Package list-member-thm: list-member-thm

Information

Files

• Package tarball list-member-thm-1.0.tgz
• Theory file list-member-thm.thy (included in the package tarball)

Theorems

⊦ ∀l n.
    Number.Natural.< n (Data.List.length l) ⇒
    Data.List.member (Data.List.nth n l) l

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

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

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

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

⊦ ∀l x.
    Data.List.member x l ⇔
    ∃i. Number.Natural.< i (Data.List.length l) ∧ x = Data.List.nth i l

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

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

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

Input Type Operators

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

Input Constants

• =
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • cond
    • F
    • T
  • List
    • Data.List.::
    • Data.List.@
    • Data.List.[]
    • Data.List.all
    • Data.List.exists
    • Data.List.filter
    • Data.List.length
    • Data.List.map
    • Data.List.member
    • Data.List.nth
• Function
  • Function.id
• Number
  • Natural
    • Number.Natural.<
    • Number.Natural.suc
  • Numeral
    • Number.Numeral.zero

Assumptions

⊦ T

⊦ ∀x. Function.id x = x

⊦ ∀n. Number.Natural.< Number.Numeral.zero (Number.Natural.suc n)

⊦ ∀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)

⊦ ∀m. m = Number.Numeral.zero ∨ ∃n. m = Number.Natural.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.
    Number.Natural.< (Number.Natural.suc m) (Number.Natural.suc n) ⇔
    Number.Natural.< 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

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

⊦ Data.List.length Data.List.[] = Number.Numeral.zero ∧
  ∀h t.
    Data.List.length (Data.List.:: h t) =
    Number.Natural.suc (Data.List.length t)

⊦ ∀P.
    P Number.Numeral.zero ∧ (∀n. P n ⇒ P (Number.Natural.suc n)) ⇒ ∀n. P n

⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)

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

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

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

⊦ (∀m. Number.Natural.< m Number.Numeral.zero ⇔ F) ∧
  ∀m n.
    Number.Natural.< m (Number.Natural.suc n) ⇔
    m = n ∨ Number.Natural.< m n

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

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

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

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

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

⊦ (∀h t. Data.List.nth Number.Numeral.zero (Data.List.:: h t) = h) ∧
  ∀h t n.
    Data.List.nth (Number.Natural.suc n) (Data.List.:: h t) =
    Data.List.nth n t

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

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

⊦ ∀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)