Package set-thm: Properties of set types
Information
| name | set-thm |
| version | 1.75 |
| description | Properties of set types |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory exported on 2019-07-04 |
| checksum | c43911526d059f66c9b310e0d8ec3d240f590cb4 |
| requires | bool function natural pair set-def |
| show | Data.Bool Data.Pair Function Number.Natural Set |
Files
- Package tarball set-thm-1.75.tgz
- Theory source file set-thm.thy (included in the package tarball)
Theorems
⊦ ¬(universe = ∅)
⊦ bigIntersect ∅ = universe
⊦ bigUnion ∅ = ∅
⊦ bigUnion universe = universe
⊦ ∀x. x ∈ universe
⊦ ∀s. disjoint ∅ s
⊦ ∀s. disjoint s ∅
⊦ ∀s. ∅ ⊆ s
⊦ ∀s. s ⊆ universe
⊦ ∀s. s ⊆ s
⊦ fromPredicate (λx. ⊥) = ∅
⊦ fromPredicate (λx. ⊤) = universe
⊦ cross universe universe = universe
⊦ ∀x. ¬(x ∈ ∅)
⊦ ∀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. universe ∩ s = s
⊦ ∀s. s ∩ ∅ = ∅
⊦ ∀s. s ∩ universe = s
⊦ ∀s. s ∩ s = s
⊦ ∀s. ∅ ∪ s = s
⊦ ∀s. universe ∪ s = universe
⊦ ∀s. s ∪ ∅ = s
⊦ ∀s. s ∪ universe = universe
⊦ ∀s. s ∪ s = s
⊦ ∀s. cross s ∅ = ∅
⊦ ∀s. cross ∅ s = ∅
⊦ ∀f. image f ∅ = ∅
⊦ ∀s. image (λx. x) s = s
⊦ universe = insert ⊤ (insert ⊥ ∅)
⊦ ∀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. s ⊆ s ∪ t
⊦ ∀s t. s ⊆ t ∪ s
⊦ ∀s t. disjoint (t \ s) s
⊦ ∀s t. s \ t ⊆ s
⊦ ∀s t. s ∩ t ⊆ s
⊦ ∀s t. t ∩ s ⊆ s
⊦ ∀s. disjoint s s ⇔ s = ∅
⊦ ∀s. universe ⊆ s ⇔ s = universe
⊦ ∀s. s ⊆ ∅ ⇔ s = ∅
⊦ ∀s. universe \ (universe \ s) = s
⊦ ∀s. bigUnion (delete s ∅) = bigUnion s
⊦ ∀s. bigUnion (insert ∅ s) = bigUnion s
⊦ ∀x s. ¬(insert x s = ∅)
⊦ ∀s t. disjoint s t ⇔ disjoint t s
⊦ ∀s t. s ∩ t = t ∩ s
⊦ ∀s t. s ∪ t = t ∪ s
⊦ ∀s t. s ⊂ t ⇒ s ⊆ t
⊦ ∀s. (∀x. x ∈ s) ⇔ s = universe
⊦ ∀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. 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 u. bigIntersect (insert s u) = s ∩ bigIntersect u
⊦ ∀s u. bigUnion (insert s u) = s ∪ bigUnion u
⊦ ∀f s. image f s = ∅ ⇔ s = ∅
⊦ ∀s t. s ⊆ t ⇒ bigIntersect t ⊆ bigIntersect s
⊦ ∀f g. f ⊆ g ⇒ bigUnion f ⊆ bigUnion g
⊦ { m. m | m < 0 } = ∅
⊦ ∀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. bigIntersect (insert s (insert t ∅)) = s ∩ t
⊦ ∀s t. bigUnion (insert s (insert t ∅)) = s ∪ t
⊦ ∀f x. image f (insert x ∅) = insert (f x) ∅
⊦ ∀f s. image f (bigUnion s) = bigUnion (image (image f) s)
