Package set: Set theory

Information

nameset
version1.11
descriptionSet theory
authorJoe Hurd <joe@gilith.com>
licenseMIT
show Data.Bool
Data.Pair
Function
Number.Natural
Set

Files

Defined Type Operator

Defined Constants

Theorems

finite

finite universe

infinite universe

¬( = universe)

¬(universe = )

size = 0

bigIntersect = universe

bigUnion =

x. x universe

s. s

s. s universe

s. s s

fromPredicate (λx. F) =

hasSize universe 2

x. ¬(x )

a. finite (insert a )

s. ¬(universe s)

s. ¬(s )

s. ¬(s s)

x. delete x =

x. insert x universe = universe

s. - s =

s. s - = s

s. s - universe =

s. s - s =

s. image id s = s

s. s s = s

s. s s = s

s. image (λx. x) s = s

universe = insert T (insert F )

x. hasSize (insert x ) 1

s. infinite s ¬finite s

s. bigIntersect (insert s ) = s

s. bigUnion (insert s ) = s

x s. x insert x s

x s. delete s x s

s t. disjoint s (t - s)

s t. disjoint (t - s) s

s t. s - t s

x. size (insert x ) = 1

s. finite s hasSize s (size s)

s. rest s = delete s (choice s)

s. infinite s ¬(s = )

s. s = disjoint s s

s. hasSize s 0 s =

s. universe s s = universe

s. s s =

s. disjoint s disjoint s

x s. ¬( = insert x s)

x s. ¬(insert x s = )

s x. finite s finite (delete s x)

s x. finite (delete s x) finite s

s x. finite (insert x s) finite s

s t. finite s finite (s - t)

s t. disjoint s t disjoint t s

s t. s t = t s

s t. s t = t s

s. (x. x s) s = universe

s. finite s a. ¬(a s)

f s. finite s finite (image f s)

p x. x fromPredicate p p x

= { x. x | F }

universe = { x. x | T }

s. size s = fold (λx n. suc n) s 0

s. ¬(s = ) choice s s

s n. hasSize s n size s = n

s. singleton s x. s = insert x

s. s universe x. ¬(x s)

s. (x. x s) ¬(s = )

x y. x insert y x = y

x s. x s insert x s = s

x s. s - insert x = delete s x

x s. insert x (insert x s) = insert x s

x s. delete (delete s x) x = delete s x

x s. insert x s = insert x s

s x. insert x s x s

s t. disjoint s t s t =

s t. s t s t = s

s t. s t s t = t

s t. s - t = s t

s t. s - t = s disjoint s t

s t. t (s - t) = t s

s t. s - t - t = s - t

s t. s - t t = s t

s t. finite t s t finite s

s u. bigIntersect (insert s u) = s bigIntersect u

s u. bigUnion (insert s u) = s bigUnion u

f s. image f s = s =

f g. f g bigIntersect g bigIntersect f

f g. f g bigUnion f bigUnion g

{ m. m | m < 0 } =

n. finite { m. m | m < n }

n. finite { m. m | m n }

s. finite s (size s = 0 s = )

x s. disjoint s (insert x ) ¬(x s)

x s. delete s x = s ¬(x s)

x s. disjoint (insert x ) s ¬(x s)

s t. finite (s t) finite s finite t

s t. finite s finite t finite (s t)

s t. infinite s finite t infinite (s - t)

s t. finite s finite t finite (s t)

s t. bigIntersect (insert s (insert t )) = s t

s t. bigUnion (insert s (insert t )) = s t

s t. finite s finite t finite (cross s t)

f x. image f (insert x ) = insert (f x)

f s. finite s size (image f s) size s

f s. image f (bigUnion s) = bigUnion (image (image f) s)

s t. bigIntersect (s t) = bigIntersect s bigIntersect t

s t. bigUnion (s t) = bigUnion s bigUnion t

finite x s. finite s finite (insert x s)

n. hasSize { m. m | m < n } n

s t x. s t s insert x t

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

s n. hasSize s n finite s size s = n

s t. s t t s s = t

s t. s (t - s) = t s t

s t. t - s s = t s t

s t. s t t s s = t

s t. bijections s t = injections s t surjections s t

(s. s = ) s. s =

(s. universe s = s) s. s universe = s

(s. s = s) s. s = s

(s. universe s = universe) s. s universe = universe

n. size { m. m | m < n } = n

s. finite s a. x. x s x a

f s x. x s f x image f s

x s. delete (insert x s) x = s ¬(x s)

