Package list: List types

Information

name	list
version	1.50
description	List types
author	Joe Hurd <joe@gilith.com>
license	MIT
requires	bool function pair natural set
show	Data.Bool Data.List Data.Pair Function Number.Natural Set

Files

Package tarball list-1.50.tgz
Theory file list.thy (included in the package tarball)

Defined Type Operator

Data
- List
  - list

Defined Constants

Data
- List
  - ::
  - @
  - []
  - all
  - case
  - concat
  - drop
  - exists
  - filter
  - foldl
  - foldr
  - fromSet
  - head
  - interval
  - last
  - length
  - map
  - member
  - nth
  - nub
  - nubReverse
  - null
  - replicate
  - reverse
  - tail
  - take
  - toSet
  - zipWith

Theorems

⊦ null []

⊦ length [] = 0

⊦ concat [] = []

⊦ nubReverse [] = []

⊦ reverse [] = []

⊦ toSet [] = ∅

⊦ map id = id

⊦ ∀l. finite (toSet l)

⊦ ∀p. all p []

⊦ ∀l. all (λx. ⊤) l

⊦ ∀x. ¬member x []

⊦ ∀p. ¬exists p []

⊦ ∀x. replicate 0 x = []

⊦ ∀m. interval m 0 = []

⊦ ∀l. reverse (reverse l) = l

⊦ ∀l. [] @ l = l

⊦ ∀l. l @ [] = l

⊦ ∀l. drop 0 l = l

⊦ ∀l. take 0 l = []

⊦ ∀f. map f [] = []

⊦ ∀P. filter P [] = []

⊦ ∀l. map (λx. x) l = l

⊦ ∀h. last (h :: []) = h

⊦ ∀l. null l ⇔ l = []

⊦ ∀l. length (reverse l) = length l

⊦ ∀l. nub (nub l) = nub l

⊦ ∀l. nubReverse (nubReverse l) = nubReverse l

⊦ ∀l. toSet (nub l) = toSet l

⊦ ∀l. toSet (nubReverse l) = toSet l

⊦ ∀l. toSet (reverse l) = toSet l

⊦ ∀l. length (nub l) ≤ length l

⊦ ∀l. length (nubReverse l) ≤ length l

⊦ ∀l. size (toSet l) ≤ length l

⊦ ∀l. drop (length l) l = []

⊦ ∀l. take (length l) l = l

⊦ ∀l. case [] (::) l = l

⊦ ∀l. foldr (::) [] l = l

⊦ ∀f. zipWith f [] [] = []

⊦ ∀h t. ¬null (h :: t)

⊦ ∀l. nub l = reverse (nubReverse (reverse l))

⊦ ∀l. null (concat l) ⇔ all null l

⊦ ∀h t. ¬(h :: t = [])

⊦ ∀h t. head (h :: t) = h

⊦ ∀h t. tail (h :: t) = t

⊦ ∀b f. case b f [] = b

⊦ ∀n x. length (replicate n x) = n

⊦ ∀m n. length (interval m n) = n

⊦ ∀f b. foldr f b [] = b

⊦ ∀f b. foldl f b [] = b

⊦ ∀l. length l = 0 ⇔ l = []

⊦ ∀l. toSet l = ∅ ⇔ l = []

⊦ ∀l. foldl (C (::)) [] l = reverse l

⊦ ∀s. finite s ⇒ toSet (fromSet s) = s

⊦ ∀h t. nth 0 (h :: t) = h

⊦ ∀l f. length (map f l) = length l

⊦ ∀p l. length (filter p l) ≤ length l

⊦ ∀p l. toSet (filter p l) ⊆ toSet l

⊦ ∀s. finite s ⇒ length (fromSet s) = size s

⊦ ∀x l. member x l ⇔ x ∈ toSet l

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

⊦ ∀l x. member x (nub l) ⇔ member x l

⊦ ∀l x. member x (nubReverse l) ⇔ member x l

⊦ ∀l x. member x (reverse l) ⇔ member x l

⊦ ∀l1 l2. foldr (::) l2 l1 = l1 @ l2

⊦ ∀l. ¬(l = []) ⇒ last l ∈ toSet l

⊦ ∀h t. toSet (h :: t) = insert h (toSet t)

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

⊦ ∀f l. reverse (map f l) = map f (reverse l)

⊦ ∀f l. toSet (map f l) = image f (toSet l)

⊦ ∀f l. map f l = [] ⇔ l = []

