Package set-finite-thm: Properties of finite sets
Information
| name | set-finite-thm |
| version | 1.68 |
| description | Properties of finite sets |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory exported on 2019-10-31 |
| checksum | d1c44780608e6904de387e5472013073d66c7c6f |
| requires | bool function natural pair set-def set-finite-def set-thm |
| show | Data.Bool Data.Pair Function Number.Natural Set |
Files
- Package tarball set-finite-thm-1.68.tgz
- Theory source file set-finite-thm.thy (included in the package tarball)
Theorems
⊦ finite universe
⊦ infinite universe
⊦ finite universe ⇔ finite universe
⊦ infinite universe ⇔ infinite universe
⊦ ∀a. finite (insert a ∅)
⊦ finite universe ⇔ finite universe ∧ finite universe
⊦ infinite universe ⇔ infinite universe ∨ infinite universe
⊦ ∀s. infinite s ⇒ ¬(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. finite s ⇒ ∃a. ¬(a ∈ s)
⊦ ∀f s. finite s ⇒ finite (image f s)
⊦ ∀s t. finite t ∧ s ⊆ t ⇒ finite s
⊦ ∀s t. infinite s ∧ s ⊆ t ⇒ infinite t
⊦ ∀n. finite { m. m | m < n }
⊦ ∀n. finite { m. m | m ≤ n }
⊦ ∀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. finite s ∧ finite t ⇒ finite (cross s t)
⊦ ∀s. finite s ⇔ ∃a. ∀x. x ∈ s ⇒ x ≤ a
⊦ ∀s. infinite s ⇔ ∀N. ∃n. N ≤ n ∧ n ∈ s
⊦ ∀s. finite s ⇒ finite { t. t | t ⊆ s }
⊦ ∀s. finite { t. t | t ⊆ s } ⇔ finite s
⊦ ∀s. finite s ⇒ (finite (bigUnion s) ⇔ ∀t. t ∈ s ⇒ finite t)
⊦ ∀s. finite (bigUnion s) ⇔ finite s ∧ ∀t. t ∈ s ⇒ finite t
⊦ ∀s p. finite s ⇒ finite { x. x | x ∈ s ∧ p x }
⊦ ∀s t. finite (cross s t) ⇔ s = ∅ ∨ t = ∅ ∨ finite s ∧ finite t
⊦ ∀s t.
infinite (cross s t) ⇔ ¬(s = ∅) ∧ infinite t ∨ infinite s ∧ ¬(t = ∅)
⊦ ∀f. (∀x y. f x = f y ⇒ x = y) ⇒ ∀s. infinite (image f s) ⇔ infinite s
⊦ ∀f s. finite s ⇒ finite { y. y | ∃x. x ∈ s ∧ y = f x }
⊦ ∀f s. finite (image f s) ⇔ ∃t. finite t ∧ t ⊆ s ∧ image f s = image f t
⊦ ∀f s. finite s ∧ s ⊆ bigUnion f ⇒ ∃g. finite g ∧ g ⊆ f ∧ s ⊆ bigUnion g
⊦ ∀s. infinite s ⇔ ∃r. (∀m n. m < n ⇒ r m < r n) ∧ image r universe = s
⊦ ∀s. infinite s ⇒ ∃r. (∀m n. m < n ⇒ r m < r n) ∧ image r universe = s
⊦ ∀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.
finite f ⇒
bigUnion { t. t | t ∈ f ∧ ∀u. u ∈ f ⇒ ¬(t ⊂ u) } = bigUnion f
⊦ ∀p.
p ∅ ∧ (∀x s. p s ∧ ¬(x ∈ s) ∧ finite s ⇒ p (insert x s)) ⇒
∀s. finite s ⇒ p s
⊦ ∀p f s.