⊦ ∀f s. image f (bigIntersect s) ⊆ bigIntersect (image (image f) s)
⊦ ∀s t. bigIntersect (s ∪ t) = bigIntersect s ∩ bigIntersect t
⊦ ∀s t. bigUnion (s ∪ t) = bigUnion s ∪ bigUnion t
⊦ ∀s t x. s ⊆ t ⇒ s ⊆ insert x t
⊦ ∀s. { x. x | x ∈ s } = s
⊦ ∀x s. x ∈ s ⇒ insert x (delete s x) = s
⊦ ∀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
⊦ { m. m | m ≤ 0 } = insert 0 ∅
⊦ { s. s | s ⊆ ∅ } = insert ∅ ∅
⊦ ∀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 ∧ ¬(t ⊆ s)
⊦ ∀s t. image fst (cross s t) = if t = ∅ then ∅ else s
⊦ ∀s t. image snd (cross s t) = if s = ∅ then ∅ else t
⊦ ∀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)
⊦ ∀a. { x. x | x = a } = insert a ∅
⊦ ∀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. (∀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)
⊦ ∀x y. cross (insert x ∅) (insert y ∅) = insert (x, y) ∅
⊦ ∀s t. s ∪ t = ∅ ⇔ s = ∅ ∧ t = ∅
⊦ ∀s t. cross s t = ∅ ⇔ s = ∅ ∨ t = ∅
⊦ ∀s t. disjoint s t ⇔ ¬∃x. x ∈ s ∧ x ∈ t
⊦ ∀s t. disjoint s (bigUnion t) ⇔ ∀x. x ∈ t ⇒ disjoint s x
⊦ ∀t u. t ⊆ bigIntersect u ⇔ ∀s. s ∈ u ⇒ t ⊆ s
⊦ ∀s. bigUnion { x. insert x ∅ | x ∈ s } = s
⊦ ∀f x s. image f (insert x s) = insert (f x) (image f 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
⊦ ∀u s. (∃t. t ∈ u ∧ t ⊆ s) ⇒ bigIntersect u ⊆ s
⊦ ∀x s. x ∈ rest s ⇔ x ∈ s ∧ ¬(x = choice s)
⊦ ∀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
⊦ ∀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)
⊦ ∀s t u. cross s (t \ u) = cross s t \ cross s u
⊦ ∀s t u. cross s (t ∩ u) = cross s t ∩ cross s u
⊦ ∀s t u. cross s (t ∪ u) = cross s t ∪ cross s u
⊦ ∀s t u. cross (t \ u) s = cross t s \ cross u s
⊦ ∀s t u. cross (t ∩ u) s = cross t s ∩ cross u s
⊦ ∀s t u. cross (t ∪ u) s = cross t s ∪ cross u s
⊦ ∀f s t. image f (s ∪ t) = image f s ∪ image f t
⊦ ∀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 x. s ⊆ insert x ∅ ⇔ s = ∅ ∨ s = insert x ∅
⊦ ∀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)
⊦ ∀s. bigIntersect s = universe \ bigUnion { t. universe \ t | t ∈ s }
⊦ ∀s. bigUnion s = universe \ bigIntersect { t. universe \ t | t ∈ s }
⊦ ∀y s f. y ∈ image f s ⇔ ∃x. y = f x ∧ x ∈ s
⊦ ∀s t. s ⊂ t ⇔ s ⊆ t ∧ ∃a. a ∈ t ∧ ¬(a ∈ s)
⊦ ∀x y s t. (x, y) ∈ cross s t ⇔ x ∈ s ∧ y ∈ t
⊦ ∀s1 s2 t1 t2. cross s1 t1 ∩ cross s2 t2 = cross (s1 ∩ s2) (t1 ∩ t2)
⊦ ∀s1 t1 s2 t2.
