Package list-map-thm: list-map-thm

Information

Files

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

Theorems

⊦ Data.List.map Function.id = Function.id

⊦ ∀l. Data.List.map (λx. x) l = l

⊦ ∀f l. Data.List.map f l = Data.List.[] ⇔ l = Data.List.[]

⊦ ∀f g l.
    Data.List.map (Function.o g f) l = Data.List.map g (Data.List.map f l)

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

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

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

Input Type Operators

• →
• bool
• Data
  • List
    • Data.List.list

Input Constants

• =
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • F
    • T
  • List
    • Data.List.::
    • Data.List.@
    • Data.List.[]
    • Data.List.map
• Function
  • Function.id
  • Function.o

Assumptions

⊦ T

⊦ Function.id = λx. x

⊦ F ⇔ ∀p. p

⊦ ∀x. Function.id x = x

⊦ ∀t. t ∨ ¬t

⊦ (¬) = λp. p ⇒ F

⊦ ∀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. ¬(Data.List.:: h t = Data.List.[])

⊦ (¬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. Function.o f 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)

⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)

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

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

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

⊦ (∀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)

⊦ ∀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)