(∀t. finite t ∧ t ⊆ image f s ⇒ p t) ⇔
∀t. finite t ∧ t ⊆ s ⇒ p (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 s.
infinite s ∧ (∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ⇒
infinite (image f s)
⊦ ∀f A. (∀x y. f x = f y ⇒ x = y) ∧ finite A ⇒ finite { x. x | f x ∈ A }
⊦ ∀f.
finite f ∧ ¬(f = ∅) ∧ (∀s t. s ∈ f ∧ t ∈ f ⇒ s ⊆ t ∨ t ⊆ s) ⇒
bigIntersect f ∈ f
⊦ ∀f.
finite f ∧ ¬(f = ∅) ∧ (∀s t. s ∈ f ∧ t ∈ f ⇒ s ⊆ t ∨ t ⊆ s) ⇒
bigUnion f ∈ f
⊦ ∀f s.
infinite s ∧ finite (image f s) ⇒
∃a. a ∈ s ∧ infinite { x. x | x ∈ s ∧ f x = f a }
⊦ ∀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 }
⊦ ∀d s t.
finite s ∧ finite t ⇒
finite { f. f | (∀x. x ∈ s ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ s) ⇒ f x = d }
⊦ ∀s t k.
finite s ∧ finite t ⇒
finite { f. f | image f s ⊆ t ∧ { x. x | ¬(f x = k x) } ⊆ s }
⊦ ∀f s.
finite s ∧ s ⊆ bigUnion f ∧ ¬(f = ∅) ∧
(∀t u. t ∈ f ∧ u ∈ f ⇒ t ⊆ u ∨ u ⊆ t) ⇒ ∃t. t ∈ f ∧ s ⊆ t
⊦ ∀f s.
finite (image f s) ⇔
∃t.
finite t ∧ t ⊆ s ∧ image f s = image f t ∧
∀x y. x ∈ t ∧ y ∈ t ⇒ (f x = f y ⇔ x = y)
⊦ ∀f t s.
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ∧ finite t ⇒
finite { x. x | x ∈ s ∧ f x ∈ t }
⊦ ∀f s t.
finite t ∧ (∀y. y ∈ t ⇒ finite { x. x | x ∈ s ∧ f x = y }) ⇒
finite { x. x | x ∈ s ∧ f x ∈ t }
⊦ ∀r s.
finite s ∧ (∀x. ¬r x x) ∧ (∀x y z. r x y ∧ r y z ⇒ r x z) ∧
(∀x. x ∈ s ⇒ ∃y. y ∈ s ∧ r x y) ⇒ s = ∅
⊦ ∀p f s.
(∀t. finite t ∧ t ⊆ image f s ⇒ p t) ⇔
∀t.
finite t ∧ t ⊆ s ∧ (∀x y. x ∈ t ∧ y ∈ t ∧ f x = f y ⇒ x = y) ⇒
p (image f t)
⊦ ∀p f s.
(∃t. finite t ∧ t ⊆ image f s ∧ p t) ⇔
∃t.
finite t ∧ t ⊆ s ∧ (∀x y. x ∈ t ∧ y ∈ t ∧ f x = f y ⇒ x = y) ∧
p (image f t)
External Type Operators
- →
- bool
- Data
- Pair
- ×
- Pair
- Number
- Natural
- natural
- Natural
- Set
- set
External Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∃!
- ∨
- ¬
- cond
- ⊥
- ⊤
- Pair
- ,
- fst
- snd
- Bool
- Function
- id
- ∘
- Number
- Natural
- +
- <
- ≤
- bit1
- max
- suc
- zero
- Natural
- Set
- ∅
- bigIntersect
- bigUnion
- cross
- delete
- \
- finite
- fromPredicate
- image
- infinite
- insert
- ∩
- ∈
- ⊂
- ⊆
- ∪
- universe
Assumptions
⊦ ⊤
⊦ finite ∅
⊦ ¬(universe = ∅)
⊦ ¬⊥ ⇔ ⊤
⊦ ¬⊤ ⇔ ⊥
⊦ bigIntersect ∅ = universe
⊦ bigUnion ∅ = ∅
⊦ ∀x. x ∈ universe
⊦ ∀t. t ⇒ t
⊦ ∀n. 0 ≤ n
⊦ ∀s. ∅ ⊆ s
⊦ ∀s. s ⊆ s
⊦ ⊥ ⇔ ∀p. p
⊦ fromPredicate (λx. ⊥) = ∅