disjoint (cross s1 t1) (cross s2 t2) ⇔ disjoint s1 s2 ∨ disjoint t1 t2
⊦ ∀s t s' t'. s ⊆ s' ∧ t ⊆ t' ⇒ cross s t ⊆ cross s' t'
⊦ ∀u s. u \ bigIntersect s = bigUnion { t. u \ t | t ∈ s }
⊦ ∀s f. cross s (bigUnion f) = bigUnion { t. cross s t | t ∈ f }
⊦ ∀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. bigUnion s ∩ t = bigUnion { x. x ∩ t | x ∈ s }
⊦ ∀s t. bigIntersect s ∪ t = bigIntersect { x. x ∪ t | x ∈ s }
⊦ ∀f t. cross (bigUnion f) t = bigUnion { s. cross s t | s ∈ f }
⊦ ∀p. p ∅ ∧ (∀a s. ¬(a ∈ s) ⇒ p (insert a s)) ⇒ ∀s. p s
⊦ ∀t. { x y. (x, y) | x ∈ ∅ ∧ y ∈ t x } = ∅
⊦ ∀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)
⊦ ∀f. (∀x y. f x = f y ⇒ x = y) ⇔ ∃g. ∀x. g (f x) = x
⊦ ∀u s. u \ bigUnion s = u ∩ bigIntersect { t. u \ t | t ∈ s }
⊦ ∀s t. (∀x. x ∈ s ⇒ ∃y. y ∈ t ∧ x ⊆ y) ⇒ bigUnion s ⊆ bigUnion t
⊦ ∀s t. (∀y. y ∈ t ⇒ ∃x. x ∈ s ∧ x ⊆ y) ⇒ bigIntersect s ⊆ bigIntersect t
⊦ ∀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
⊦ ∀p. bigUnion { x. insert x ∅ | p x } = { x. x | p 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)
⊦ ∀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 }
⊦ ∀p f s. (∀t. t ⊆ image f s ⇒ p t) ⇔ ∀t. t ⊆ s ⇒ p (image f t)
⊦ ∀p f s. (∃t. t ⊆ image f s ∧ p t) ⇔ ∃t. t ⊆ s ∧ p (image f t)
⊦ ∀p f. { x. f x | p x } = image f { x. x | p 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 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
⊦ ∀s f.
s ∩ bigIntersect f =
if f = ∅ then s else bigIntersect { t. s ∩ t | t ∈ f }
⊦ ∀f s.
bigIntersect f ∩ s =
if f = ∅ then s else bigIntersect { t. t ∩ s | t ∈ f }
⊦ ∀p a b. (a, b) ∈ { x y. (x, y) | p x y } ⇔ p a b
⊦ ∀p. { x. x | p x } = { a b. (a, b) | p (a, b) }
⊦ ∀n. { m. m | m < suc n } = insert 0 { m. suc m | m < n }
⊦ ∀n. { m. m | m ≤ suc n } = insert 0 { m. suc m | m ≤ n }
⊦ ∀n. { m. m | m ≤ suc n } = insert (suc n) { m. m | m ≤ n }
⊦ ∀f g.
cross (bigUnion f) (bigUnion g) =
bigUnion { s t. cross s t | s ∈ f ∧ t ∈ g }
⊦ ∀p x t. (∀s. s ⊆ insert x t ⇒ p s) ⇔ ∀s. s ⊆ t ⇒ p s ∧ p (insert x s)
⊦ ∀p x t. (∃s. s ⊆ insert x t ∧ p s) ⇔ ∃s. s ⊆ t ∧ (p s ∨ p (insert x s))
⊦ ∀s f.
cross s (bigIntersect f) =
if f = ∅ then cross s universe
else bigIntersect { t. cross s t | t ∈ f }
⊦ ∀f s.
cross (bigIntersect f) s =
if f = ∅ then cross universe s
else bigIntersect { t. cross t s | t ∈ f }
⊦ ∀f. (∀s t. image f s = image f t ⇒ s = t) ⇔ ∀x y. f x = f y ⇒ x = y
⊦ ∀s t s' t'. cross s t ⊆ cross s' t' ⇔ s = ∅ ∨ t = ∅ ∨ s ⊆ s' ∧ t ⊆ t'
⊦ ∀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 t u. (∀s. s ⊆ t ∪ u ⇒ p s) ⇔ ∀t' u'. t' ⊆ t ∧ u' ⊆ u ⇒ p (t' ∪ u')
⊦ ∀p t u. (∃s. s ⊆ t ∪ u ∧ p s) ⇔ ∃t' u'. t' ⊆ t ∧ u' ⊆ u ∧ p (t' ∪ u')
⊦ ∀p s t. (∀z. z ∈ cross s t ⇒ p z) ⇔ ∀x y. x ∈ s ∧ y ∈ t ⇒ p (x, y)
⊦ ∀p s t. (∃z. z ∈ cross s t ∧ p z) ⇔ ∃x y. x ∈ s ∧ y ∈ t ∧ p (x, y)
⊦ ∀f s t.
