Package set-size-thm: Properties of finite set cardinality
Information
| name | set-size-thm |
| version | 1.53 |
| description | Properties of finite set cardinality |
| author | Joe Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2012-08-06 |
| requires | bool natural pair set-def set-finite set-fold set-size-def set-thm |
| show | Data.Bool Data.Pair Number.Natural Set |
Files
- Package tarball set-size-thm-1.53.tgz
- Theory file set-size-thm.thy (included in the package tarball)
Theorems
⊦ size ∅ = 0
⊦ hasSize universe 2
⊦ ∀x. hasSize (insert x ∅) 1
⊦ ∀x. size (insert x ∅) = 1
⊦ ∀s. finite s ⇔ hasSize s (size s)
⊦ ∀s. hasSize s 0 ⇔ s = ∅
⊦ ∀s n. hasSize s n ⇒ size s = n
⊦ ∀s. finite s ⇒ (size s = 0 ⇔ s = ∅)
⊦ ∀f s. finite s ⇒ size (image f s) ≤ size s
⊦ ∀n. hasSize { m. m | m < n } n
⊦ ∀n. size { m. m | m < n } = n
⊦ ∀a b. a ⊂ b ∧ finite b ⇒ size a < size b
⊦ ∀a b. a ⊆ b ∧ finite b ⇒ size a ≤ size b
⊦ ∀n. hasSize { m. m | m ≤ n } (n + 1)
⊦ ∀n. size { m. m | m ≤ n } = n + 1
⊦ ∀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
⊦ ∀x s.
finite s ⇒ size (insert x s) = if x ∈ s then size s else suc (size s)
⊦ ∀s t. finite s ∧ t ⊆ s ⇒ size (s \ t) = size s - size 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)
⊦ ∀x s. finite s ⇒ size (delete s x) = if x ∈ s then size s - 1 else size s
⊦ ∀s. finite s ⇒ size { t. t | t ⊆ s } = 2 ↑ size s
⊦ ∀s n. hasSize s n ⇒ hasSize { t. t | t ⊆ s } (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
⊦ ∀s n. hasSize s (suc n) ⇔ ¬(s = ∅) ∧ ∀a. a ∈ s ⇒ hasSize (delete s a) n
⊦ ∀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
⊦ ∀s n. hasSize s (suc n) ⇔ ∃a t. hasSize t n ∧ ¬(a ∈ t) ∧ s = insert a t
⊦ ∀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)
⊦ ∀s t u. finite u ∧ disjoint s t ∧ s ∪ t = u ⇒ size s + size t = size u
⊦ ∀s f. finite s ∧ image f s = s ⇒ ∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y
⊦ ∀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.
finite s ⇒
(size (image f s) = size s ⇔ ∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y)
⊦ ∀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 t m n.
hasSize s m ∧ hasSize t n ⇒
hasSize { x y. (x, y) | x ∈ s ∧ y ∈ t } (m * n)
⊦ ∀f s t.
finite s ∧ (∀x. x ∈ s ⇒ f x ∈ t) ∧ (∀y. y ∈ t ⇒ ∃!x. x ∈ s ∧ f x = y) ⇒
size t = size s
⊦ ∀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.
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)
⊦ ∀d s t.
finite s ∧ finite t ⇒
size { f. f | (∀x. x ∈ s ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ s) ⇒ f x = d } =
size t ↑ size s
⊦ ∀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 }
(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 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)
⊦ ∀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
Input Type Operators
- →
- bool
- Data
- Pair
- ×
- Pair
- Number
- Natural
- natural
- Natural
- Set
- set
Input Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∃!
