (∀a1'.
                      a1' = ind_type$CONSTR (SUC 0) ARB (λn. ind_type$BOTTOM) ∨
                      (∃a0 a1.
                           a1' =
                           (λa0 a1.
                                ind_type$CONSTR (SUC (SUC 0)) ARB
                                  (ind_type$FCONS a0
                                     (ind_type$FCONS a1 (λn. ind_type$BOTTOM))))
                             a0 a1 ∧ 'tree' a0 ∧
                           '@temp @ind_typeEncode0list' a1) ⇒
                      '@temp @ind_typeEncode0list' a1') ⇒
                 'tree' a0') rep

tree_case_def

⊢ ∀a0 a1 f. tree_CASE (Node a0 a1) f = f a0 a1

tree_size_def

⊢ (∀f a0 a1. tree_size f (Node a0 a1) = 1 + (f a0 + tree1_size f a1)) ∧
  (∀f. tree1_size f [] = 0) ∧
  ∀f a0 a1. tree1_size f (a0::a1) = 1 + (tree_size f a0 + tree1_size f a1)

Theorems

biprefix_refl

⊢ ∀x. biprefix x x

biprefix_sym

⊢ ∀x y. biprefix x y ⇒ biprefix y x

biprefix_append

⊢ ∀a b c d. biprefix (a ++ b) (c ++ d) ⇒ biprefix a c

biprefix_cons

⊢ ∀a b c d. biprefix (a::b) (c::d) ⇔ a = c ∧ biprefix b d

biprefix_appends

⊢ ∀a b c. biprefix (a ++ b) (a ++ c) ⇔ biprefix b c

wf_encoder_alt

⊢ wf_encoder p e ⇔ ∀x y. p x ∧ p y ∧ biprefix (e x) (e y) ⇒ x = y

wf_encoder_eq

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

wf_encoder_total

⊢ ∀p e. wf_encoder (K T) e ⇒ wf_encoder p e

wf_encode_unit

⊢ ∀p. wf_encoder p encode_unit

wf_encode_bool

⊢ ∀p. wf_encoder p encode_bool

encode_prod_alt

⊢ ∀xb yb p. encode_prod xb yb p = xb (FST p) ++ yb (SND p)

wf_encode_prod

⊢ ∀p1 p2 e1 e2.
      wf_encoder p1 e1 ∧ wf_encoder p2 e2 ⇒
      wf_encoder (lift_prod p1 p2) (encode_prod e1 e2)

wf_encode_sum

⊢ ∀p1 p2 e1 e2.
      wf_encoder p1 e1 ∧ wf_encoder p2 e2 ⇒
      wf_encoder (lift_sum p1 p2) (encode_sum e1 e2)

wf_encode_option

⊢ ∀p e. wf_encoder p e ⇒ wf_encoder (lift_option p) (encode_option e)

wf_encode_list

⊢ ∀p e. wf_encoder p e ⇒ wf_encoder (EVERY p) (encode_list e)

encode_list_cong

⊢ ∀l1 l2 f1 f2.
      l1 = l2 ∧ (∀x. MEM x l2 ⇒ f1 x = f2 x) ⇒
      encode_list f1 l1 = encode_list f2 l2

encode_blist_def_compute

⊢ (∀e l. encode_blist 0 e l = []) ∧
  (∀m e l.
       encode_blist (NUMERAL (BIT1 m)) e l =
       e (HD l) ++ encode_blist (NUMERAL (BIT1 m) − 1) e (TL l)) ∧
  ∀m e l.
      encode_blist (NUMERAL (BIT2 m)) e l =
      e (HD l) ++ encode_blist (NUMERAL (BIT1 m)) e (TL l)

lift_blist_suc

⊢ ∀n p h t. lift_blist (SUC n) p (h::t) ⇔ p h ∧ lift_blist n p t

wf_encode_blist

⊢ ∀m p e. wf_encoder p e ⇒ wf_encoder (lift_blist m p) (encode_blist m e)

