Package set-thm: Properties of set types

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Files

• Package tarball set-thm-1.43.tgz
• Theory file set-thm.thy (included in the package tarball)

Theorems

⊦ ¬(universe = ∅)

⊦ bigIntersect ∅ = universe

⊦ bigUnion ∅ = ∅

⊦ ∀x. x ∈ universe

⊦ ∀s. disjoint ∅ s

⊦ ∀s. disjoint s ∅

⊦ ∀s. ∅ ⊆ s

⊦ ∀s. s ⊆ universe

⊦ ∀s. s ⊆ s

⊦ fromPredicate (λx. ⊥) = ∅

⊦ ∀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

⊦ ∀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 = ∅

⊦ ∀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. (∀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 = ∅

⊦ ∀f g. f ⊆ g ⇒ bigIntersect g ⊆ bigIntersect f

⊦ ∀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)

⊦ ∀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

⊦ ∀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

⊦ { 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. 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 | a = x } = insert a ∅

⊦ ∀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)

⊦ ∀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 f. t ⊆ bigIntersect f ⇔ ∀s. s ∈ f ⇒ t ⊆ 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

⊦ ∀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)

⊦ ∀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)

⊦ ∀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

⊦ ∀u s. u \ bigIntersect s = bigUnion { t. u \ t | t ∈ s }

⊦ ∀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 }

⊦ ∀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

⊦ ∀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 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 }

⊦ ∀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 }

⊦ ∀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. { ab. ab | p ab } = { a b. a, b | p (a, b) }

⊦ ∀n. { m. m | m < suc n } = insert 0 { m. suc m | m < n }

⊦ ∀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 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)

⊦ ∀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. 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 }

⊦ ∀f s t. (∀y. y ∈ t ⇒ ∃x. f x = y) ∧ (∀x. f x ∈ t ⇔ x ∈ s) ⇒ image f s = 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)

⊦ ∀f s t.
    (∀y. y ∈ t ⇒ ∃x. x ∈ s ∧ f x = y) ⇔
    ∃g. ∀y. y ∈ t ⇒ g y ∈ s ∧ f (g y) = y

⊦ ∀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

⊦ ∀a t.
    { s. s | s ⊆ insert a t } =
    { s. s | s ⊆ t } ∪ image (λs. insert a s) { s. s | s ⊆ t }

⊦ ∀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 }

⊦ ∀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 }

⊦ ∀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 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

⊦ ∀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 })

Input Type Operators

• →
• bool
• Data
  • Pair
    • ×
• Number
  • Natural
    • natural
• Set
  • set

Input Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • cond
    • ⊥
    • ⊤
  • Pair
    • ,
• Function
  • id
  • ∘
• Number
  • Natural
    • <
    • suc
    • zero
• Set
  • ∅
  • bigIntersect
  • bigUnion
  • choice
  • cross
  • delete
  • \
  • disjoint
  • fromPredicate
  • image
  • insert
  • ∩
  • ∈
  • ⊂
  • rest
  • ⊆
  • ∪
  • universe

Assumptions

⊦ ⊤

⊦ id = λx. x

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ ∀t. t ⇒ t

⊦ ⊥ ⇔ ∀p. p

⊦ ∀t. t ∨ ¬t

⊦ ∀n. 0 < suc n

⊦ (¬) = λp. p ⇒ ⊥

⊦ (∃) = λp. p ((select) 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

⊦ ∀n. ¬(suc n = 0)

⊦ ∀m. m < 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

⊦ ∀p x. p x ⇒ p ((select) p)

⊦ ∀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

⊦ ∀a b. (a ⇔ b) ⇒ a ⇒ b

⊦ ∀p x. x ∈ fromPredicate p ⇔ p x

⊦ ∅ = { x. x | ⊥ }

⊦ universe = { x. x | ⊤ }

⊦ ∀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. (∀xy. p xy) ⇔ ∀x y. p (x, y)

⊦ ∀p. (∃xy. p xy) ⇔ ∃x y. p (x, y)

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g

⊦ ∀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. ∀x y. fn (x, y) = f x y

⊦ ∀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