Package natural-distance: Natural number distance function

Information

namenatural-distance
version1.6
descriptionNatural number distance function
authorJoe Hurd <joe@gilith.com>
licenseMIT
showData.Bool
Number.Natural

Files

Defined Constant

Theorems

n. distance 0 n = n

n. distance n 0 = n

n. distance n n = 0

m n. distance m n = distance n m

m n. distance m n m + n

m n. distance m (m + n) = n

m n. distance (m + n) m = n

m n. distance m n = 0 m = n

m n p. distance m p distance m n + distance n p

m n p. distance (m + n) (m + p) = distance n p

p m n. distance (m + p) (n + p) = distance m n

m n. distance m n = if m n then n - m else m - n

m n p. m * distance n p = distance (m * n) (m * p)

p m n. distance m n * p = distance (m * p) (n * p)

m n p q. distance m p distance (m + n) (p + q) + distance n q

m n p q. distance (m + n) (p + q) distance m p + distance n q

m n p q. distance m n + distance n p q distance m p q

m n p. distance m n p m n + p n m + p

P x y. P (distance x y) d. (x = y + d P d) (y = x + d P d)

m n p q r s.
    distance m n r distance p q s
    distance m p distance n q + (r + s)

Input Type Operators

Input Constants

Assumptions

T

n. 0 n

n. n n

F p. p

t. t ¬t

(¬) = λp. p F

t. (x. t) t

() = λp. p = λx. T

x. x = x T

n. ¬(suc n = 0)

m. m - 0 = m

n. n - n = 0

m n. m m + n

m n. n m + n

() = λp q. p q p

(¬T F) (¬F T)

x y. x = y y = x

m n. m * n = n * m

m n. m + n = n + m

m n. m n n m

m n. m + n - m = n

m n. ¬(m < n) n m

() = λp q. (λf. f p q) = λf. f T T

P. ¬(x. P x) x. ¬P x

() = λP. q. (x. P x q) q

m n. m + n = m n = 0

m n. m n d. n = m + d

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

() = λp q. r. (p r) (q r) r

m n. m n n m m = n

m n. m < n d. n = m + suc d

m n p. m + (n + p) = m + n + p

m n p. m + n = m + p n = p

m n p. m + n m + p n p

m n p. m n n p m p

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

m n. n m (m - n = 0 m = n)

m n p. p n m + n - (m + p) = n - p

m n p. m * n m * p m = 0 n p

(n. 0 * n = 0) m n. suc m * n = m * n + n

m n p q. m p n q m + n p + q

m n p. p n m * (n - p) = m * n - m * p

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

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)

m n p.
    m + n = n + m m + n + p = m + (n + p) m + (n + p) = n + (m + p)

(n. 0 + n = n) (m. m + 0 = m) (m n. suc m + n = suc (m + n))
  m n. m + suc n = suc (m + n)