- ∨
- ¬
- cond
- ⊥
- ⊤
- Pair
- ,
- Bool
- Number
- Natural
- *
- +
- -
- <
- ≤
- ↑
- bit0
- bit1
- suc
- zero
- Natural
- Set
- ∅
- bigUnion
- cross
- delete
- \
- disjoint
- finite
- fold
- fromPredicate
- hasSize
- image
- insert
- ∩
- ∈
- ⊂
- size
- ⊆
- ∪
- universe
Assumptions
⊦ ⊤
⊦ finite ∅
⊦ ¬⊥ ⇔ ⊤
⊦ ¬⊤ ⇔ ⊥
⊦ bit0 0 = 0
⊦ bigUnion ∅ = ∅
⊦ ∀t. t ⇒ t
⊦ ∀n. 0 ≤ n
⊦ ∀n. n ≤ n
⊦ ∀s. ∅ ⊆ s
⊦ ∀s. s ⊆ s
⊦ ⊥ ⇔ ∀p. p
⊦ ∀x. ¬(x ∈ ∅)
⊦ ∀a. finite (insert a ∅)
⊦ ∀t. t ∨ ¬t
⊦ ∀m. ¬(m < 0)
⊦ ∀n. ¬(n < n)
⊦ ∀n. n < suc n
⊦ (¬) = λp. p ⇒ ⊥
⊦ ∀t. (∀x. t) ⇔ t
⊦ ∀t. (∃x. t) ⇔ t
⊦ ∀t. (λx. t x) = t
⊦ (∀) = λp. p = λx. ⊤
⊦ ∀t. ¬¬t ⇔ t
⊦ ∀t. (⊤ ⇔ t) ⇔ t
⊦ ∀t. (t ⇔ ⊤) ⇔ t
⊦ ∀t. ⊥ ∧ t ⇔ ⊥
⊦ ∀t. ⊤ ∧ t ⇔ t
⊦ ∀t. t ∧ ⊥ ⇔ ⊥
⊦ ∀t. t ∧ ⊤ ⇔ t
⊦ ∀t. t ∧ t ⇔ t
⊦ ∀t. ⊥ ⇒ t ⇔ ⊤
⊦ ∀t. ⊤ ⇒ t ⇔ t
⊦ ∀t. t ⇒ ⊤ ⇔ ⊤
⊦ ∀t. ⊥ ∨ t ⇔ t
⊦ ∀t. ⊤ ∨ t ⇔ ⊤
⊦ ∀t. t ∨ ⊥ ⇔ t
⊦ ∀t. t ∨ ⊤ ⇔ ⊤
⊦ ∀n. ¬(suc n = 0)
⊦ ∀n. 0 * n = 0
⊦ ∀n. 0 + n = n
⊦ ∀m. m - 0 = m
⊦ ∀n. n - n = 0
⊦ ∀s. ∅ ∪ s = s
⊦ ∀f. image f ∅ = ∅
⊦ universe = insert ⊤ (insert ⊥ ∅)
⊦ size = fold (λx n. suc n) 0
⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t
⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t
⊦ ∀t. t ⇒ ⊥ ⇔ ¬t
⊦ ∀n. bit1 n = suc (bit0 n)
⊦ ∀m. m ↑ 0 = 1
⊦ ∀x s. delete s x ⊆ s
⊦ ∀s t. disjoint s (t \ s)
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)
⊦ ∀m. suc m = m + 1
⊦ ∀m. m ≤ 0 ⇔ m = 0
⊦ ∀n. suc n - 1 = n
⊦ ∀t1 t2. (if ⊥ then t1 else t2) = t2
⊦ ∀t1 t2. (if ⊤ then t1 else t2) = t1
⊦ ∀x s. ¬(insert x s = ∅)
⊦ ∀p x. p x ⇒ p ((select) p)
⊦ ∀n. bit0 (suc n) = suc (suc (bit0 n))
⊦ ∀f y. (let x ← y in f x) = f y
⊦ ∀xy. ∃x y. xy = (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
⊦ ∀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
⊦ ∀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. s ∪ t = t ∪ s
⊦ ∀f s. finite s ⇒ finite (image f s)
⊦ ∀p x. x ∈ fromPredicate p ⇔ p x
⊦ ∀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
⊦ ∀s. (∃x. x ∈ s) ⇔ ¬(s = ∅)
⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤
⊦ ∀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 ⇔ t2 ⇒ t1
⊦ ∀m n. m + suc n = suc (m + n)
⊦ ∀m n. suc m + n = suc (m + n)
⊦ ∀m n. suc m = suc n ⇔ m = n
⊦ ∀m n. suc m ≤ suc n ⇔ m ≤ n
⊦ ∀m n. m + n = m ⇔ n = 0
⊦ ∀s t. disjoint s t ⇔ s ∩ t = ∅
⊦ ∀s t. s ⊆ t ⇔ s ∪ t = t
⊦ ∀s t. s \ t = ∅ ⇔ s ⊆ t
⊦ ∀s t. s \ t = s ⇔ disjoint s t
⊦ ∀s t. s \ t \ t = s \ t
⊦ ∀s t. s \ t ∪ t = s ∪ t
⊦ ∀s t. finite t ∧ s ⊆ t ⇒ finite s
⊦ ∀s u. bigUnion (insert s u) = s ∪ bigUnion u
⊦ { m. m | m < 0 } = ∅
⊦ ∀x s. delete s x = s ⇔ ¬(x ∈ s)
⊦ ∀t1 t2. ¬(t1 ∧ t2) ⇔ ¬t1 ∨ ¬t2
⊦ ∀m n. m * suc n = m + m * n
⊦ ∀m n. m ↑ suc n = m * m ↑ n
⊦ ∀m n. suc m * n = m * n + n
⊦ ∀s t. finite (s ∪ t) ⇔ finite s ∧ finite t
⊦ ∀s t. finite s ∨ finite t ⇒ finite (s ∩ t)
⊦ ∀p. (∀xy. p xy) ⇔ ∀x y. p (x, y)
⊦ ∀m n. m ≤ n ⇔ ∃d. n = m + d
⊦ ∀s t x. s ⊆ t ⇒ s ⊆ insert x t
⊦ ∀f g. (∀x. f x = g x) ⇔ f = g