⊦ ∀x n. replicate (suc n) x = x :: replicate n x

⊦ ∀l n. n < length l ⇒ member (nth n l) l

⊦ ∀l m. null (l @ m) ⇔ null l ∧ null m

⊦ ∀l m. length (l @ m) = length l + length m

⊦ ∀l m. reverse (l @ m) = reverse m @ reverse l

⊦ ∀l1 l2. toSet (l1 @ l2) = toSet l1 ∪ toSet l2

⊦ ∀l1 l2. foldl (C (::)) l2 l1 = reverse l1 @ l2

⊦ ∀l. l = [] ∨ ∃h t. l = h :: t

⊦ ∀l. ¬(l = []) ⇒ head l :: tail l = l

⊦ ∀P l. ¬all P l ⇔ exists (λx. ¬P x) l

⊦ ∀P l. ¬exists P l ⇔ all (λx. ¬P x) l

⊦ ∀x l. reverse (x :: l) = reverse l @ x :: []

⊦ ∀m n. interval m (suc n) = m :: interval (suc m) n

⊦ ∀n l. n ≤ length l ⇒ length (take n l) = n

⊦ ∀l i. i < length l ⇒ nth i l ∈ toSet l

⊦ ∀l k. foldl (λn x. suc n) k l = length l + k

⊦ ∀l k. foldr (λx n. suc n) k l = length l + k

⊦ ∀s. finite s ⇒ ∀x. member x (fromSet s) ⇔ x ∈ s

⊦ ∀P l. (∀x. member x l ⇒ P x) ⇔ all P l

⊦ ∀P l. (∃x. P x ∧ member x l) ⇔ exists P l

⊦ ∀h t. last (h :: t) = if t = [] then h else last t

⊦ ∀h k t. last (h :: k :: t) = last (k :: t)

⊦ ∀n x i. i < n ⇒ nth i (replicate n x) = x

⊦ ∀l h t. (h :: t) @ l = h :: t @ l

⊦ ∀l m n. l @ m @ n = (l @ m) @ n

⊦ ∀l. ¬(l = []) ⇒ length (tail l) = length l - 1

⊦ ∀f g l. map (g ∘ f) l = map g (map f l)

⊦ ∀P l. all P l ⇔ ∀x. x ∈ toSet l ⇒ P x

⊦ ∀P l. exists P l ⇔ ∃x. x ∈ toSet l ∧ P x

⊦ ∀P f l. all P (map f l) ⇔ all (P ∘ f) l

⊦ ∀P f l. exists P (map f l) ⇔ exists (P ∘ f) l

⊦ ∀f b l. foldr f b (reverse l) = foldl (C f) b l

⊦ ∀f b l. foldl f b l = foldr (C f) b (reverse l)

⊦ ∀f b l. foldl f b (reverse l) = foldr (C f) b l

⊦ ∀b f h t. case b f (h :: t) = f h t

⊦ ∀n x. toSet (replicate n x) = if n = 0 then ∅ else insert x ∅

⊦ ∀l m. head (l @ m) = if l = [] then head m else head l

⊦ ∀p q. last (p @ q) = if q = [] then last p else last q

⊦ ∀l m. l @ m = [] ⇔ l = [] ∧ m = []

⊦ ∀l. ¬(l = []) ⇒ last l = nth (length l - 1) l

⊦ ∀s. finite s ⇒ toSet (fromSet s) = s ∧ length (fromSet s) = size s

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

⊦ ∀p h t. all p (h :: t) ⇔ p h ∧ all p t

⊦ ∀p h t. exists p (h :: t) ⇔ p h ∨ exists p t

⊦ ∀P l x. member x (filter P l) ⇔ P x ∧ member x l

⊦ ∀P l. (∀x. all (P x) l) ⇔ all (λs. ∀x. P x s) l

⊦ ∀P l. (∃x. exists (P x) l) ⇔ exists (λs. ∃x. P x s) l

⊦ ∀n l. n ≤ length l ⇒ length (drop n l) = length l - n

⊦ ∀n l. n ≤ length l ⇒ take n l @ drop n l = l

⊦ ∀x h t. member x (h :: t) ⇔ x = h ∨ member x t

⊦ ∀x l1 l2. member x (l1 @ l2) ⇔ member x l1 ∨ member x l2

⊦ ∀m n i. i < n ⇒ nth i (interval m n) = m + i

⊦ ∀f l1 l2. map f (l1 @ l2) = map f l1 @ map f l2

⊦ ∀f. (∀m. ∃l. map f l = m) ⇔ ∀y. ∃x. f x = y

⊦ ∀P l1 l2. all P (l1 @ l2) ⇔ all P l1 ∧ all P l2

⊦ ∀P l1 l2. filter P (l1 @ l2) = filter P l1 @ filter P l2

⊦ ∀P l. all P l ⇔ ∀i. i < length l ⇒ P (nth i l)