(∀x y. f x = f y ⇒ x = y) ⇒ image f (s ∩ t) = image f s ∩ image f t
⊦ ∀f. (∀k. ∃t. { x. x | f x ∈ t } = k) ⇔ ∀x y. f x = f y ⇒ x = y
⊦ ∀p f. bigIntersect { x. f x | p x } = { a. a | ∀x. p x ⇒ a ∈ f x }
⊦ ∀p f. bigUnion { x. f x | p x } = { a. a | ∃x. p x ∧ a ∈ f x }
⊦ ∀p s t. (∀x. x ∈ s ∪ t ⇒ p x) ⇔ (∀x. x ∈ s ⇒ p x) ∧ ∀x. x ∈ t ⇒ p x
⊦ ∀p s t. (∃x. x ∈ s ∪ t ∧ p x) ⇔ (∃x. x ∈ s ∧ p x) ∨ ∃x. x ∈ t ∧ p x
⊦ ∀f s a.
(∀x. x ∈ s ∧ f x = f a ⇒ x = a) ⇒
image f (delete s a) = delete (image f s) (f a)
⊦ ∀f s t. (∀y. y ∈ t ⇒ ∃x. f x = y) ∧ (∀x. f x ∈ t ⇔ x ∈ s) ⇒ image f s = t
⊦ ∀f g s t.
image (λ(x, y). (f x, g y)) (cross s t) = cross (image f s) (image g t)
⊦ ∀f. (∀y. ∃x. f x = y) ⇔ ∀p. image f { x. x | p (f x) } = { x. x | p x }
⊦ ∀d t.
{ f. f | (∀x. x ∈ ∅ ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ ∅) ⇒ f x = d } =
insert (λx. d) ∅
⊦ ∀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 s.
∃t.
t ⊆ s ∧ image f s = image f t ∧
∀x y. x ∈ t ∧ y ∈ t ∧ f x = f y ⇒ x = y
⊦ ∀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 ∈ { x y. f x y | p x y } ∧ q z) ⇔ ∃x y. p x y ∧ q (f x y)
⊦ ∀s1 t1 s2 t2.
cross s1 t1 = cross s2 t2 ⇔
(s1 = ∅ ∨ t1 = ∅) ∧ (s2 = ∅ ∨ t2 = ∅) ∨ s1 = s2 ∧ t1 = t2
⊦ ∀f s t.
(∀y. y ∈ t ⇒ ∃x. x ∈ s ∧ f x = y) ⇔
∃g. ∀y. y ∈ t ⇒ g y ∈ s ∧ f (g y) = y
⊦ ∀f.
(∀t t'. { x. x | f x ∈ t } = { x. x | f x ∈ t' } ⇒ t = t') ⇔
image f universe = universe
⊦ ∀p f.
bigIntersect { x y. f x y | p x y } =
{ a. a | ∀x y. p x y ⇒ a ∈ f x y }
⊦ ∀p f.
bigUnion { x y. f x y | p x y } = { a. a | ∃x y. p x y ∧ a ∈ f x y }
⊦ ∀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.
(∀x y. x ∈ s ∧ y ∈ t ∧ f x = f y ⇒ x = y) ⇒
image f (s \ t) = image f s \ image f t
⊦ ∀p f s.
(∀x y. x ∈ image f s ∧ y ∈ image f s ⇒ p x y) ⇔
∀x y. x ∈ s ∧ y ∈ s ⇒ p (f x) (f y)
⊦ ∀f s.
