Package parser: Stream parsers

Information

name	parser
version	1.158
description	Stream parsers
author	Joe Leslie-Hurd <joe@gilith.com>
license	MIT
homepage	http://opentheory.gilith.com/?pkg=parser
haskell-category	Parsing
haskell-int-file	haskell.int
haskell-src-file	haskell.art
haskell-test-file	haskell-test.art
haskell-equality-type	"Parser.Stream.stream"
haskell-arbitrary-type	"Parser.Stream.stream"
checksum	597bac8e2b26aa505ab7d1ba22bafc0c26b3ae0b
requires	base
show	Data.Bool Data.List Data.Option Data.Pair Data.Sum Function Number.Natural Parser Parser.Stream Relation

Files

Package tarball parser-1.158.tgz
Theory source file parser.thy (included in the package tarball)

Defined Type Operators

Parser
- parser
- Stream
  - stream

Defined Constants

Parser
- any
- apply
- dest
- filter
- fold
  - fold.prs
- foldN
- invariant
- inverse
- map
- mapPartial
  - mapPartial.prs
- mk
- none
- orelse
  - orelse.prs
- pair
- parse
- sequence
  - sequence.prs
- some
- strongInverse
- token
  - token.prs
- Stream
  - append
  - case
  - cons
  - eof
  - error
  - fromList
  - isProperSuffix
  - isSuffix
  - length
  - map
  - toList

Theorems

⊦ wellFounded isProperSuffix

⊦ ¬(error = eof)

⊦ length eof = 0

⊦ length error = 0

⊦ fromList [] = eof

⊦ ∀xs. isSuffix xs xs

⊦ ∀p. invariant (dest p)

⊦ ∀f. invariant (token.prs f)

⊦ ∀p. invariant (sequence.prs p)

⊦ any = some (const ⊤)

⊦ none = token (const none)

⊦ ∀xs. ¬isProperSuffix xs eof

⊦ ∀xs. ¬isProperSuffix xs error

⊦ ∀xs. ¬isProperSuffix xs xs

⊦ inverse any (λx. x :: [])

⊦ strongInverse any (λx. x :: [])

⊦ toList eof = ([], ⊥)

⊦ toList error = ([], ⊤)

⊦ ∀xs. append [] xs = xs

⊦ ∀xs. apply none xs = none

⊦ ∀f. map f eof = eof

⊦ ∀f. map f error = error

⊦ ∀p. parse p eof = eof

⊦ ∀p. parse p error = error

⊦ ∀a. mk (dest a) = a

⊦ ∀p. apply p eof = none

⊦ ∀p. apply p error = none

⊦ ∀p₁ p₂. invariant (orelse.prs p₁ p₂)

⊦ ∀p f. invariant (mapPartial.prs p f)

⊦ ∀f s. invariant (fold.prs f s)

⊦ ∀l. fromList l = append l eof

⊦ ∀l. length (fromList l) = length l

⊦ ∀p. filter any p = some p

⊦ ∀f. token f = mk (token.prs f)

⊦ ∀f. mapPartial any f = token f

⊦ ∀f. dest (token f) = token.prs f

⊦ ∀p. sequence p = mk (sequence.prs p)

⊦ ∀p. dest (sequence p) = sequence.prs p

⊦ ∀l. toList (fromList l) = (l, ⊥)

⊦ ∀xs. length xs = length (fst (toList xs))

⊦ ∀xs. isSuffix xs eof ⇔ xs = eof

⊦ ∀xs. isSuffix xs error ⇔ xs = error

⊦ ∀f. fold f = mk ∘ fold.prs f

⊦ ∀x xs. ¬(eof = cons x xs)

⊦ ∀x xs. ¬(error = cons x xs)

⊦ ∀x xs. dest none x xs = none

⊦ ∀r. invariant r ⇔ dest (mk r) = r

⊦ ∀xs ys. isProperSuffix xs ys ⇒ isSuffix xs ys

⊦ ∀p xs. length (parse p xs) ≤ length xs

⊦ ∀x xs. length (cons x xs) = suc (length xs)