⊦ ∀P l. exists P l ⇔ ∃i. i < length l ∧ P (nth i l)

⊦ ∀P f l. filter P (map f l) = map f (filter (P ∘ f) l)

⊦ ∀h t.
nubReverse (h :: t) =
if member h t then nubReverse t else h :: nubReverse t

⊦ ∀p q l. all p (filter q l) ⇔ all (λx. q x ⇒ p x) l

⊦ ∀p q l. exists p (filter q l) ⇔ exists (λx. q x ∧ p x) l

⊦ ∀l x. member x l ⇔ ∃i. i < length l ∧ x = nth i l

⊦ ∀P Q l. (∀x. P x ⇒ Q x) ⇒ all P l ⇒ all Q l

⊦ ∀P. P [] ∧ (∀a0 a1. P a1 ⇒ P (a0 :: a1)) ⇒ ∀x. P x

⊦ ∀f b h t. foldr f b (h :: t) = f h (foldr f b t)

⊦ ∀f b h t. foldl f b (h :: t) = foldl f (f b h) t

⊦ ∀h t n. n < length t ⇒ nth (suc n) (h :: t) = nth n t

⊦ ∀x l. x ∈ toSet l ⇔ ∃i. i < length l ∧ x = nth i l

⊦ ∀n h t. n ≤ length t ⇒ drop (suc n) (h :: t) = drop n t

⊦ ∀f l i. i < length l ⇒ nth i (map f l) = f (nth i l)

⊦ ∀f y l. member y (map f l) ⇔ ∃x. member x l ∧ y = f x

⊦ ∀f g l. all (λx. f x = g x) l ⇒ map f l = map g l

⊦ ∀P Q l. all P l ∧ all Q l ⇔ all (λx. P x ∧ Q x) l

⊦ ∀P Q l. all (λx. P x ⇒ Q x) l ∧ all P l ⇒ all Q l

⊦ ∀f b l1 l2. foldr f b (l1 @ l2) = foldr f (foldr f b l2) l1

⊦ ∀f b l1 l2. foldl f b (l1 @ l2) = foldl f (foldl f b l1) l2

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

⊦ ∀n h t. n ≤ length t ⇒ take (suc n) (h :: t) = h :: take n t

⊦ ∀P h t. filter P (h :: t) = if P h then h :: filter P t else filter P t

⊦ ∀l n. length l = suc n ⇔ ∃h t. l = h :: t ∧ length t = n

⊦ ∀NIL' CONS'.
∃fn. fn [] = NIL' ∧ ∀a0 a1. fn (a0 :: a1) = CONS' a0 a1 (fn a1)

⊦ ∀n l i. n + i < length l ⇒ nth i (drop n l) = nth (n + i) l

⊦ ∀n l i. n ≤ length l ∧ i < n ⇒ nth i (take n l) = nth i l

⊦ ∀P Q l. (∀x. member x l ∧ P x ⇒ Q x) ∧ all P l ⇒ all Q l

⊦ ∀P Q l. (∀x. member x l ∧ P x ⇒ Q x) ∧ exists P l ⇒ exists Q l

⊦ ∀f l1 l2 n. length l1 = n ∧ length l2 = n ⇒ length (zipWith f l1 l2) = n

⊦ ∀l m.
length l = length m ∧ (∀i. i < length l ⇒ nth i l = nth i m) ⇒ l = m

⊦ ∀f. (∀l m. map f l = map f m ⇒ l = m) ⇔ ∀x y. f x = f y ⇒ x = y

⊦ ∀f h1 h2 t1 t2.
length t1 = length t2 ⇒
zipWith f (h1 :: t1) (h2 :: t2) = f h1 h2 :: zipWith f t1 t2

⊦ ∀k l m.
k < length l + length m ⇒
nth k (l @ m) = if k < length l then nth k l else nth (k - length l) m

⊦ ∀g f b l1 l2.
(∀s. g b s = s) ∧ (∀x s1 s2. g (f x s1) s2 = f x (g s1 s2)) ⇒
foldr f b (l1 @ l2) = g (foldr f b l1) (foldr f b l2)

⊦ ∀g f b l1 l2.
(∀s. g s b = s) ∧ (∀s1 s2 x. g s1 (f s2 x) = f (g s1 s2) x) ⇒
foldl f b (l1 @ l2) = g (foldl f b l1) (foldl f b l2)

Input Type Operators

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

Input Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∃!
  - ∨
  - ¬
  - cond
  - ⊥
  - ⊤
- Pair
  - ,
Function
- id
- ∘
- C
Number
- Natural
  - *
  - +
  - -
  - <
  - ≤
  - bit0
  - bit1
  - even
  - exp
  - suc
  - zero
Set
- ∅
- finite
- image
- insert
- ∈
- size
- ⊆
- ∪