¬(s = ∅) ∧
(∀x y. x ∈ bigUnion s ∧ y ∈ bigUnion s ∧ f x = f y ⇒ x = y) ⇒
image f (bigIntersect s) = bigIntersect (image (image f) s)
⊦ ∀a t.
{ s. s | s ⊆ insert a t } =
{ s. s | s ⊆ t } ∪ image (λs. insert a s) { s. s | s ⊆ t }
⊦ ∀f s a.
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ∧ a ∈ s ⇒
image f (delete s a) = delete (image f s) (f a)
⊦ ∀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
⊦ ∀f s t.
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ∧ t ⊆ s ⇒
image f (s \ t) = image f s \ image f t
⊦ ∀f g s.
¬(g = ∅) ∧ (∀t. t ∈ delete g s ⇒ { x. x | f x ∈ image f t } ⊆ t) ⇒
image f (bigIntersect g) = bigIntersect (image (image f) g)
⊦ ∀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 ∈ { 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 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 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 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)
⊦ ∀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)
⊦ ∀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
⊦ ∀f s.
bigIntersect { x. bigUnion (f x) | x ∈ s } =
bigUnion { g. bigIntersect { x. g x | x ∈ s } | ∀x. x ∈ s ⇒ g x ∈ f x }
⊦ ∀f s.
bigUnion { x. bigIntersect (f x) | x ∈ s } =
bigIntersect { g. bigUnion { x. g x | x ∈ s } | ∀x. x ∈ s ⇒ g x ∈ f x }
⊦ ∀p s t.
{ x. x | x ∈ s \ t ∧ p x } =
{ x. x | x ∈ s ∧ p x } \ { x. x | x ∈ t ∧ p x }
⊦ ∀p s t.
{ x. x | x ∈ s ∩ t ∧ p x } =
{ x. x | x ∈ s ∧ p x } ∩ { x. x | x ∈ t ∧ p x }
⊦ ∀p s t.
{ x. x | x ∈ s ∪ t ∧ p x } =
{ x. x | x ∈ s ∧ p x } ∪ { x. x | x ∈ t ∧ p x }
⊦ ∀f s t.
{ x. x | f x ∈ image f s } ⊆ s ∨ { x. x | f x ∈ image f t } ⊆ t ⇒
image f (s ∩ t) = image f s ∩ 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 ⇔ ⊥) }
⊦ ∀p s a.
{ x. x | x ∈ insert a s ∧ p x } =
if p a then insert a { x. x | x ∈ s ∧ p x } else { x. x | x ∈ s ∧ p x }
⊦ ∀f s u.
(∀t1 t2.
t1 ⊆ u ∧ t2 ⊆ u ∧
{ x. x | x ∈ s ∧ f x ∈ t1 } = { x. x | x ∈ s ∧ f x ∈ t2 } ⇒
t1 = t2) ⇔ u ⊆ image f s
⊦ ∀f g s u.
¬(g = ∅) ∧ (∀t. t ∈ g ⇒ t ⊆ u) ∧
(∀t. t ∈ delete g s ⇒ { x. x | x ∈ u ∧ f x ∈ image f t } ⊆ t) ⇒
image f (bigIntersect g) = bigIntersect (image (image f) g)
⊦ ∀f s u.
(∀k. k ⊆ s ⇒ ∃t. t ⊆ u ∧ { x. x | x ∈ s ∧ f x ∈ t } = k) ⇔
image f s ⊆ u ∧ ∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y
⊦ ∀f g.
cross (bigIntersect f) (bigIntersect g) =
if f = ∅ then bigIntersect { t. cross universe t | t ∈ g }
else if g = ∅ then bigIntersect { s. cross s universe | s ∈ f }
else bigIntersect { s t. cross s t | s ∈ f ∧ t ∈ g }
⊦ ∀f s t u.