⊦ cross universe universe = universe
⊦ ∀x. ¬(x ∈ ∅)
⊦ ∀x. id x = x
⊦ ∀t. t ∨ ¬t
⊦ ∀m. ¬(m < 0)
⊦ ∀n. ¬(n < n)
⊦ ∀n. n < suc n
⊦ ∀n. n ≤ suc n
⊦ ∀s. ¬(s ⊂ s)
⊦ (¬) = 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 ∨ ⊤ ⇔ ⊤
⊦ ∀t. t ∨ t ⇔ t
⊦ ∀s. s ∩ universe = s
⊦ ∀s. ∅ ∪ s = s
⊦ ∀s. s ∪ ∅ = s
⊦ ∀s. cross s ∅ = ∅
⊦ ∀s. cross ∅ s = ∅
⊦ universe = insert ⊤ (insert ⊥ ∅)
⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t
⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t
⊦ ∀t. t ⇒ ⊥ ⇔ ¬t
⊦ ∀s. infinite s ⇔ ¬finite s
⊦ ∀x s. delete s x ⊆ s
⊦ ∀m n. m ≤ max m n
⊦ ∀m n. n ≤ max m n
⊦ ∀s t. s ⊆ s ∪ t
⊦ ∀s t. s ⊆ t ∪ s
⊦ ∀s t. s \ t ⊆ s
⊦ ∀s t. s ∩ t ⊆ s
⊦ ∀s t. t ∩ s ⊆ s
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)
⊦ ∀m. suc m = m + 1
⊦ ∀s. s ⊆ ∅ ⇔ s = ∅
⊦ ∀a b. fst (a, b) = a
⊦ ∀a b. snd (a, b) = b
⊦ ∀x s. ¬(insert x s = ∅)
⊦ ∀p x. p x ⇒ p ((select) p)
⊦ ∀x y. x = y ⇔ y = x
⊦ ∀x s. finite s ⇒ finite (insert x s)
⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1
⊦ ∀a b. (a ⇔ b) ⇒ a ⇒ b
⊦ ∀m n. m < n ⇒ m ≤ n
⊦ ∀s t. s ∩ t = t ∩ s
⊦ ∀s t. s ∪ t = t ∪ s
⊦ ∀s t. s ⊂ t ⇒ s ⊆ t
⊦ ∀p x. x ∈ fromPredicate p ⇔ p x
⊦ ∀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
⊦ ∀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
⊦ ∀x y. x ∈ insert y ∅ ⇔ x = y
⊦ ∀x s. x ∈ s ⇔ insert x s = s
⊦ ∀x s. insert x ∅ ∪ s = insert x s
⊦ ∀t1 t2. ¬(t1 ⇒ t2) ⇔ t1 ∧ ¬t2
⊦ ∀t1 t2. ¬t1 ⇒ ¬t2 ⇔ t2 ⇒ t1
⊦ ∀s t. s ⊆ t ⇔ s ∩ t = s
⊦ ∀s t. s ⊆ t ⇔ s ∪ t = t
⊦ ∀s u. bigIntersect (insert s u) = s ∩ bigIntersect u
⊦ ∀s u. bigUnion (insert s u) = s ∪ bigUnion u
⊦ ∀f g. f ⊆ g ⇒ bigUnion f ⊆ bigUnion g
⊦ { m. m | m < 0 } = ∅
⊦ ∀f g x. (f ∘ g) x = f (g x)
⊦ ∀x s. delete s x = s ⇔ ¬(x ∈ s)
⊦ ∀p. (∃x. p x) ⇔ ∃a b. p (a, b)
⊦ ∀f g. (∀x. f x = g x) ⇔ f = g
⊦ ∀p a. (∀x. x = a ⇒ p x) ⇔ p a
⊦ ∀p a. (∃x. a = x ∧ p x) ⇔ p a
⊦ (∨) = 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
⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n
⊦ ∀s t. s ⊆ t ∧ t ⊆ s ⇔ s = t
⊦ ∀f. ∃fn. ∀a b. fn (a, b) = f a b
⊦ ∀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
⊦ ∀p q. p ∨ (∀x. q x) ⇔ ∀x. p ∨ q x
⊦ ∀p q. p ∨ (∃x. q x) ⇔ ∃x. p ∨ q x
⊦ ∀m n. m < suc 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
⊦ ∀x s t. s ⊆ insert x t ⇔ delete s x ⊆ t
⊦ ∀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 ∧ n < p ⇒ m < p