⊦ ∀p₁ p₂. orelse p₁ p₂ = mk (orelse.prs p₁ p₂)

⊦ ∀p₁ p₂. dest (orelse p₁ p₂) = orelse.prs p₁ p₂

⊦ ∀p f. mapPartial p f = mk (mapPartial.prs p f)

⊦ ∀p f. dest (mapPartial p f) = mapPartial.prs p f

⊦ ∀f s. fold f s = mk (fold.prs f s)

⊦ ∀f s. dest (fold f s) = fold.prs f s

⊦ apply any = case none none (λx xs. some (x, xs))

⊦ ∀e b f. case e b f eof = b

⊦ ∀e b f. case e b f error = e

⊦ ∀h t. fromList (h :: t) = cons h (fromList t)

⊦ ∀x xs. dest any x xs = some (x, xs)

⊦ ∀l₁ l₂. fromList (l₁ @ l₂) = append l₁ (fromList l₂)

⊦ ∀xs ys. isProperSuffix xs ys ⇒ length xs < length ys

⊦ ∀xs ys. isSuffix xs ys ⇒ length xs ≤ length ys

⊦ ∀p. some p = token (λx. if p x then some x else none)

⊦ ∀p₁ p₂. dest p₁ = dest p₂ ⇔ p₁ = p₂

⊦ ∀p f. map p f = mapPartial p (λx. some (f x))

⊦ ∀l xs. length (append l xs) = length l + length xs

⊦ ∀xs y ys. isProperSuffix xs (cons y ys) ⇔ isSuffix xs ys

⊦ ∀xs ys. isSuffix xs ys ⇔ xs = ys ∨ isProperSuffix xs ys

⊦ ∀p x xs. apply p (cons x xs) = dest p x xs

⊦ ∀h t xs. append (h :: t) xs = cons h (append t xs)

⊦ ∀xs ys zs. append (xs @ ys) zs = append xs (append ys zs)

⊦ ∀xs ys zs.
isProperSuffix xs ys ∧ isProperSuffix ys zs ⇒ isProperSuffix xs zs

⊦ ∀xs ys zs. isProperSuffix xs ys ∧ isSuffix ys zs ⇒ isProperSuffix xs zs

⊦ ∀xs ys zs. isSuffix xs ys ∧ isProperSuffix ys zs ⇒ isProperSuffix xs zs

⊦ ∀xs ys zs. isSuffix xs ys ∧ isSuffix ys zs ⇒ isSuffix xs zs

⊦ ∀p f xs. parse (map p f) xs = map f (parse p xs)

⊦ ∀p f. filter p f = mapPartial p (λx. if f x then some x else none)

⊦ ∀p₁ p₂. pair p₁ p₂ = sequence (map p₁ (λx. map p₂ (λy. (x, y))))

⊦ ∀xs ys e. toList xs = (ys, e) ⇒ length xs = length ys

⊦ ∀f x xs. map f (cons x xs) = cons (f x) (map f xs)

⊦ surjective (λ(l, b). append l (if b then error else eof))

⊦ ∀f s. apply (fold f s) = case none none (λx xs. fold.prs f s x xs)

⊦ ∀xs. xs = error ∨ xs = eof ∨ ∃x xt. xs = cons x xt

⊦ ∀p.
apply (some p) =
case none none (λx xs. if p x then some (x, xs) else none)

⊦ ∀f x xs.
token.prs f x xs = case f x of none → none | some y → some (y, xs)

⊦ ∀f.
apply (token f) =
case none none (λx xs. case f x of none → none | some y → some (y, xs))

⊦ ∀p. (∀xs. (∀ys. isProperSuffix ys xs ⇒ p ys) ⇒ p xs) ⇒ ∀xs. p xs

⊦ ∀e b f x xs. case e b f (cons x xs) = f x xs

⊦ ∀p x xs. dest (some p) x xs = if p x then some (x, xs) else none

⊦ ∀p e l. inverse p e ⇒ parse p (fromList (concat (map e l))) = fromList l

⊦ ∀f e. (∀x. f (e x) = some x) ⇒ inverse (token f) (λx. e x :: [])