{ x. x | x ∈ u ∧ f x ∈ image f s } ⊆ s ∧ t ⊆ u ∨
{ x. x | x ∈ u ∧ f x ∈ image f t } ⊆ t ∧ s ⊆ u ⇒
image f (s ∩ t) = image f s ∩ image f t
⊦ ∀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 })
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
- <
- ≤
- suc
- zero
- Natural
- Set
- ∅
- bigIntersect
- bigUnion
- choice
- cross
- delete
- \
- disjoint
- fromPredicate
- image
- insert
- ∩
- ∈
- ⊂
- rest
- ⊆
- ∪
- universe
Assumptions
⊦ ⊤
⊦ id = λx. x
⊦ ¬⊥ ⇔ ⊤
⊦ ¬⊤ ⇔ ⊥
⊦ ∀t. t ⇒ t
⊦ ⊥ ⇔ ∀p. p
⊦ ∀t. t ∨ ¬t
⊦ ∀m. ¬(m < 0)
⊦ ∀n. 0 < suc n
⊦ (¬) = λp. p ⇒ ⊥
⊦ (∃) = 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. ¬¬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
⊦ ∀n. ¬(suc n = 0)
⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t
⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t
⊦ ∀t. t ⇒ ⊥ ⇔ ¬t
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)
⊦ ∀s. rest s = delete s (choice s)
⊦ ∀t1 t2. (if ⊥ then t1 else t2) = t2
⊦ ∀t1 t2. (if ⊤ then t1 else t2) = t1
⊦ ∀a b. fst (a, b) = a
⊦ ∀a b. snd (a, b) = b
⊦ ∀p x. p x ⇒ p ((select) p)
⊦ ∀x. ∃a b. x = (a, b)
⊦ ∀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
⊦ ∀p x. x ∈ fromPredicate p ⇔ p x
⊦ ∅ = { x. x | ⊥ }
⊦ universe = { x. x | ⊤ }
⊦ ∀m n. m < suc n ⇔ m ≤ n
⊦ ∀m. m = 0 ∨ ∃n. m = suc n
⊦ (∧) = 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) ⇔ t1 ∧ ¬t2
⊦ ∀t1 t2. ¬t1 ⇒ ¬t2 ⇔ t2 ⇒ t1
⊦ ∀m n. suc m < suc n ⇔ m < n
⊦ ∀s t. disjoint s t ⇔ s ∩ t = ∅
⊦ ∀f g x. (f ∘ g) x = f (g x)
⊦ ∀t1 t2. ¬(t1 ∧ t2) ⇔ ¬t1 ∨ ¬t2
⊦ ∀t1 t2. ¬(t1 ∨ t2) ⇔ ¬t1 ∧ ¬t2
⊦ ∀p. (∀x. p x) ⇔ ∀a b. p (a, b)
⊦ ∀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
⊦ ∀p a. (∃x. x = a ∧ p x) ⇔ p a
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀f. ∃fn. ∀a b. fn (a, b) = f a b
⊦ ∀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. 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
⊦ ∀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
⊦ ∀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
⊦ ∀s t. s ⊆ t ⇔ ∀x. x ∈ s ⇒ x ∈ t
⊦ ∀s t. (∀x. x ∈ s ⇔ x ∈ t) ⇒ s = t
⊦ ∀r. (∀x. ∃y. r x y) ⇔ ∃f. ∀x. r x (f x)
⊦ ∀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
⊦ ∀p q r. p ∨ q ⇒ r ⇔ (p ⇒ r) ∧ (q ⇒ r)
⊦ ∀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
⊦ ∀s. bigIntersect s = { x. x | ∀u. u ∈ s ⇒ x ∈ u }
⊦ ∀s. bigUnion s = { x. x | ∃u. u ∈ s ∧ x ∈ u }
⊦ ∀a b a' b'. (a, b) = (a', b') ⇔ a = a' ∧ b = b'
⊦ ∀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 x. delete s x = { y. y | y ∈ s ∧ ¬(y = x) }
⊦ ∀s t. s \ t = { x. x | x ∈ s ∧ ¬(x ∈ t) }
⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)
⊦ ∀f s. image f s = { y. y | ∃x. x ∈ s ∧ y = f x }
⊦ ∀s t. cross s t = { x y. (x, y) | x ∈ s ∧ y ∈ t }