encode_num_def

⊢ encode_num n = if n = 0 then [T; T]
  else if EVEN n then F::encode_num ((n − 2) DIV 2)
  else T::F::encode_num ((n − 1) DIV 2)

encode_num_ind

⊢ ∀P.
      (∀n.
           (n ≠ 0 ∧ EVEN n ⇒ P ((n − 2) DIV 2)) ∧
           (n ≠ 0 ∧ ¬EVEN n ⇒ P ((n − 1) DIV 2)) ⇒
           P n) ⇒
      ∀v. P v

wf_encode_num

⊢ ∀p. wf_encoder p encode_num

encode_bnum_def_compute

⊢ (∀n. encode_bnum 0 n = []) ∧
  (∀m n.
       encode_bnum (NUMERAL (BIT1 m)) n =
       ¬EVEN n::encode_bnum (NUMERAL (BIT1 m) − 1) (n DIV 2)) ∧
  ∀m n.
      encode_bnum (NUMERAL (BIT2 m)) n =
      ¬EVEN n::encode_bnum (NUMERAL (BIT1 m)) (n DIV 2)

wf_pred_bnum_total

⊢ ∀m. wf_pred_bnum m (λx. x < 2 ** m)

wf_pred_bnum

⊢ ∀m p. wf_pred_bnum m p ⇒ collision_free m p

encode_bnum_length

⊢ ∀m n. LENGTH (encode_bnum m n) = m

encode_bnum_inj

⊢ ∀m x y. x < 2 ** m ∧ y < 2 ** m ∧ encode_bnum m x = encode_bnum m y ⇒ x = y

wf_encode_bnum_collision_free

⊢ ∀m p. wf_encoder p (encode_bnum m) ⇔ collision_free m p

wf_encode_bnum

⊢ ∀m p. wf_pred_bnum m p ⇒ wf_encoder p (encode_bnum m)

datatype_tree

⊢ DATATYPE (tree Node)

tree_11

⊢ ∀a0 a1 a0' a1'. Node a0 a1 = Node a0' a1' ⇔ a0 = a0' ∧ a1 = a1'

tree_nchotomy

⊢ ∀tt. ∃a l. tt = Node a l

tree_Axiom

⊢ ∀f0 f1 f2.
      ∃fn0 fn1.
          (∀a0 a1. fn0 (Node a0 a1) = f0 a0 a1 (fn1 a1)) ∧ fn1 [] = f1 ∧
          ∀a0 a1. fn1 (a0::a1) = f2 a0 a1 (fn0 a0) (fn1 a1)

tree_induction

⊢ ∀P0 P1.
      (∀l. P1 l ⇒ ∀a. P0 (Node a l)) ∧ P1 [] ∧ (∀t l. P0 t ∧ P1 l ⇒ P1 (t::l)) ⇒
      (∀t. P0 t) ∧ ∀l. P1 l

tree_case_cong

⊢ ∀M M' f.
      M = M' ∧ (∀a0 a1. M' = Node a0 a1 ⇒ f a0 a1 = f' a0 a1) ⇒
      tree_CASE M f = tree_CASE M' f'

tree_case_eq

⊢ tree_CASE x f = v ⇔ ∃a l. x = Node a l ∧ f a l = v

tree_ind

⊢ ∀p. (∀a ts. (∀t. MEM t ts ⇒ p t) ⇒ p (Node a ts)) ⇒ ∀t. p t

encode_tree_def

⊢ encode_tree e (Node a ts) = e a ++ encode_list (encode_tree e) ts

lift_tree_def

⊢ lift_tree p (Node a ts) ⇔ p a ∧ EVERY (lift_tree p) ts

wf_encode_tree

⊢ ∀p e. wf_encoder p e ⇒ wf_encoder (lift_tree p) (encode_tree e)