s t. s t s t ¬(s = t)

s t. s t s t ¬(t s)

a b. a b finite b size a < size b

a b. a b finite b size a size b

s. bigIntersect s = universe t. t s t = universe

s. bigUnion s = t. t s t =

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

x y s. delete (delete s x) y = delete (delete s y) x

x s t. s insert x t delete s x t

x s t. insert x s t = insert x (s t)

s c. image (λx. c) s = if s = then else insert c

s t x. disjoint (delete s x) t disjoint (delete t x) s

s t x. s - insert x t = delete s x - t

s t x. delete s x t = delete (s t) x

s t u. s t u s - t u

s t u. s t u s - u t

t u s. s - t - u = s - (t u)

t u s. s - t - u = s - u - t

s t u. s t u = s (t u)

s t u. s t u = s (t u)

s t u. s t t u s u

s t u. s t t u s u

s t u. s t t u s u

s t u. s t t u s u

s t. s = t x. x s x t

s t. s t x. x s x t

s t. (x. x s x t) s = t

s. finite s finite { t. t | t s }

f s t. s t image f s image f t

f g s. image (f o g) s = image f (image g s)

s t. s t = s = t =

s t. cross s t = s = t =

s. finite s (finite (bigUnion s) t. t s finite t)

s. finite (bigUnion s) finite s t. t s finite t

n. hasSize { m. m | m n } (n + 1)

s t. disjoint s t ¬x. x s x t

s t. disjoint s (bigUnion t) x. x t disjoint s x

t f. t bigIntersect f s. s f t s

s x. x bigIntersect s t. t s x t

s x. x bigUnion s t. t s x t

f t. bigUnion f t s. s f s t

x s. x rest s x s ¬(x = choice s)

p x. x { y. y | p y } p x

(s t. s s t) s t. s t s

(s t. s t s) s t. t s s

x y s. x insert y s x = y x s

x s t. insert x s t x t s t

n. size { m. m | m n } = n + 1

s t x. x s t x s x t

s t x. x s t x s x t

s t u. s t - u s t disjoint s u

s t u. s t u s t s u

s t u. disjoint (s t) u disjoint s u disjoint t u

s t u. s t u s u t u

s t u. s (t u) = s t s u

s t u. s t u = (s t) (s u)

s t u. (s t) u = s u t u

s t u. s t u = (s u) (t u)

s t. ¬(s = t) x. x t ¬(x s)

f s t. image f (s t) = image f s image f t

f. (y. x. f x = y) g. y. f (g y) = y

f. (t. s. image f s = t) y. x. f x = y

s P. { x. x | x s P x } s

f s. { x. f x | x s } = image f s

x s t. s delete t x s t ¬(x s)

x s t. disjoint (insert x s) t ¬(x t) disjoint s t

x s. ¬(x s) t. s insert x t s t

s x y. x delete s y x s ¬(x = y)

s x. x s t. s = insert x t ¬(x t)

s t x. x s - t x s ¬(x t)

s t. s t x. ¬(x s) insert x s t

s. s = x t. s = insert x t ¬(x t)

f s t. finite t s image f t size s size t

s t. finite s finite t size (s t) size s + size t

s t. finite t s t (size s = size t s = t)

a b. finite b a b size a = size b a = b

a b. finite b a b size b size a a = b

s t. finite s finite t size (cross s t) = size s * size t

s n. finite s n size s t. t s hasSize t n

s n. (finite s n size s) t. t s hasSize t n

a. finite a a = x s. a = insert x s finite s

y s f. y image f s x. y = f x x s

s t. finite s t s size (s - t) = size s - size t

s P. finite s finite { x. x | x s P x }

s. bigIntersect s = { x. x | u. u s x u }

s. bigUnion s = { x. x | u. u s x u }

s t. s t s t a. a t ¬(a s)

x y s t. (x, y) cross s t x s y t

x s. insert x s = { y. y | y = x y s }

s t. s t = { x. x | x s x t }

s t. s t = { x. x | x s x t }

s t. finite s finite t size (s t) = size s + size (t - s)

s t m n. hasSize s m hasSize t n hasSize (cross s t) (m * n)

f s t. s image f t u. u t s = image f u

s t. t bigUnion s = bigUnion { x. t x | x s }

s t. t bigIntersect s = bigIntersect { x. t x | x s }

s t. bigUnion s t = bigUnion { x. x t | x s }

s t. bigIntersect s t = bigIntersect { x. x t | x s }

P. P (a s. ¬(a s) P (insert a s)) s. P s

x s. finite s size (delete s x) = if x s then size s - 1 else size s

s x. delete s x = { y. y | y s ¬(y = x) }

s t. s - t = { x. x | x s ¬(x t) }

t. { x y. x, y | x y t x } =

FINITE'.
    FINITE' (x s. FINITE' s FINITE' (insert x s))
    a. finite a FINITE' a

