Theory fcp

⊢ $FCP = (λg. @f. ∀i. i < dimindex (:β) ⇒ f ' i = g i)

⊢ ∀a b.
    FCP_CONCAT a b =
    FCP i. if i < dimindex (:γ) then b ' i else a ' (i − dimindex (:γ))

⊢ ∀h v. FCP_CONS h v = (0 :+ h) (FCP i. v ' (PRE i))

⊢ ∀P v. FCP_EVERY P v ⇔ ∀i. dimindex (:α) ≤ i ∨ P (v ' i)

⊢ ∀P v. FCP_EXISTS P v ⇔ ∃i. i < dimindex (:α) ∧ P (v ' i)

⊢ ∀f i v. FCP_FOLD f i v = FOLDL f i (V2L v)

⊢ ∀v. FCP_FST v = FCP i. v ' (i + dimindex (:γ))

⊢ ∀v. FCP_HD v = v ' 0

⊢ ∀f v. FCP_MAP f v = FCP i. f (v ' i)

⊢ ∀v. FCP_SND v = FCP i. v ' i

⊢ ∀v. FCP_TL v = FCP i. v ' (SUC i)

⊢ ∀a b. (a :+ b) = (λm. FCP c. if a = c then b else m ' c)

⊢ ∀a b. FCP_ZIP a b = FCP i. (a ' i,b ' i)

⊢ ∀L. L2V L = FCP i. L❲i❳

⊢ ∀v. V2L v = GENLIST ($' v) (dimindex (:β))

⊢ ∃rep.
    TYPE_DEFINITION
      (λa0.
           ∀ $var$('bit0').
             (∀a0.
                (∃a. a0 = (λa. ind_type$CONSTR 0 a (λn. ind_type$BOTTOM)) a) ∨
                (∃a. a0 =
                     (λa. ind_type$CONSTR (SUC 0) a (λn. ind_type$BOTTOM))
                       a) ⇒
                $var$('bit0') a0) ⇒
             $var$('bit0') a0) rep

⊢ (∀a f f1. bit0_CASE (BIT0A a) f f1 = f a) ∧
  ∀a f f1. bit0_CASE (BIT0B a) f f1 = f1 a

⊢ (∀f a. bit0_size f (BIT0A a) = 1 + f a) ∧
  ∀f a. bit0_size f (BIT0B a) = 1 + f a

⊢ ∃rep.
    TYPE_DEFINITION
      (λa0.
           ∀ $var$('bit1').
             (∀a0.
                (∃a. a0 = (λa. ind_type$CONSTR 0 a (λn. ind_type$BOTTOM)) a) ∨
                (∃a. a0 =
                     (λa. ind_type$CONSTR (SUC 0) a (λn. ind_type$BOTTOM))
                       a) ∨
                a0 =
                ind_type$CONSTR (SUC (SUC 0)) ARB (λn. ind_type$BOTTOM) ⇒
                $var$('bit1') a0) ⇒
             $var$('bit1') a0) rep

⊢ (∀a f f1 v. bit1_CASE (BIT1A a) f f1 v = f a) ∧
  (∀a f f1 v. bit1_CASE (BIT1B a) f f1 v = f1 a) ∧
  ∀f f1 v. bit1_CASE BIT1C f f1 v = v

⊢ (∀f a. bit1_size f (BIT1A a) = 1 + f a) ∧
  (∀f a. bit1_size f (BIT1B a) = 1 + f a) ∧ ∀f. bit1_size f BIT1C = 0

⊢ ∃rep. TYPE_DEFINITION (λf. T) rep

⊢ (∀a. mk_cart (dest_cart a) = a) ∧
  ∀r. (λf. T) r ⇔ dest_cart (mk_cart r) = r

⊢ dimindex (:α) = if FINITE 𝕌(:α) then CARD 𝕌(:α) else 1

⊢ ∀h f. fcp_CASE (mk_cart h) f = f h

⊢ ∀x i. x ' i = dest_cart x (finite_index i)

⊢ ∃rep. TYPE_DEFINITION (λx. x = ARB ∨ FINITE 𝕌(:α)) rep

⊢ (∀a. mk_finite_image (dest_finite_image a) = a) ∧
  ∀r. (λx. x = ARB ∨ FINITE 𝕌(:α)) r ⇔
      dest_finite_image (mk_finite_image r) = r

