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

Information

Files

• Package tarball list-map-1.11.tgz
• Theory file list-map.thy (included in the package tarball)

Defined Constant

• Data
  • List
    • map

Theorems

⊦ map id = id

⊦ ∀l. map (λx. x) l = l

⊦ ∀l f. length (map f l) = length l

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

⊦ ∀f l. map f l = [] ⇔ l = []

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

⊦ ∀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

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

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

Input Type Operators

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

Input Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ~
    • F
    • T
  • List
    • ::
    • @
    • []
    • length
    • toSet
• Function
  • id
  • o
• Number
  • Natural
    • Number.Natural.suc
    • Number.Natural.zero
• Set
  • Set.∅
  • Set.image
  • Set.insert

Assumptions

⊦ T

⊦ id = λx. x

⊦ F ⇔ ∀p. p

⊦ ∀x. id x = x

⊦ ∀t. t ∨ ¬t

⊦ (~) = λp. p ⇒ F

⊦ (∃) = λP. P ((select) P)

⊦ ∀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)

⊦ ∀h t. ¬(h :: t = [])

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ (∧) = 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

⊦ ∀f g x. (f o g) x = f (g x)

⊦ ∀f g. f = g ⇔ ∀x. f x = g x

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

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

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

⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)

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

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

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

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

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

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

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