Package set-fold-thm: set-fold-thm

Information

nameset-fold-thm
version1.9
descriptionset-fold-thm
authorJoe Hurd <joe@gilith.com>
licenseHOLLight
provenanceHOL Light theory extracted on 2011-07-18
showData.Bool

Files

Theorems

f b.
    (x y s. ¬(x = y) f x (f y s) = f y (f x s))
    Set.fold f Set.∅ b = b
    x s.
      Set.finite s
      Set.fold f s b =
      if Set.∈ x s then f x (Set.fold f (Set.delete s x) b)
      else Set.fold f (Set.delete s x) b

s f g b.
    Set.finite s (x. Set.∈ x s f x = g x)
    (x y s. ¬(x = y) f x (f y s) = f y (f x s))
    (x y s. ¬(x = y) g x (g y s) = g y (g x s))
    Set.fold f s b = Set.fold g s b

Input Type Operators

Input Constants

Assumptions

T

F p. p

t. t ¬t

(~) = λp. p F

t. (x. t) t

() = λp. p = λx. T

x. x = x T

() = λp q. p q p

t. (t T) (t F)

(¬T F) (¬F T)

x y. x = y y = x

s x. Set.finite s Set.finite (Set.delete s x)

() = λp q. (λf. f p q) = λf. f T T

x s. Set.delete s x = s ¬Set.∈ x s

() = λp q. r. (p r) (q r) r

x s. Set.∈ x s Set.insert x (Set.delete s x) = s

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

p q r. p q r p q r

t1 t2. (if T then t1 else t2) = t1 (if F then t1 else t2) = t2

x y s. Set.∈ x (Set.insert y s) x = y Set.∈ x s

s x y. Set.∈ x (Set.delete s y) Set.∈ x s ¬(x = y)

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

P.
    P Set.∅
    (x s. P s ¬Set.∈ x s Set.finite s P (Set.insert x s))
    s. Set.finite s P s

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)

f b.
    (x y s. ¬(x = y) f x (f y s) = f y (f x s))
    Set.fold f Set.∅ b = b
    x s.
      Set.finite s
      Set.fold f (Set.insert x s) b =
      if Set.∈ x s then Set.fold f s b else f x (Set.fold f s b)