⊦ ∀m n p. m < n ∧ n ≤ p ⇒ m < p
⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p
⊦ ∀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. s ⊆ t ⇔ ∀x. x ∈ s ⇒ x ∈ t
⊦ ∀s t. (∀x. x ∈ s ⇔ x ∈ t) ⇔ s = t
⊦ ∀f s t. s ⊆ t ⇒ image f s ⊆ image f t
⊦ ∀f g s. image (f ∘ g) s = image f (image g s)
⊦ ∀r. (∀x. ∃y. r x y) ⇔ ∃f. ∀x. r x (f x)
⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n
⊦ ∀s x. x ∈ bigUnion s ⇔ ∃t. t ∈ s ∧ x ∈ t
⊦ ∀p x. x ∈ { y. y | p y } ⇔ p x
⊦ ∀x y s. x ∈ insert y s ⇔ x = y ∨ x ∈ s
⊦ ∀p q r. p ∨ q ⇒ r ⇔ (p ⇒ r) ∧ (q ⇒ r)
⊦ ∀s t x. x ∈ s ∪ t ⇔ x ∈ s ∨ x ∈ t
⊦ (∃!) = λp. (∃) p ∧ ∀x y. p x ∧ p y ⇒ x = y
⊦ ∀s p. { x. x | x ∈ s ∧ p x } ⊆ s
⊦ ∀f s. { x. f x | x ∈ s } = image f s
⊦ ∀p. (∀n. (∀m. m < n ⇒ p m) ⇒ p n) ⇒ ∀n. p n
⊦ ∀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
⊦ ∀p. (∃n. p n) ⇔ ∃n. p n ∧ ∀m. m < n ⇒ ¬p m
⊦ ∀y s f. y ∈ image f s ⇔ ∃x. y = f x ∧ x ∈ s
⊦ ∀a b a' b'. (a, b) = (a', b') ⇔ a = a' ∧ b = b'
⊦ ∀x y s t. (x, y) ∈ cross s t ⇔ x ∈ s ∧ y ∈ t
⊦ ∀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 } = ∅
⊦ ∀p. p ∅ ∧ (∀x s. p s ⇒ p (insert x s)) ⇒ ∀a. finite a ⇒ p a
⊦ ∀s t. (∀x. x ∈ s ⇒ ∃y. y ∈ t ∧ x ⊆ y) ⇒ bigUnion s ⊆ bigUnion t
⊦ ∀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. cross s t = { x y. (x, y) | x ∈ s ∧ y ∈ t }
⊦ ∀f s. bigUnion (image f s) = { y. y | ∃x. x ∈ s ∧ y ∈ f x }
⊦ ∀n. { m. m | m < suc n } = insert n { m. m | m < n }
⊦ ∀p a s. (∀x. x ∈ insert a s ⇒ p x) ⇔ p a ∧ ∀x. x ∈ s ⇒ p x
⊦ ∀p a b. (a, b) ∈ { x y. (x, y) | p x y } ⇔ p a b
⊦ ∀p. (∃n. p n) ∧ (∃m. ∀n. p n ⇒ n ≤ m) ⇔ ∃m. p m ∧ ∀n. p n ⇒ n ≤ m
⊦ ∀p f q. (∀z. z ∈ { x. f x | p x } ⇒ q z) ⇔ ∀x. p x ⇒ q (f x)
⊦ ∀p f q. (∃z. z ∈ { x. f x | p x } ∧ q z) ⇔ ∃x. p x ∧ q (f x)
⊦ ∀p f. bigUnion { x. f x | p x } = { a. a | ∃x. p x ∧ a ∈ f x }
⊦ ∀d t.
{ f. f | (∀x. x ∈ ∅ ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ ∅) ⇒ f x = d } =
insert (λx. d) ∅
⊦ ∀p f q. (∀z. z ∈ { x y. f x y | p x y } ⇒ q z) ⇔ ∀x y. p x y ⇒ q (f x y)
⊦ ∀f s t.
s ⊆ image f t ⇔
∃u. u ⊆ t ∧ (∀x y. x ∈ u ∧ y ∈ u ∧ f x = f y ⇒ x = y) ∧ s = image f u
⊦ ∀p f s.
(∃t. t ⊆ image f s ∧ p t) ⇔
∃t. t ⊆ s ∧ (∀x y. x ∈ t ∧ y ∈ t ∧ f x = f y ⇒ x = y) ∧ p (image f t)
⊦ ∀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 ⇔ ⊥) }
⊦ ∀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 })