x y s.
    delete (insert x s) y =
    if x = y then delete s y else insert x (delete s y)

x s t. insert x s t = if x t then insert x (s t) else s t

x s t. insert x s t = if x t then s t else insert x (s t)

s t x. insert x s - t = if x t then s - t else insert x (s - t)

s. finite s size { t. t | t s } = exp 2 (size s)

f. (x y. f x = f y x = y) g. x. g (f x) = x

s n. hasSize s n hasSize { t. t | t s } (exp 2 n)

s t.
    finite s finite t (size (s t) = size s + size t disjoint s t)

s t. finite s finite t disjoint s t size (s t) = size s + size t

f s. (x y. f x = f y x = y) infinite s infinite (image f s)

f. (x y. f x = f y x = y) s. infinite (image f s) infinite s

s x x'. (x s x' s) (x delete s x' x' delete s x)

s x x'. (x delete s x' x' delete s x) x s x' s

s n. hasSize s (suc n) ¬(s = ) a. a s hasSize (delete s a) n

f s. finite s finite { y. y | x. x s y = f x }

f s. image f s = { y. y | x. x s y = f x }

P f s. (y. y image f s P y) x. x s P (f x)

P f s. (y. y image f s P y) x. x s P (f x)

s t. finite s finite t size (s t) = size s + size t - size (s t)

s t. finite s finite t size (s t) + size (s t) = size s + size t

s t.
    finite s finite t size (s t) < size s + size t ¬disjoint s t

(f. image f = )
  f x s. image f (insert x s) = insert (f x) (image f s)

s t. cross s t = { x y. x, y | x s y t }

P f s. (t. t image f s P t) t. t s P (image f t)

f s. bigIntersect (image f s) = { y. y | x. x s y f x }

f s. bigUnion (image f s) = { y. y | x. x s y f x }

s t m n.
    hasSize s m hasSize t n disjoint s t hasSize (s t) (m + n)

s t m n. hasSize s m hasSize t n t s hasSize (s - t) (m - n)

P f. { x. f x | P x } = image f { x. x | P x }

size = 0
  x s.
    finite s size (insert x s) = if x s then size s else suc (size s)

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

n. { m. m | m < suc n } = insert n { m. m | m < n }

s t u. finite u disjoint s t s t = u size s + size t = size u

P a s. (x. x insert a s P x) P a x. x s P x

P a s. (x. x insert a s P x) P a x. x s P x

P s. (x. x bigUnion s P x) t x. t s x t P x

P s. (x. x bigUnion s P x) t x. t s x t P x

P a b. (a, b) { x y. x, y | P x y } P a b

P. { p. p | P p } = { a b. a, b | P (a, b) }

f s t.
    finite t t image f s s'. finite s' s' s t = image f s'

f s t.
    finite t t image f s s'. finite s' s' s t image f s'

s t. finite s finite t finite { x y. x, y | x s y t }

f s a.
    (x. f x = f a x = a)
    image f (delete s a) = delete (image f s) (f a)

f. (s t. image f s = image f t s = t) x y. f x = f y x = y

P.
    P (x s. P s ¬(x s) finite s P (insert x s))
    s. finite s P s

f s t.
    (x y. f x = f y x = y) image f (s - t) = image f s - image f t

f s t.
    (x y. f x = f y x = y) image f (s t) = image f s image f t

P f s.
    (t. finite t t image f s P t)
    t. finite t t s P (image f t)

f s.
    (x y. x s y s f x = f y x = y)
    (finite (image f s) finite s)

f A. (x y. f x = f y x = y) finite A finite { x. x | f x A }

s f. finite s image f s = s x y. x s y s f x = f y x = y

f s t. (y. y t x. f x = y) (x. f x t x s) image f s = t

(a. { x. x | x = a } = insert a ) a. { x. x | a = x } = insert a

f. (y. x. f x = y) P. image f { x. x | P (f x) } = { x. x | P x }

s t.
    finite s finite t
    size { x y. x, y | x s y t } = size s * size t

s n.
    hasSize s n
    f. (m. m < n f m s) x. x s ∃!m. m < n f m = x

f s.
    (x y. x s y s f x = f y x = y) finite s
    size (image f s) = size s

P.
    P (s. finite s ¬(s = ) x. x s (P (delete s x) P s))
    s. finite s P s

f s n.
    (x y. x s y s f x = f y x = y)
    (hasSize (image f s) n hasSize s n)