⊦ ∀p₁ p₂ xs.
apply (orelse p₁ p₂) xs =
case apply p₁ xs of none → apply p₂ xs | some yys → some yys

⊦ ∀p x xs y ys. dest p x xs = some (y, ys) ⇒ isSuffix ys xs

⊦ ∀x xs y ys. cons x xs = cons y ys ⇔ x = y ∧ xs = ys

⊦ ∀p e. inverse p e ⇔ ∀x xs. apply p (append (e x) xs) = some (x, xs)

⊦ ∀p e x xs. inverse p e ⇒ parse p (append (e x) xs) = cons x (parse p xs)

⊦ ∀p. p error ∧ p eof ∧ (∀x xs. p xs ⇒ p (cons x xs)) ⇒ ∀xs. p xs

⊦ ∀l xs ys e. toList (append l xs) = (l @ ys, e) ⇔ toList xs = (ys, e)

⊦ ∀f xs ys e. toList xs = (ys, e) ⇒ toList (map f xs) = (map f ys, e)

⊦ length error = 0 ∧ length eof = 0 ∧
∀x xs. length (cons x xs) = length xs + 1

⊦ (∀xs. append [] xs = xs) ∧
∀h t xs. append (h :: t) xs = cons h (append t xs)

⊦ ∀p₁ p₂ x xs.
orelse.prs p₁ p₂ x xs =
case dest p₁ x xs of none → dest p₂ x xs | some yys → some yys

⊦ ∀p xs.
apply p xs = none ∨
∃y ys. apply p xs = some (y, ys) ∧ isProperSuffix ys xs

⊦ ∀x xs. toList (cons x xs) = let (l, e) ← toList xs in (x :: l, e)

⊦ ∀p f g e.
inverse p e ∧ (∀x. f (g x) = x) ⇒ inverse (map p f) (λx. e (g x))

⊦ ∀p.
invariant p ⇔
∀x xs. case p x xs of none → ⊤ | some (y, ys) → isSuffix ys xs

⊦ ∀p xs.
apply (sequence p) xs =
case apply p xs of none → none | some (y, ys) → apply y ys

⊦ ∀p f g e.
inverse p e ∧ (∀x. f (g x) = some x) ⇒
inverse (mapPartial p f) (λx. e (g x))

⊦ ∀p x xs.
dest p x xs = none ∨ ∃y ys. dest p x xs = some (y, ys) ∧ isSuffix ys xs

⊦ ∀p x xs.
sequence.prs p x xs =
case dest p x xs of none → none | some (q, ys) → apply q ys

⊦ ∀p x xs.
invariant p ⇒
p x xs = none ∨ ∃y ys. p x xs = some (y, ys) ∧ isSuffix ys xs

⊦ ∀e b f.
∃fn. fn error = e ∧ fn eof = b ∧ ∀x xs. fn (cons x xs) = f x xs (fn xs)

⊦ ∀p e xs ys ye.
strongInverse p e ∧ toList (parse p xs) = (ys, ye) ⇒
ye ∨ xs = fromList (concat (map e ys))

⊦ (∀p. apply p error = none) ∧ (∀p. apply p eof = none) ∧
∀p x xs. apply p (cons x xs) = dest p x xs

⊦ ∀p f xs.
apply (map p f) xs =
case apply p xs of none → none | some (y, ys) → some (f y, ys)

⊦ ∀p e.
    strongInverse p e ⇔
    inverse p e ∧
    ∀xs y ys. apply p xs = some (y, ys) ⇒ xs = append (e y) ys

⊦ ∀p x xs.
parse p (cons x xs) =
case dest p x xs of none → error | some (y, ys) → cons y (parse p ys)

