Package list: Basic theory of list types

Information

Files

• Package tarball list-1.16.tgz
• Theory file list.thy (included in the package tarball)

Defined Type Operator

• Data
  • List
    • list

Defined Constants

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

Theorems

⊦ map id = id

⊦ ∀l. Set.finite (toSet l)

⊦ ∀l. all (λx. T) l

⊦ ∀l. reverse (reverse l) = l

⊦ ∀l. l @ [] = l

⊦ ∀l. drop 0 l = l

⊦ ∀l. take 0 l = []

⊦ ∀l. map (λx. x) l = l

⊦ ∀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. Set.size (toSet l) ≤ length l

⊦ ∀l. drop (length l) l = []

⊦ ∀l. take (length l) l = l

⊦ ∀l. case.[].:: [] (::) l = l

⊦ ∀f. zipWith f [] [] = []

⊦ ∀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

⊦ ∀n x. length (replicate n x) = n

⊦ ∀m n. length (interval m n) = n

⊦ ∀l. length l = 0 ⇔ l = []

⊦ ∀l. toSet l = Set.∅ ⇔ l = []

⊦ ∀s. Set.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. Set.⊆ (toSet (filter p l)) (toSet l)

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

⊦ ∀x l. member x l ⇔ Set.∈ x (toSet l)

⊦ ∀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

⊦ ∀l. ¬(l = []) ⇒ Set.∈ (last l) (toSet l)

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

⊦ ∀f l. map f l = [] ⇔ l = []

⊦ ∀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) = Set.∪ (toSet l1) (toSet l2)

⊦ ∀l. l = [] ∨ ∃h t. l = h :: t

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

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

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

⊦ ∀l i. i < length l ⇒ Set.∈ (nth i l) (toSet l)

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

⊦ (null [] ⇔ T) ∧ ∀h t. null (h :: t) ⇔ F

⊦ ∀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

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

⊦ ∀l m n. l @ m @ n = (l @ m) @ n

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

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

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

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

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

⊦ ∀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 = []

⊦ length [] = 0 ∧ ∀h t. length (h :: t) = suc (length t)

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

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

⊦ ∀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

⊦ concat [] = [] ∧ ∀h t. concat (h :: t) = h @ concat t

⊦ toSet [] = Set.∅ ∧ ∀h t. toSet (h :: t) = Set.insert h (toSet 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 f l. filter P (map f l) = map f (filter (P o f) 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

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

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

⊦ ∀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

⊦ (∀x. replicate 0 x = []) ∧ ∀x n. replicate (suc n) x = x :: replicate n x

⊦ ∀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

⊦ (∀m. interval m 0 = []) ∧
  ∀m n. interval m (suc n) = m :: interval (suc m) n

⊦ ∀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 ≤ 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

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

⊦ (∀l. [] @ l = l) ∧ ∀l h t. (h :: t) @ l = h :: t @ l

⊦ (∀f. map f [] = []) ∧ ∀f h t. map f (h :: t) = f h :: map f t

⊦ (∀P. all P [] ⇔ T) ∧ ∀P h t. all P (h :: t) ⇔ P h ∧ all P t

⊦ (∀P. exists P [] ⇔ F) ∧ ∀P h t. exists P (h :: t) ⇔ P h ∨ exists P t

⊦ (∀h. last (h :: []) = h) ∧ ∀h k t. last (h :: k :: t) = last (k :: t)

⊦ (∀x. member x [] ⇔ F) ∧ ∀x h t. member x (h :: t) ⇔ x = h ∨ member x t

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

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

⊦ (∀l. drop 0 l = l) ∧
  ∀n h t. n ≤ length t ⇒ drop (suc n) (h :: t) = drop n t

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

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

⊦ (∀l. take 0 l = []) ∧
  ∀n h t. n ≤ length t ⇒ take (suc n) (h :: t) = h :: take n t

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

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

⊦ (∀h t. nth 0 (h :: t) = h) ∧
  ∀h t n. n < length t ⇒ nth (suc n) (h :: t) = nth n t

⊦ ∀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

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

Input Type Operators

• →
• bool
• Data
  • Pair
    • Data.Pair.*
• Number
  • Natural
    • natural
• Set
  • Set.set

Input Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∃!
    • ∨
    • ~
    • cond
    • F
    • T
  • Pair
    • Data.Pair.,
• Function
  • id
  • o
• Number
  • Natural
    • *
    • +
    • -
    • <
    • ≤
    • bit0
    • bit1
    • even
    • exp
    • suc
    • zero
• Set
  • Set.∅
  • Set.finite
  • Set.image
  • Set.∈
  • Set.insert
  • Set.size
  • Set.⊆
  • Set.∪

Assumptions

⊦ T

⊦ Set.finite Set.∅

⊦ id = λx. x

⊦ Set.size Set.∅ = 0

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ ∀s. Set.⊆ s s

⊦ F ⇔ ∀p. p

⊦ 1 = suc 0

⊦ ∀x. ¬Set.∈ x Set.∅

⊦ ∀x. id x = x

⊦ ∀t. t ∨ ¬t

⊦ ∀n. ¬(n < n)

⊦ ∀n. 0 < suc n

⊦ ∀n. n < suc n

⊦ ∀n. n ≤ suc n

⊦ (~) = λp. p ⇒ F

⊦ (∃) = 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

⊦ ∀x. x = x ⇔ T

⊦ ∀n. ¬(suc n = 0)

⊦ ∀m. m + 0 = m

⊦ ∀m. m - 0 = m

⊦ ∀n. n - n = 0

⊦ ∀n. even (2 * n)

