Package sum-def: sum-def

Information

namesum-def
version1.0
descriptionsum-def
authorJoe Hurd <joe@gilith.com>
licenseHOLLight
provenanceHOL Light theory extracted on 2011-02-19
showData.Bool

Files

Defined Type Operator

Defined Constants

Theorems

x. Data.Sum.destLeft (Data.Sum.left x) = x

y. Data.Sum.destRight (Data.Sum.right y) = y

P. (a. P (Data.Sum.left a)) (a. P (Data.Sum.right a)) x. P x

INL' INR'.
    fn.
      (a. fn (Data.Sum.left a) = INL' a)
      a. fn (Data.Sum.right a) = INR' a

Input Type Operators

Input Constants

Assumptions

T

n. Number.Natural.≤ Number.Numeral.zero n

F p. p

(¬) = λp. p F

() = λP. P ((select) P)

a. ∃!x. x = a

t. (x. t) t

t. (x. t) t

t. (λx. t x) = t

() = λP. P = λx. T

x. x = x T

n. ¬(Number.Natural.suc n = Number.Numeral.zero)

n.
    Number.Natural.even
      (Number.Natural.*
         (Number.Numeral.bit0 (Number.Numeral.bit1 Number.Numeral.zero)) n)

n. Number.Numeral.bit0 n = Number.Natural.+ n n

() = λp q. p q p

t. (t T) (t F)

n. Number.Numeral.bit1 n = Number.Natural.suc (Number.Natural.+ n n)

x y. Data.Pair.fst (Data.Pair., x y) = x

x y. Data.Pair.snd (Data.Pair., x y) = y

(¬T F) (¬F T)

t1 t2. t1 t2 t2 t1

n.
    Number.Natural.*
      (Number.Numeral.bit0 (Number.Numeral.bit1 Number.Numeral.zero)) n =
    Number.Natural.+ n n

m n. ¬(Number.Natural.< m n Number.Natural.≤ n m)

m n. ¬(Number.Natural.≤ m n Number.Natural.< n m)

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

() = λp q. (λf. f p q) = λf. f T T

() = λP. q. (x. P x q) q

m n. Number.Natural.suc m = Number.Natural.suc n m = n

m n.
    Number.Natural.even (Number.Natural.* m n)
    Number.Natural.even m Number.Natural.even n

m n.
    Number.Natural.even (Number.Natural.+ m n) Number.Natural.even m
    Number.Natural.even n

f g. f = g x. f x = g x

P a. (x. a = x P x) P a

() = λp q. r. (p r) (q r) r

(Number.Natural.even Number.Numeral.zero T)
  n. Number.Natural.even (Number.Natural.suc n) ¬Number.Natural.even n

m n. Number.Natural.≤ m n Number.Natural.< m n m = n

m n. Number.Natural.≤ m n Number.Natural.≤ n m m = n

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

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

m n p.
    Number.Natural.* m (Number.Natural.* n p) =
    Number.Natural.* (Number.Natural.* m n) p

m n p. Number.Natural.+ m p = Number.Natural.+ n p m = n

P x. (y. P y y = x) (select) P = x

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

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

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

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

(t. ¬¬t t) (¬T F) (¬F T)

m n.
    Number.Natural.exp m n = Number.Numeral.zero
    m = Number.Numeral.zero ¬(n = Number.Numeral.zero)

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

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

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

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

e f.
    fn.
      fn Number.Numeral.zero = e
      n. fn (Number.Natural.suc n) = f (fn n) n

m n p.
    Number.Natural.* m n = Number.Natural.* m p
    m = Number.Numeral.zero n = p

m n p.
    Number.Natural.≤ (Number.Natural.* m n) (Number.Natural.* m p)
    m = Number.Numeral.zero Number.Natural.≤ n p

m n p.
    Number.Natural.< (Number.Natural.* m n) (Number.Natural.* m p)
    ¬(m = Number.Numeral.zero) Number.Natural.< n p

A B C D. (A B) (C D) A C B D

A B C D. (A B) (C D) A C B D

P. (x. ∃!y. P x y) f. x y. P x y f x = y

(m.
     Number.Natural.exp m Number.Numeral.zero =
     Number.Numeral.bit1 Number.Numeral.zero)
  m n.
    Number.Natural.exp m (Number.Natural.suc n) =
    Number.Natural.* m (Number.Natural.exp m n)

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

P. (∃!x. P x) (x. P x) x x'. P x P x' x = x'

(m. Number.Natural.≤ m Number.Numeral.zero m = Number.Numeral.zero)
  m n.
    Number.Natural.≤ m (Number.Natural.suc n)
    m = Number.Natural.suc n Number.Natural.≤ m n

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

(n. Number.Natural.+ Number.Numeral.zero n = n)
  (m. Number.Natural.+ m Number.Numeral.zero = m)
  (m n.
     Number.Natural.+ (Number.Natural.suc m) n =
     Number.Natural.suc (Number.Natural.+ m n))
  m n.
    Number.Natural.+ m (Number.Natural.suc n) =
    Number.Natural.suc (Number.Natural.+ m n)