⊢ finite_index = @f. ∀x. ∃!n. n < dimindex (:α) ∧ f n = x

⊢ ∀m a. (a :+ m ' a) m = m

⊢ ∀P. FINITE 𝕌(:'N) ∧ (∀i. i < dimindex (:'N) ⇒ FINITE {x | P i x}) ⇒
      CARD {v | (∀i. i < dimindex (:'N) ⇒ P i (v ' i))} =
      nproduct {0 .. dimindex (:'N) − 1} (λi. CARD {x | P i x})

⊢ FINITE 𝕌(:α) ∧ FINITE 𝕌(:'N) ⇒
  CARD 𝕌(:α['N]) = CARD 𝕌(:α) ** dimindex (:'N)

⊢ ∀x y. x = y ⇔ ∀i. i < dimindex (:β) ⇒ x ' i = y ' i

⊢ ∀b x y. (if b then x else y) ' i = if b then x ' i else y ' i

⊢ 1 ≤ dimindex (:α)

⊢ 0 < dimindex (:α)

⊢ dimindex (:α) ≠ 0

⊢ ∀i v. i < dimindex (:β) ⇒ (V2L v)❲i❳ = v ' i

⊢ ∀m a w b.
    (a :+ w) m ' b =
    if b < dimindex (:β) then if a = b then w else m ' b
    else FAIL $' $var$(index out of range) ((a :+ w) m) b

⊢ ∀i. i < dimindex (:β) ⇒ $FCP g ' i = g i

⊢ ∀a b c d.
    FINITE 𝕌(:β) ∧ FINITE 𝕌(:γ) ∧ FCP_CONCAT a b = FCP_CONCAT c d ⇒
    a = c ∧ b = d

⊢ ∀x. FINITE 𝕌(:β) ∧ FINITE 𝕌(:γ) ⇒ FCP_CONCAT (FCP_FST x) (FCP_SND x) = x

⊢ ∀a b.
    FINITE 𝕌(:β) ∧ FINITE 𝕌(:γ) ⇒
    FCP_FST (FCP_CONCAT a b) = a ∧ FCP_SND (FCP_CONCAT a b) = b

⊢ ∀a v. FCP_CONS a v = L2V (a::V2L v)

⊢ ∀g. (FCP i. g ' i) = g

⊢ ∀P v. FCP_EVERY P v ⇔ EVERY P (V2L v)

⊢ ∀P v. FCP_EXISTS P v ⇔ EXISTS P (V2L v)

⊢ ∀v. FCP_HD v = HD (V2L v)

⊢ ∀f v. FCP_MAP f v = L2V (MAP f (V2L v))

⊢ ∀v. 1 < dimindex (:β) ∧ dimindex (:γ) = dimindex (:β) − 1 ⇒
      FCP_TL v = L2V (TL (V2L v))

⊢ ∀f g. (∀i. i < dimindex (:β) ⇒ f ' i = g i) ⇔ $FCP g = f

⊢ ∀m a b c d. a ≠ b ⇒ (a :+ c) ((b :+ d) m) = (b :+ d) ((a :+ c) m)

⊢ ∀m a b c. (a :+ c) ((a :+ b) m) = (a :+ c) m

⊢ ∀m a v. m ' a = v ⇒ (a :+ v) m = m

⊢ FINITE 𝕌(:α) ∧ FINITE 𝕌(:'N) ⇒ FINITE 𝕌(:α['N])

⊢ ∀P m.
    FINITE 𝕌(:'N) ∧ (∀i. i < dimindex (:'N) ⇒ {x | P i x} HAS_SIZE m i) ⇒
    {v | ∀i. i < dimindex (:'N) ⇒ P i (v ' i)} HAS_SIZE
    nproduct {0 .. dimindex (:'N) − 1} m

⊢ ∀m. 𝕌(:α) HAS_SIZE m ∧ FINITE 𝕌(:'N) ⇒
      𝕌(:α['N]) HAS_SIZE m ** dimindex (:'N)

⊢ ∀v. LENGTH (V2L v) = dimindex (:β)

⊢ INFINITE 𝕌(:α) ⇒ dimindex (:α) = 1

⊢ ∀v. ¬NULL (V2L v)

⊢ ∀i a. i < dimindex (:β) ⇒ L2V a ' i = a❲i❳

⊢ ∀i a. i < dimindex (:β) ⇒ FCP_TL a ' i = a ' (SUC i)