⊦ ∀n. bit0 n = n + n

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ ∀n. bit1 n = suc (n + n)

⊦ ∀n. suc n - 1 = n

⊦ ∀x s. ¬(Set.insert x s = Set.∅)

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀f y. (let x ← y ∈ f x) = f y

⊦ ∀x y. x = y ⇒ y = x

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

⊦ ∀m n. m + n - m = n

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

⊦ ∀s t. Set.∪ s t = Set.∪ 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 T T

⊦ ∀P. ¬(∀x. P x) ⇔ ∃x. ¬P x

⊦ ∀P. ¬(∃x. P x) ⇔ ∀x. ¬P x

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

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

⊦ ∀x s. Set.∪ (Set.insert x Set.∅) s = Set.insert x s

⊦ ∀m n. m + suc 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 o g) x = f (g x)

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

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

⊦ ∀m n. m ≤ n ⇔ ∃d. n = m + d

⊦ ∀f g. f = g ⇔ ∀x. f x = g x

⊦ ∀P a. (∃x. a = x ∧ P x) ⇔ P a

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

⊦ (even 0 ⇔ T) ∧ ∀n. even (suc n) ⇔ ¬even n

⊦ ∀m n. m ≤ n ⇔ m < n ∨ m = n

⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n

⊦ (∀s. Set.∪ Set.∅ s = s) ∧ ∀s. Set.∪ s Set.∅ = 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. (∀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

⊦ ∀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 < m + p ⇔ n < p

⊦ ∀m n p. m < n ∧ n ≤ p ⇒ m < p

⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p

⊦ ∀s t u. Set.∪ (Set.∪ s t) u = Set.∪ s (Set.∪ t u)

⊦ ∀s t u. Set.⊆ s t ∧ Set.⊆ t u ⇒ Set.⊆ s u

⊦ ∀s t. s = t ⇔ ∀x. Set.∈ x s ⇔ Set.∈ x t

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

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

⊦ ∀t1 t2. (if T then t1 else t2) = t1 ∧ (if F then t1 else t2) = t2

⊦ ∀m n. n ≤ m ⇒ suc m - suc n = m - n

⊦ ∀m n. m * n = 0 ⇔ m = 0 ∨ n = 0

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

⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)

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

⊦ (∀s t. Set.⊆ s (Set.∪ s t)) ∧ ∀s t. Set.⊆ s (Set.∪ t s)

⊦ ∀x y s. Set.∈ x (Set.insert y s) ⇔ x = y ∨ Set.∈ x s

⊦ ∀s t u. Set.⊆ (Set.∪ s t) u ⇔ Set.⊆ s u ∧ Set.⊆ t u

⊦ ∀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

⊦ (∀n. 0 + n = n) ∧ ∀m n. suc m + n = suc (m + n)

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

⊦ ∀x y a b. Data.Pair., x y = Data.Pair., a b ⇔ x = a ∧ y = b

⊦ ∀A B C D. (A ⇒ B) ∧ (C ⇒ D) ⇒ A ∧ C ⇒ B ∧ D

⊦ ∀A B C D. (A ⇒ B) ∧ (C ⇒ D) ⇒ A ∨ C ⇒ B ∨ D

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

⊦ (∀m. exp m 0 = 1) ∧ ∀m n. exp m (suc n) = m * exp m n

⊦ ∀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'

⊦ (∀m. m < 0 ⇔ F) ∧ ∀m n. m < suc n ⇔ m = n ∨ m < n

⊦ ∀t1 t2. (¬(t1 ∧ t2) ⇔ ¬t1 ∨ ¬t2) ∧ (¬(t1 ∨ t2) ⇔ ¬t1 ∧ ¬t2)

⊦ (∀f. Set.image f Set.∅ = Set.∅) ∧
  ∀f x s. Set.image f (Set.insert x s) = Set.insert (f x) (Set.image f s)

⊦ Set.size Set.∅ = 0 ∧
  ∀x s.
    Set.finite s ⇒
    Set.size (Set.insert x s) =
    if Set.∈ x s then Set.size s else suc (Set.size s)

⊦ (∀m. m ≤ 0 ⇔ m = 0) ∧ ∀m n. m ≤ suc n ⇔ m = suc n ∨ m ≤ n

⊦ ∀t. ((T ⇔ t) ⇔ t) ∧ ((t ⇔ T) ⇔ t) ∧ ((F ⇔ t) ⇔ ¬t) ∧ ((t ⇔ F) ⇔ ¬t)

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

⊦ ∀t. (T ∧ t ⇔ t) ∧ (t ∧ T ⇔ t) ∧ (F ∧ t ⇔ F) ∧ (t ∧ F ⇔ F) ∧ (t ∧ t ⇔ t)

⊦ ∀t. (T ∨ t ⇔ T) ∧ (t ∨ T ⇔ T) ∧ (F ∨ t ⇔ t) ∧ (t ∨ F ⇔ t) ∧ (t ∨ t ⇔ t)

⊦ ∀t. (T ⇒ t ⇔ t) ∧ (t ⇒ T ⇔ T) ∧ (F ⇒ t ⇔ T) ∧ (t ⇒ t ⇔ T) ∧ (t ⇒ F ⇔ ¬t)

⊦ (∀n. 0 + n = n) ∧ (∀m. m + 0 = m) ∧ (∀m n. suc m + n = suc (m + n)) ∧
  ∀m n. m + suc n = suc (m + n)

⊦ ∀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)