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀x s. x ∈ s ⇒ insert x (delete s x) = s
⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n
⊦ ∀s n. hasSize s n ⇔ finite s ∧ size s = n
⊦ ∀s t. s ∪ (t \ s) = t ⇔ s ⊆ t
⊦ ∀s t. s ⊆ t ∧ t ⊆ s ⇒ s = t
⊦ ∀f. ∃fn. ∀x y. fn (x, y) = f x y
⊦ ∀f s x. x ∈ s ⇒ f x ∈ image f s
⊦ ∀p. (∀x y. p x y) ⇔ ∀y x. p x y
⊦ ∀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
⊦ ∀m n. m < n ⇔ m ≤ n ∧ ¬(m = n)
⊦ ∀m n. m < suc n ⇔ m = n ∨ m < n
⊦ ∀s t. s ⊂ t ⇔ s ⊆ t ∧ ¬(s = t)
⊦ ∀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
⊦ ∀s. bigUnion s = ∅ ⇔ ∀t. t ∈ s ⇒ t = ∅
⊦ ∀x y z. x = y ∧ y = z ⇒ x = z
⊦ ∀p q r. p ⇒ q ⇒ r ⇔ p ∧ q ⇒ r
⊦ ∀t1 t2 t3. (t1 ∧ t2) ∧ t3 ⇔ t1 ∧ t2 ∧ t3
⊦ ∀t1 t2 t3. (t1 ∨ t2) ∨ t3 ⇔ t1 ∨ t2 ∨ t3
⊦ ∀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
⊦ ∀s t. s ⊆ t ⇔ ∀x. x ∈ s ⇒ x ∈ t
⊦ ∀s t. (∀x. x ∈ s ⇔ x ∈ t) ⇔ s = t
⊦ ∀m n. n < m ⇒ suc (m - suc n) = m - n
⊦ ∀s. finite s ⇒ (finite (bigUnion s) ⇔ ∀t. t ∈ s ⇒ finite t)
⊦ ∀s t. disjoint s (bigUnion t) ⇔ ∀x. x ∈ t ⇒ disjoint s x
⊦ ∀f x s. image f (insert x s) = insert (f x) (image f s)
⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n
⊦ ∀p x. x ∈ { y. y | p y } ⇔ p x
⊦ ∀x y s. x ∈ insert y s ⇔ x = y ∨ x ∈ s
⊦ ∀x s t. insert x s ⊆ t ⇔ x ∈ t ∧ s ⊆ t
⊦ ∀s t x. x ∈ s ∩ t ⇔ x ∈ s ∧ x ∈ t
⊦ ∀s t x. x ∈ s ∪ t ⇔ x ∈ s ∨ x ∈ t
⊦ (∃!) = λp. (∃) p ∧ ∀x y. p x ∧ p y ⇒ x = y
⊦ ∀f s. { x. f x | x ∈ s } = image f s
⊦ ∀x s t. disjoint (insert x s) t ⇔ ¬(x ∈ t) ∧ disjoint s t
⊦ ∀s x y. x ∈ delete s y ⇔ x ∈ s ∧ ¬(x = y)
⊦ ∀s t x. x ∈ s \ t ⇔ x ∈ s ∧ ¬(x ∈ t)
⊦ ∀s. s = ∅ ∨ ∃x t. s = insert x t ∧ ¬(x ∈ t)
⊦ ∀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
⊦ ∀y s f. y ∈ image f s ⇔ ∃x. y = f x ∧ x ∈ s
⊦ ∀x y a b. (x, y) = (a, b) ⇔ x = a ∧ y = b
⊦ ∀x y s t. (x, y) ∈ cross s t ⇔ x ∈ s ∧ y ∈ t
⊦ ∀m n p q. m ≤ p ∧ n ≤ q ⇒ m + n ≤ p + q
⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)
⊦ ∀t. { x y. (x, y) | x ∈ ∅ ∧ y ∈ t x } = ∅
⊦ ∀x s t. insert x s ∪ t = if x ∈ t then s ∪ t else insert x (s ∪ t)
⊦ ∀p. (∃!x. p x) ⇔ (∃x. p x) ∧ ∀x x'. p x ∧ p x' ⇒ x = x'
⊦ ∀p f s. (∀y. y ∈ image f s ⇒ p y) ⇔ ∀x. x ∈ s ⇒ p (f x)
⊦ ∀s t. cross s t = { x y. (x, y) | x ∈ s ∧ y ∈ t }
⊦ ∀n. { m. m | m < suc n } = insert n { m. m | m < n }
⊦ ∀p a b. (a, b) ∈ { x y. (x, y) | p x y } ⇔ p a b
⊦ ∀p.
p ∅ ∧ (∀x s. p s ∧ ¬(x ∈ s) ∧ finite s ⇒ p (insert x s)) ⇒
∀s. finite s ⇒ p s
⊦ ∀f s.
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ⇒
(finite (image f s) ⇔ finite s)
⊦ ∀d t.
{ f. f | (∀x. x ∈ ∅ ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ ∅) ⇒ f x = d } =
insert (λx. d) ∅
⊦ ∀f s t.
(∀y. y ∈ t ⇒ ∃x. x ∈ s ∧ f x = y) ⇔
∃g. ∀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 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 b (insert x s) =
if x ∈ s then fold f b s else f x (fold f b s)
⊦ ∀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 })