⊦ ∀p xs.
    parse p xs =
    case apply p xs of
      none → case xs of error → error | eof → eof | cons y ys → error
    | some (y, ys) → cons y (parse p ys)

⊦ (∀f. map f error = error) ∧ (∀f. map f eof = eof) ∧
∀f x xs. map f (cons x xs) = cons (f x) (map f xs)

⊦ ∀p₁ p₂ e₁ e₂.
inverse p₁ e₁ ∧ inverse p₂ e₂ ⇒
inverse (pair p₁ p₂) (λ(x₁, x₂). e₁ x₁ @ e₂ x₂)

⊦ ∀p₁ p₂ e₁ e₂.
strongInverse p₁ e₁ ∧ strongInverse p₂ e₂ ⇒
strongInverse (pair p₁ p₂) (λ(x₁, x₂). e₁ x₁ @ e₂ x₂)

⊦ ∀p f xs.
    apply (filter p f) xs =
    case apply p xs of
      none → none
    | some (y, ys) → if f y then some (y, ys) else none

⊦ ∀p f x xs.
dest (map p f) x xs =
case dest p x xs of none → none | some (y, ys) → some (f y, ys)

⊦ ∀p f xs.
    apply (mapPartial p f) xs =
    case apply p xs of
      none → none
    | some (y, ys) → case f y of none → none | some z → some (z, ys)

⊦ ∀f s x xs.
    fold.prs f s x xs =
    case f x s of
      none → none
    | some y →
        case y of
          left z → some (z, xs)
        | right t →
            case xs of
              error → none
            | eof → none
            | cons z zs → fold.prs f t z zs

⊦ ∀h.
    (∀f g xs.
       (∀ys. isProperSuffix ys xs ⇒ f ys = g ys) ⇒ h f xs = h g xs) ⇒
    ∃fn. ∀xs. fn xs = h fn xs

⊦ ∀p f x xs.
    mapPartial.prs p f x xs =
    case dest p x xs of
      none → none
    | some (y, ys) → case f y of none → none | some z → some (z, ys)

⊦ ∀p f x xs.
    dest (filter p f) x xs =
    case dest p x xs of
      none → none
    | some (y, ys) → if f y then some (y, ys) else none

⊦ toList error = ([], ⊤) ∧ toList eof = ([], ⊥) ∧
∀x xs. toList (cons x xs) = let (l, e) ← toList xs in (x :: l, e)

⊦ ∀f e.
    (∀x. f (e x) = some x) ∧
    (∀y₁ y₂ x. f y₁ = some x ∧ f y₂ = some x ⇒ y₁ = y₂) ⇒
    strongInverse (token f) (λx. e x :: [])

⊦ ∀f n s.
    foldN f n s =
    fold
      (λx (m, t).
         map (λu. if m = 0 then left u else right (m - 1, u)) (f x t))
      (n, s)

⊦ ∀p f g e.
    strongInverse p e ∧ (∀x. f (g x) = x) ∧
    (∀y₁ y₂ x. f y₁ = x ∧ f y₂ = x ⇒ y₁ = y₂) ⇒
    strongInverse (map p f) (λx. e (g x))

⊦ (∀e b f. case e b f error = e) ∧ (∀e b f. case e b f eof = b) ∧
∀e b f x xs. case e b f (cons x xs) = f x xs

⊦ ∀p₁ p₂ xs.
    apply (pair p₁ p₂) xs =
    case apply p₁ xs of
      none → none
    | some (y, ys) →
        case apply p₂ ys of none → none | some (z, zs) → some ((y, z), zs)

⊦ (∀p. parse p error = error) ∧ (∀p. parse p eof = eof) ∧
  ∀p x xs.
    parse p (cons x xs) =
    case dest p x xs of none → error | some (y, ys) → cons y (parse p ys)

⊦ ∀p f g e.
    strongInverse p e ∧ (∀x. f (g x) = some x) ∧
    (∀y₁ y₂ x. f y₁ = some x ∧ f y₂ = some x ⇒ y₁ = y₂) ⇒
    strongInverse (mapPartial p f) (λx. e (g x))

