Package list-member-thm: list-member-thm

Information

namelist-member-thm
version1.0
descriptionlist-member-thm
authorJoe Hurd <joe@gilith.com>
licenseHOLLight
provenanceHOL Light theory extracted on 2011-03-15
showData.Bool

Files

Theorems

l n.
    Number.Natural.< n (Data.List.length l)
    Data.List.member (Data.List.nth n l) l

P l. (x. Data.List.member x l P x) Data.List.all P l

P l. (x. P x Data.List.member x l) Data.List.exists P l

P l x.
    Data.List.member x (Data.List.filter P l) P x Data.List.member x l

x l1 l2.
    Data.List.member x (Data.List.@ l1 l2)
    Data.List.member x l1 Data.List.member x l2

l x.
    Data.List.member x l
    i. Number.Natural.< i (Data.List.length l) x = Data.List.nth i l

f y l.
    Data.List.member y (Data.List.map f l)
    x. Data.List.member x l y = f x

P Q l.
    (x. Data.List.member x l P x Q x) Data.List.all P l
    Data.List.all Q l

P Q l.
    (x. Data.List.member x l P x Q x) Data.List.exists P l
    Data.List.exists Q l

Input Type Operators

Input Constants

Assumptions

T

x. Function.id x = x

n. Number.Natural.< Number.Numeral.zero (Number.Natural.suc n)

t. (x. t) t

t. (x. t) t

t. (λx. t x) = t

() = λP. P = λx. T

x. x = x T

() = λp q. p q p

t. (t T) (t F)

(¬T F) (¬F T)

m. m = Number.Numeral.zero n. m = Number.Natural.suc n

() = λp q. (λf. f p q) = λf. f T T

P. ¬(x. P x) x. ¬P x

P. ¬(x. P x) x. ¬P x

() = λP. q. (x. P x q) q

m n.
    Number.Natural.< (Number.Natural.suc m) (Number.Natural.suc n)
    Number.Natural.< m n

() = λp q. r. (p r) (q r) r

P Q. P (x. Q x) x. P Q x

P Q. P (x. Q x) x. P Q x

P Q. (x. P x) Q x. P x Q

P Q. (x. P x) Q x. P x Q

t1 t2 t3. t1 t2 t3 (t1 t2) t3

P. (x. y. P x y) y. x. P x (y x)

Data.List.length Data.List.[] = Number.Numeral.zero
  h t.
    Data.List.length (Data.List.:: h t) =
    Number.Natural.suc (Data.List.length t)

P.
    P Number.Numeral.zero (n. P n P (Number.Natural.suc n)) n. P n

(t. ¬¬t t) (¬T F) (¬F T)

P Q. (x. P x) (x. Q x) x. P x Q x

P. P Data.List.[] (a0 a1. P a1 P (Data.List.:: a0 a1)) x. P x

P c x y. P (if c then x else y) (c P x) (¬c P y)

(m. Number.Natural.< m Number.Numeral.zero F)
  m n.
    Number.Natural.< m (Number.Natural.suc n)
    m = n Number.Natural.< m n

(l. Data.List.@ Data.List.[] l = l)
  h t l.
    Data.List.@ (Data.List.:: h t) l = Data.List.:: h (Data.List.@ t l)

(P. Data.List.all P Data.List.[] T)
  h P t. Data.List.all P (Data.List.:: h t) P h Data.List.all P t

(P. Data.List.exists P Data.List.[] F)
  h P t.
    Data.List.exists P (Data.List.:: h t) P h Data.List.exists P t

(f. Data.List.map f Data.List.[] = Data.List.[])
  f h t.
    Data.List.map f (Data.List.:: h t) =
    Data.List.:: (f h) (Data.List.map f t)

(x. Data.List.member x Data.List.[] F)
  h x t.
    Data.List.member x (Data.List.:: h t) x = h Data.List.member x t

(h t. Data.List.nth Number.Numeral.zero (Data.List.:: h t) = h)
  h t n.
    Data.List.nth (Number.Natural.suc n) (Data.List.:: h t) =
    Data.List.nth n t

t. ((T t) t) ((t T) t) ((F t) ¬t) ((t F) ¬t)

(P. Data.List.filter P Data.List.[] = Data.List.[])
  h P t.
    Data.List.filter P (Data.List.:: h t) =
    (if P h then Data.List.:: h (Data.List.filter P t)
     else Data.List.filter P t)

t. (T t t) (t T t) (F t F) (t F F) (t t t)

t. (T t T) (t T T) (F t t) (t F t) (t t t)

t. (T t t) (t T T) (F t T) (t t T) (t F ¬t)

p q r.
    (p q q p) ((p q) r p q r) (p q r q p r)
    (p p p) (p p q p q)