Package natural-add: Definitions and theorems about natural number addition

Information

namenatural-add
version1.0
description Definitions and theorems about natural number addition
authorJoe Hurd <joe@gilith.com>
licenseMIT
showData.Bool
Number.Natural
Number.Numeral

Files

Defined Constant

Theorems

m. m + 0 = m

n. bit0 n = n + n

m n. m m + n

m n. n m + n

m. suc m = m + 1

n. bit1 n = suc (n + n)

m n. m + n = n + m

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

m n. m < m + n 0 < n

m n. n < m + n 0 < m

m n. m + n = m n = 0

m n. m + n = n m = 0

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

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 + p = n + p m = n

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

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

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

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

m n. m + n = 0 m = 0 n = 0

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

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

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

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

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

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)

Input Type Operators

Input Constants

Assumptions

T

n. n n

F p. p

t. t ¬t

(¬) = 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. ¬(suc n = 0)

n. bit1 n = suc (bit0 n)

() = λp q. p q p

t. (t T) (t F)

(¬T F) (¬F T)

t1 t2. t1 t2 t2 t1

m n. m < n m n

() = 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. suc m = suc n m = n

m n. suc m < suc n m < n

() = λp q. r. (p r) (q r) r

bit0 0 = 0 n. bit0 (suc n) = suc (suc (bit0 n))

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

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

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

P. P 0 (n. P n P (suc n)) n. P n

(t. ¬¬t t) (¬T F) (¬F T)

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 0 = e n. fn (suc n) = f (fn n) n

(m. m < 0 F) m n. m < suc n m = n m < n

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

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)