Assumptions

⊦ ⊤

⊦ finite ∅

⊦ id = λx. x

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ bit0 0 = 0

⊦ size ∅ = 0

⊦ ∀t. t ⇒ t

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ ∀s. s ⊆ s

⊦ ⊥ ⇔ ∀p. p

⊦ ∀x. ¬(x ∈ ∅)

⊦ ∀x. id x = x

⊦ ∀t. t ∨ ¬t

⊦ ∀n. ¬(n < n)

⊦ ∀n. 0 < suc n

⊦ ∀n. n < suc n

⊦ ∀n. n ≤ suc n

⊦ (¬) = λp. p ⇒ ⊥

⊦ (∃) = λp. p ((select) p)

⊦ ∀a. ∃x. x = a

⊦ ∀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 ∨ ⊤ ⇔ ⊤

⊦ ∀n. ¬(suc n = 0)

⊦ ∀m. m < 0 ⇔ ⊥

⊦ ∀n. 0 + n = n

⊦ ∀m. m + 0 = m

⊦ ∀m. m - 0 = m

⊦ ∀n. n - n = 0

⊦ ∀s. ∅ ∪ s = s

⊦ ∀f. image f ∅ = ∅

⊦ ∀f. C (C f) = f

⊦ C = λf x y. f y x

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀n. even (2 * n)

⊦ ∀n. bit1 n = suc (bit0 n)

⊦ ∀m. exp m 0 = 1

⊦ ∀m n. m ≤ m + n

⊦ ∀s t. s ⊆ s ∪ t

⊦ ∀s t. s ⊆ t ∪ s

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀n. even (suc n) ⇔ ¬even n

⊦ ∀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 = ∅)

⊦ ∀n. bit0 (suc n) = suc (suc (bit0 n))

⊦ ∀f y. (let x ← y in f x) = f 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 - m = n

⊦ ∀s x. finite (insert x s) ⇔ finite s

⊦ ∀s t. s ∪ t = t ∪ s

⊦ ∀n. 2 * n = n + n

⊦ ∀m n. ¬(m < n ∧ n ≤ m)

⊦ ∀m n. ¬(m ≤ n ∧ n < m)

⊦ ∀m n. ¬(m < n) ⇔ n ≤ m

⊦ ∀m n. suc m ≤ 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

⊦ ∀x s. insert x (insert x s) = insert x s

⊦ ∀x s. insert x ∅ ∪ s = insert x s

⊦ ∀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. suc m ≤ suc n ⇔ m ≤ n

⊦ ∀f g x. (f ∘ g) x = f (g x)

⊦ ∀f x y. C f x y = f y x

⊦ ∀t1 t2. ¬(t1 ∧ t2) ⇔ ¬t1 ∨ ¬t2

⊦ ∀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. exp m (suc n) = m * exp m n

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g

⊦ ∀p a. (∃x. a = x ∧ p x) ⇔ p a

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

⊦ ∀m n. m ≤ n ⇔ m < n ∨ m = n

⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n

⊦ ∀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. (∀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

⊦ ∀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 + p = n + p ⇔ m = n

⊦ ∀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 ∧ t ⊆ u ⇒ s ⊆ u

⊦ ∀s t. (∀x. x ∈ s ⇔ x ∈ t) ⇔ s = t

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

⊦ ∀p. (∀x. ∃y. p x y) ⇔ ∃y. ∀x. p x (y x)

⊦ ∀m n. n ≤ m ⇒ suc m - suc n = m - n

⊦ ∀m n. m ≤ suc n ⇔ m = suc n ∨ m ≤ n

⊦ ∀m n. m * n = 0 ⇔ m = 0 ∨ n = 0

⊦ ∀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

⊦ ∀m n. exp m n = 0 ⇔ m = 0 ∧ ¬(n = 0)

⊦ ∀x y s. x ∈ insert y s ⇔ x = y ∨ x ∈ s

⊦ ∀s t u. s ∪ t ⊆ u ⇔ s ⊆ u ∧ t ⊆ u

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

⊦ ∀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

⊦ ∀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

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

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

⊦ ∀x s.
finite s ⇒ size (insert x s) = if x ∈ s then size s else suc (size s)

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

⊦ ∀x y a b. (x, y) = (a, b) ⇔ x = a ∧ y = 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)

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

⊦ ∀p.
p ∅ ∧ (∀x s. p s ∧ ¬(x ∈ s) ∧ finite s ⇒ p (insert x s)) ⇒
∀s. finite s ⇒ p s