Theory: ind_type

Parents

• numeral

Types

• recspace(Arity = 1)

Constants

• FNIL  :num -> 'a
• NUMFST  :num -> num
• INJA  :'a -> num -> 'a -> bool
• INJF  :(num -> num -> 'a -> bool) -> num -> 'a -> bool
• INJN  :num -> num -> 'a -> bool
• INJP  :(num -> 'a -> bool) -> (num -> 'a -> bool) -> num -> 'a -> bool
• NUMSND  :num -> num
• NUMSUM  :bool -> num -> num
• ZCONSTR  :num -> 'a -> (num -> num -> 'a -> bool) -> num -> 'a -> bool
• ZRECSPACE  :(num -> 'a -> bool) -> bool
• NUMLEFT  :num -> bool
• BOTTOM  :'a recspace
• NUMPAIR  :num -> num -> num
• NUMRIGHT  :num -> num
• dest_rec  :'a recspace -> num -> 'a -> bool
• ZBOT  :num -> 'a -> bool
• mk_rec  :(num -> 'a -> bool) -> 'a recspace
• ISO  :('a -> 'b) -> ('b -> 'a) -> bool
• CONSTR  :num -> 'a -> (num -> 'a recspace) -> 'a recspace
• FCONS  :'a -> (num -> 'a) -> num -> 'a

Definitions

BOTTOM

|- BOTTOM = mk_rec ZBOT

CONSTR

|- !c i r. CONSTR c i r = mk_rec (ZCONSTR c i (\n. dest_rec (r n)))

FCONS

|- (!a f. FCONS a f 0 = a) /\ !a f n. FCONS a f (SUC n) = f n

FNIL

|- !n. FNIL n = @x. T

INJA

|- !a. INJA a = (\n b. b = a)

INJF

|- !f. INJF f = (\n. f (NUMFST n) (NUMSND n))

INJN

|- !m. INJN m = (\n a. n = m)

INJP

|- !f1 f2.
     INJP f1 f2 =
     (\n a. (if NUMLEFT n then f1 (NUMRIGHT n) a else f2 (NUMRIGHT n) a))

ISO

|- !f g. ISO f g = (!x. f (g x) = x) /\ !y. g (f y) = y

NUMPAIR

|- !x y. NUMPAIR x y = 2 ** x * (2 * y + 1)

NUMPAIR_DEST

|- !x y. (NUMFST (NUMPAIR x y) = x) /\ (NUMSND (NUMPAIR x y) = y)

NUMSUM

|- !b x. NUMSUM b x = (if b then SUC (2 * x) else 2 * x)

NUMSUM_DEST

|- !x y. (NUMLEFT (NUMSUM x y) = x) /\ (NUMRIGHT (NUMSUM x y) = y)

recspace_repfns

|- (!a. mk_rec (dest_rec a) = a) /\
   !r. ZRECSPACE r = (dest_rec (mk_rec r) = r)

recspace_TY_DEF

|- ?rep. TYPE_DEFINITION ZRECSPACE rep

ZBOT

|- ZBOT = INJP (INJN 0) @z. T

ZCONSTR

|- !c i r. ZCONSTR c i r = INJP (INJN (SUC c)) (INJP (INJA i) (INJF r))

ZRECSPACE

|- ZRECSPACE =
   (\a0.
      !ZRECSPACE'.
        (!a0.
           (a0 = ZBOT) \/
           (?c i r. (a0 = ZCONSTR c i r) /\ !n. ZRECSPACE' (r n)) ==>
           ZRECSPACE' a0) ==>
        ZRECSPACE' a0)

Theorems

CONSTR_BOT

|- !c i r. ~(CONSTR c i r = BOTTOM)

CONSTR_IND

|- !P. P BOTTOM /\ (!c i r. (!n. P (r n)) ==> P (CONSTR c i r)) ==> !x. P x

CONSTR_INJ

|- !c1 i1 r1 c2 i2 r2.
     (CONSTR c1 i1 r1 = CONSTR c2 i2 r2) = (c1 = c2) /\ (i1 = i2) /\ (r1 = r2)

CONSTR_REC

|- !Fn. ?f. !c i r. f (CONSTR c i r) = Fn c i r (\n. f (r n))

DEST_REC_INJ

|- !x y. (dest_rec x = dest_rec y) = (x = y)

INJ_INVERSE2

|- !P.
     (!x1 y1 x2 y2. (P x1 y1 = P x2 y2) = (x1 = x2) /\ (y1 = y2)) ==>
     ?X Y. !x y. (X (P x y) = x) /\ (Y (P x y) = y)

INJA_INJ

|- !a1 a2. (INJA a1 = INJA a2) = (a1 = a2)

INJF_INJ

|- !f1 f2. (INJF f1 = INJF f2) = (f1 = f2)

INJN_INJ

|- !n1 n2. (INJN n1 = INJN n2) = (n1 = n2)

INJP_INJ

|- !f1 f1' f2 f2'. (INJP f1 f2 = INJP f1' f2') = (f1 = f1') /\ (f2 = f2')

ISO_FUN

|- ISO f f' /\ ISO g g' ==> ISO (\h a'. g (h (f' a'))) (\h a. g' (h (f a)))

ISO_REFL

|- ISO (\x. x) (\x. x)

ISO_USAGE

|- ISO f g ==>
   (!P. (!x. P x) = !x. P (g x)) /\ (!P. (?x. P x) = ?x. P (g x)) /\
   !a b. (a = g b) = (f a = b)

MK_REC_INJ

|- !x y. (mk_rec x = mk_rec y) ==> ZRECSPACE x /\ ZRECSPACE y ==> (x = y)

NUMPAIR_INJ

|- !x1 y1 x2 y2. (NUMPAIR x1 y1 = NUMPAIR x2 y2) = (x1 = x2) /\ (y1 = y2)

NUMPAIR_INJ_LEMMA

|- !x1 y1 x2 y2. (NUMPAIR x1 y1 = NUMPAIR x2 y2) ==> (x1 = x2)

NUMSUM_INJ

|- !b1 x1 b2 x2. (NUMSUM b1 x1 = NUMSUM b2 x2) = (b1 = b2) /\ (x1 = x2)

ZCONSTR_ZBOT

|- !c i r. ~(ZCONSTR c i r = ZBOT)