Package list-member: Definitions and theorems about the list member function

Information

Files

• Package tarball list-member-1.13.tgz
• Theory file list-member.thy (included in the package tarball)

Defined Constant

• Data
  • List
    • member

Theorems

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

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

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

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

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

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

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

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

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

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

⊦ ∀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

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

Input Type Operators

• →
• bool
• Data
  • List
    • list
• Number
  • Natural
    • natural
• Set
  • Set.set

Input Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ~
    • cond
    • F
    • T
  • List
    • ::
    • @
    • []
    • all
    • exists
    • filter
    • fromSet
    • length
    • map
    • nth
    • reverse
    • toSet
• Function
  • Function.id
• Number
  • Natural
    • <
    • suc
    • zero
• Set
  • Set.∅
  • Set.finite
  • Set.∈
  • Set.insert

Assumptions

⊦ T

⊦ F ⇔ ∀p. p

⊦ ∀x. ¬Set.∈ x Set.∅

⊦ ∀x. Function.id x = x

⊦ ∀n. 0 < suc n

⊦ (~) = λp. p ⇒ F

⊦ (∃) = λP. P ((select) P)

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (∃x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

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

⊦ ∀x y. x = y ⇒ y = x

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

⊦ ∀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

⊦ ∀m n. suc m < suc n ⇔ m < n

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

⊦ ∀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

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

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

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

⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ toSet [] = Set.∅ ∧ ∀h t. toSet (h :: t) = Set.insert h (toSet t)

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

⊦ ∀P Q. (∃x. P x) ∨ (∃x. Q x) ⇔ ∃x. P x ∨ Q x

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

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

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

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

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

⊦ (∀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

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

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

⊦ ∀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)

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

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

⊦ ∀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)