⊢ ∀x. dimindex (:β) = LENGTH x ⇒ V2L (L2V x) = x

⊢ (∀a a'. BIT0A a = BIT0A a' ⇔ a = a') ∧ ∀a a'. BIT0B a = BIT0B a' ⇔ a = a'

⊢ ∀f0 f1. ∃fn. (∀a. fn (BIT0A a) = f0 a) ∧ ∀a. fn (BIT0B a) = f1 a

⊢ ∀M M' f f1.
    M = M' ∧ (∀a. M' = BIT0A a ⇒ f a = f' a) ∧
    (∀a. M' = BIT0B a ⇒ f1 a = f1' a) ⇒
    bit0_CASE M f f1 = bit0_CASE M' f' f1'

⊢ bit0_CASE x f f1 = v ⇔
  (∃a. x = BIT0A a ∧ f a = v) ∨ ∃a. x = BIT0B a ∧ f1 a = v

⊢ ∀a' a. BIT0A a ≠ BIT0B a'

⊢ ∀P. (∀a. P (BIT0A a)) ∧ (∀a. P (BIT0B a)) ⇒ ∀b. P b

⊢ ∀bb. (∃a. bb = BIT0A a) ∨ ∃a. bb = BIT0B a

⊢ (∀a a'. BIT1A a = BIT1A a' ⇔ a = a') ∧ ∀a a'. BIT1B a = BIT1B a' ⇔ a = a'

⊢ ∀f0 f1 f2. ∃fn.
    (∀a. fn (BIT1A a) = f0 a) ∧ (∀a. fn (BIT1B a) = f1 a) ∧ fn BIT1C = f2

⊢ ∀M M' f f1 v.
    M = M' ∧ (∀a. M' = BIT1A a ⇒ f a = f' a) ∧
    (∀a. M' = BIT1B a ⇒ f1 a = f1' a) ∧ (M' = BIT1C ⇒ v = v') ⇒
    bit1_CASE M f f1 v = bit1_CASE M' f' f1' v'

⊢ bit1_CASE x f f1 v = v' ⇔
  (∃a. x = BIT1A a ∧ f a = v') ∨ (∃a. x = BIT1B a ∧ f1 a = v') ∨
  x = BIT1C ∧ v = v'

⊢ (∀a' a. BIT1A a ≠ BIT1B a') ∧ (∀a. BIT1A a ≠ BIT1C) ∧ ∀a. BIT1B a ≠ BIT1C

⊢ ∀P. (∀a. P (BIT1A a)) ∧ (∀a. P (BIT1B a)) ∧ P BIT1C ⇒ ∀b. P b

⊢ ∀bb. (∃a. bb = BIT1A a) ∨ (∃a. bb = BIT1B a) ∨ bb = BIT1C

⊢ FINITE 𝕌(:α) ⇒ CARD 𝕌(:α) = dimindex (:α)

⊢ DATATYPE (bit0 BIT0A BIT0B)

⊢ DATATYPE (bit1 BIT1A BIT1B BIT1C)

⊢ ∀f. ∃g. ∀h. g (mk_cart h) = f h

⊢ ∀P. (∀f. P (mk_cart f)) ⇒ ∀a. P a

⊢ ∀a b f. (x :+ y) ($FCP f) = FCP c. if x = c then y else f c

⊢ FINITE 𝕌(:α bit0) ⇔ FINITE 𝕌(:α)

⊢ FINITE 𝕌(:α bit1) ⇔ FINITE 𝕌(:α)

⊢ FINITE 𝕌(:unit)

⊢ FINITE 𝕌(:α + β) ⇔ FINITE 𝕌(:α) ∧ FINITE 𝕌(:β)

⊢ dimindex (:α bit0) = if FINITE 𝕌(:α) then 2 * dimindex (:α) else 1

⊢ dimindex (:α bit1) = if FINITE 𝕌(:α) then 2 * dimindex (:α) + 1 else 1

⊢ ∀f n.
    $FCP f ' n =
    if n < dimindex (:β) then f n
    else FAIL $' $var$(FCP out of bounds) ($FCP f) n

⊢ dimindex (:unit) = 1

⊢ dimindex (:α + β) =
  if FINITE 𝕌(:α) ∧ FINITE 𝕌(:β) then dimindex (:α) + dimindex (:β) else 1