f s n.
    (x y. x s y s f x = f y x = y) hasSize s n
    hasSize (image f s) n

(s. hasSize s 0 s = )
  s n. hasSize s (suc n) a t. hasSize t n ¬(a t) s = insert a t

d t.
    { f. f | (x. x f x t) x. ¬(x ) f x = d } =
    insert (λx. d)

s t m n.
    hasSize s m hasSize t n
    hasSize { x y. x, y | x s y t } (m * n)

f s.
    (x y. x s y s f x = f y x = y) g. x. x s g (f x) = x

f u v.
    (t. t v s. s u image f s = t)
    y. y v x. x u f x = y

f t.
    finite t (y. y t finite { x. x | f x = y })
    finite { x. x | f x t }

f s t.
    finite s (x. x s finite (t x))
    finite { x y. f x y | x s y t x }

f s t.
    (y. y t x. x s f x = y)
    g. y. y t g y s f (g y) = y

(P. (x. x P x) T)
  P a s. (x. x insert a s P x) P a x. x s P x

(P. (x. x P x) F)
  P a s. (x. x insert a s P x) P a x. x s P x

f.
    (x y. f x = f y x = y) (y. x. f x = y)
    g. (y. f (g y) = y) x. g (f x) = x

f s t.
    finite s (x. x s f x t) (y. y t ∃!x. x s f x = y)
    size t = size s

d s t.
    finite s finite t
    finite { f. f | (x. x s f x t) x. ¬(x s) f x = d }

s t f.
    finite s size s = size t image f s = t
    x y. x s y s f x = f y x = y

s t.
    surjections s t =
    { f. f | (x. x s f x t) x. x t y. y s f y = x }

s t.
    finite s finite t size s size t
    f. image f s t x y. x s y s f x = f y x = y

s t m n.
    hasSize s m (x. x s finite (t x) size (t x) n)
    size (bigUnion { x. t x | x s }) m * n

s t m n.
    hasSize s m (x. x s hasSize (t x) n)
    hasSize { x y. x, y | x s y t x } (m * n)

s t.
    injections s t =
    { f. f |
      (x. x s f x t) x y. x s y s f x = f y x = y }

f A s.
    (x y. x s y s f x = f y x = y) finite A
    finite { x. x | x s f x A }

d s t.
    finite s finite t
    size { f. f | (x. x s f x t) x. ¬(x s) f x = d } =
    exp (size t) (size s)

f s t.
    finite t (y. y t finite { x. x | x s f x = y })
    finite { x. x | x s f x t }

f u.
    (s t. s u t u image f s = image f t s = t)
    x y. x u y u f x = f y x = y

d s t m n.
    hasSize s m hasSize t n
    hasSize { f. f | (x. x s f x t) x. ¬(x s) f x = d }
      (exp n m)

s t f g n.
    (x. x s f x t g (f x) = x)
    (y. y t g y s f (g y) = y) hasSize s n hasSize t n

s t f g.
    (x. x s f x t g (f x) = x)
    (y. y t g y s f (g y) = y) n. hasSize s n hasSize t n

s f.
    finite s image f s s
    ((y. y s x. x s f x = y)
     x y. x s y s f x = f y x = y)

s t f g.
    (finite s finite t) (x. x s f x t g (f x) = x)
    (y. y t g y s f (g y) = y) size s = size t

s t.
    finite s finite t size s = size t
    f g.
      (x. x s f x t g (f x) = x)
      y. y t g y s f (g y) = y

s t a.
    { x y. x, y | x insert a s y t x } =
    image ((,) a) (t a) { x y. x, y | x s y t x }

s.
    { t. t | t s } =
    image (λp. { x. x | p x })
      { p. p | (x. x s p x universe) x. ¬(x s) (p x F) }

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

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

s t f.
    finite s finite t size s = size t image f s t
    ((y. y t x. x s f x = y)
     x y. x s y s f x = f y x = y)

s t m n.
    hasSize s m (x. x s hasSize (t x) n)
    (x y. x s y s ¬(x = y) disjoint (t x) (t y))
    hasSize (bigUnion { x. t x | x s }) (m * n)