⊦ ∀p₁ p₂ x xs.
    dest (pair p₁ p₂) x xs =
    case dest p₁ x xs of
      none → none
    | some (y, ys) →
        case apply p₂ ys of none → none | some (z, zs) → some ((y, z), zs)

External Type Operators

→
bool
Data
- List
  - list
- Option
  - option
- Pair
  - ×
- Sum
  - +
Number
- Natural
  - natural

External Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∃!
  - ∨
  - ¬
  - cond
  - ⊥
  - ⊤
- List
  - ::
  - @
  - []
  - concat
  - length
  - map
- Option
  - case
  - map
  - none
  - some
- Pair
  - ,
  - fst
  - snd
- Sum
  - case
  - left
  - right
Function
- const
- ∘
- surjective
Number
- Natural
  - *
  - +
  - -
  - <
  - ≤
  - ↑
  - bit0
  - bit1
  - even
  - suc
  - zero
Relation
- irreflexive
- measure
- subrelation
- wellFounded

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ length [] = 0

⊦ bit0 0 = 0

⊦ concat [] = []

⊦ ∀t. t ⇒ t

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ ∀m. wellFounded (measure m)

⊦ ⊥ ⇔ ∀p. p

⊦ ∀t. t ∨ ¬t

⊦ (¬) = 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. ⊤

⊦ ∀a. ¬(some a = none)

⊦ ∀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 = n

⊦ ∀m. m + 0 = m

⊦ ∀l. [] @ l = l

⊦ ∀f. map f [] = []

⊦ ∀r. wellFounded r ⇒ irreflexive r

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀n. even (2 * n)

⊦ ∀n. bit1 n = suc (bit0 n)

⊦ ∀m. m ↑ 0 = 1

⊦ ∀x y. const x y = x

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀m. suc m = m + 1

⊦ ∀n. even (suc n) ⇔ ¬even n

⊦ ∀m. m ≤ 0 ⇔ m = 0

⊦ ∀x. (fst x, snd x) = x

⊦ ∀t₁ t₂. (if ⊥ then t₁ else t₂) = t₂

⊦ ∀t₁ t₂. (if ⊤ then t₁ else t₂) = t₁

⊦ ∀h t. ¬(h :: t = [])

⊦ ∀b f. case b f none = b

⊦ ∀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

⊦ ∀x. ∃a b. x = (a, b)

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀t₁ t₂. t₁ ∨ t₂ ⇔ t₂ ∨ t₁

⊦ ∀m n. m + n = n + m

⊦ ∀m n. m < n ⇒ m ≤ n

⊦ ∀r x. irreflexive r ⇒ ¬r x x

⊦ ∀n. 2 * n = n + n

⊦ ∀h t. length (h :: t) = suc (length t)

⊦ ∀m n. ¬(m < n ∧ n ≤ m)

⊦ ∀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

⊦ ∀x. x = none ∨ ∃a. x = some a

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ ∀f. surjective f ⇔ ∀y. ∃x. y = f x

⊦ ∀p. ¬(∃x. p x) ⇔ ∀x. ¬p x

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀a b. some a = some b ⇔ a = b

⊦ ∀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

⊦ ∀h t. concat (h :: t) = h @ concat t

⊦ ∀r s. subrelation r s ∧ wellFounded s ⇒ wellFounded r

⊦ ∀b f a. case b f (some a) = f a

⊦ ∀f g a. case f g (left a) = f a

⊦ ∀f g b. case f g (right b) = g b

⊦ ∀f g x. (f ∘ g) x = f (g x)

⊦ ∀t₁ t₂. ¬(t₁ ∧ t₂) ⇔ ¬t₁ ∨ ¬t₂

⊦ ∀m n. even (m * n) ⇔ even m ∨ even n

⊦ ∀m n. even (m + n) ⇔ even m ⇔ even n

⊦ ∀m n. m ↑ suc n = m * m ↑ n

⊦ ∀p. (∃x. p x) ⇔ ∃a b. p (a, b)

⊦ ∀l. l = [] ∨ ∃h t. l = h :: t

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g

⊦ ∀p a. (∃x. a = x ∧ p x) ⇔ p a

⊦ ∀x. (∃a. x = left a) ∨ ∃b. x = right b

⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r

⊦ ∀m n. m ≤ n ⇔ m < n ∨ m = n

⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n

⊦ ∀f. ∃fn. ∀a b. fn (a, b) = f a b

⊦ ∀m n. m < n ⇔ ∃d. n = m + suc d

⊦ ∀m x y. measure m x y ⇔ m x < m y

⊦ ∀p q. (∃x. p ∧ q x) ⇔ p ∧ ∃x. q x

⊦ ∀t₁ t₂ t₃. (t₁ ∧ t₂) ∧ t₃ ⇔ t₁ ∧ t₂ ∧ t₃

⊦ ∀m n p. m * (n * p) = m * n * p

⊦ ∀p m n. m + p = n + p ⇔ m = n

⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p

⊦ ∀l h t. (h :: t) @ l = h :: t @ l

⊦ ∀p x. (∀y. p y ⇔ y = x) ⇒ (select) p = x

⊦ ∀r. (∀x. ∃y. r x y) ⇔ ∃f. ∀x. r x (f x)

⊦ ∀m n. m ≤ suc n ⇔ m = suc n ∨ m ≤ n

⊦ ∀m n. m * n = 0 ⇔ m = 0 ∨ n = 0

⊦ ∀f h t. map f (h :: t) = f h :: map f t

⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n

⊦ ∀m n. m ↑ n = 0 ⇔ m = 0 ∧ ¬(n = 0)

⊦ ∀r s. subrelation r s ⇔ ∀x y. r x y ⇒ s x y

⊦ (∃!) = λp. (∃) p ∧ ∀x y. p x ∧ p y ⇒ x = y

⊦ ∀p g h. ∃f. ∀x. f x = if p x then f (g x) else h 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

⊦ ∀p. p [] ∧ (∀h t. p t ⇒ p (h :: t)) ⇒ ∀l. p l

⊦ ∀m n p. m * n = m * p ⇔ m = 0 ∨ n = p

⊦ ∀m n p. m * n ≤ m * p ⇔ m = 0 ∨ n ≤ p

⊦ ∀m n p. m * n < m * p ⇔ ¬(m = 0) ∧ n < p

⊦ ∀h₁ h₂ t₁ t₂. h₁ :: t₁ = h₂ :: t₂ ⇔ h₁ = h₂ ∧ t₁ = t₂

⊦ ∀a b a' b'. (a, b) = (a', b') ⇔ a = a' ∧ b = b'

⊦ ∀p₁ p₂ q₁ q₂. (p₁ ⇒ p₂) ∧ (q₁ ⇒ q₂) ⇒ p₁ ∧ q₁ ⇒ p₂ ∧ q₂

⊦ ∀p₁ p₂ q₁ q₂. (p₁ ⇒ p₂) ∧ (q₁ ⇒ q₂) ⇒ p₁ ∨ q₁ ⇒ p₂ ∨ q₂

⊦ ∀p. (∀x. ∃!y. p x y) ⇔ ∃f. ∀x y. p x y ⇔ f x = y

⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)

⊦ ∀b f. ∃fn. fn [] = b ∧ ∀h t. fn (h :: t) = f h t (fn t)

⊦ ∀p. (∃!x. p x) ⇔ (∃x. p x) ∧ ∀x x'. p x ∧ p x' ⇒ x = x'

⊦ ∀r. wellFounded r ⇔ ∀p. (∀x. (∀y. r y x ⇒ p y) ⇒ p x) ⇒ ∀x. p x

⊦ ∀r.
    wellFounded r ⇒
    ∀h.
      (∀f g x. (∀z. r z x ⇒ f z = g z) ⇒ h f x = h g x) ⇒
      ∃f. ∀x. f x = h f x