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

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

s t.
    finite s finite t size s = size t
    f.
      (x. x s f x t) (y. y t x. x s f x = y)
      x y. x s y s f x = f y x = y

s f g b.
    finite s (x. 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))
    fold f s b = fold g s b

f s t.
    (x. x s f x t)
    ((x y. x s y s f x = f y x = y)
     (y. y t x. x s f x = y)
     g.
       (y. y t g y s) (y. y t f (g y) = y)
       x. x s g (f x) = x)

d a s t.
    { f. f |
      (x. x insert a s f x t) x. ¬(x insert a s) f x = d } =
    image (λ(b, g) x. if x = a then b else g x)
      (cross t { f. f | (x. x s f x t) x. ¬(x s) f x = d })

(P f Q. (z. z { x. f x | P x } Q z) x. P x Q (f x))
  (P f Q.
     (z. z { x y. f x y | P x y } Q z) x y. P x y Q (f x y))
  P f Q.
    (z. z { w x y. f w x y | P w x y } Q z)
    w x y. P w x y Q (f w x y)

(P f Q. (z. z { x. f x | P x } Q z) x. P x Q (f x))
  (P f Q.
     (z. z { x y. f x y | P x y } Q z) x y. P x y Q (f x y))
  P f Q.
    (z. z { w x y. f w x y | P w x y } Q z)
    w x y. P w x y Q (f w x y)

(P f. bigIntersect { x. f x | P x } = { a. a | x. P x a f x })
  (P f.
     bigIntersect { x y. f x y | P x y } =
     { a. a | x y. P x y a f x y })
  P f.
    bigIntersect { x y z. f x y z | P x y z } =
    { a. a | x y z. P x y z a f x y z }

(P f. bigUnion { x. f x | P x } = { a. a | x. P x a f x })
  (P f.
     bigUnion { x y. f x y | P x y } =
     { a. a | x y. P x y a f x y })
  P f.
    bigUnion { x y z. f x y z | P x y z } =
    { a. a | x y z. P x y z a f x y z }

Input Type Operators

Input Constants

Assumptions

T

id = λx. x

n. 0 n

n. n n

F p. p

1 = suc 0

t. t ¬t

n. ¬(n < n)

n. n < suc n

(~) = λp. p F

() = λP. P ((select) P)

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)

m. m - 0 = m

n. n - n = 0

n. bit0 n = n + n

m n. m max m n

m n. n max m n

() = λp q. p q p

t. (t T) (t F)

n. bit1 n = suc (n + n)

m. suc m = m + 1

n. suc n - 1 = n

p x. p x p ((select) p)

(¬T F) (¬F T)

f y. (let xyf x) = f y

p. x y. p = (x, y)

x y. x = y y = x

x y. x = y y = x

t1 t2. t1 t2 t2 t1

t1 t2. t1 t2 t2 t1

a b. (a b) a b

m n. m * n = n * m

m n. m + n = n + m

m n. m = n m n

m n. m < n m n

m n. m + n - n = m

m n. ¬(m n) n < m

m n. m < suc n m n

m n. suc m n m < n

m. m = 0 n. m = 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

t1 t2. ¬(t1 t2) t1 ¬t2

t1 t2. ¬t1 ¬t2 t2 t1

m n. suc m = suc n m = n

m n. suc m suc n m n

m n. m + n = m n = 0

f g x. (f o g) x = f (g x)

m n. m * suc n = m + m * n

P. (p. P p) p1 p2. P (p1, p2)

P. (p. P p) p1 p2. P (p1, p2)

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

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

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

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

PAIR'. fn. a0 a1. fn (a0, a1) = PAIR' a0 a1

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

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

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

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

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

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

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

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

m n. m < n m n ¬(m = n)

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

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

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

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

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

x y z. x = y y = z x = z

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

p q r. p q r p q r

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

p q r. p q r p q r

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

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

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

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

m n p. m n n p m p

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. n < m suc (m - suc n) = m - n

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

(t. ¬¬t t) (¬T F) (¬F T)

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

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

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

(∃!) = λP. () P x y. P x P y x = y

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 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

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

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

x y a b. (x, y) = (a, b) x = a y = b

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

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

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

(m. exp m 0 = 1) m n. exp m (suc n) = m * exp m n

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

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

t1 t2. (¬(t1 t2) ¬t1 ¬t2) (¬(t1 t2) ¬t1 ¬t2)

(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)

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

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)

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)

(n. 0 * n = 0) (m. m * 0 = 0) (n. 1 * n = n) (m. m * 1 = m)
  (m n. suc m * n = m * n + n) m n. m * suc n = m + m * n