Package function: Basic theory of functions

Information

Files

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

Defined Constants

• Function
  • id
  • injective
  • o
  • surjective

Theorems

⊦ id = λx. x

⊦ ∀x. id x = x

⊦ ∀f g. (f o g) = λx. f (g x)

⊦ ∀f. surjective f ⇔ ∀y. ∃x. y = f x

⊦ ∀f. (id o f) = f ∧ (f o id) = f

⊦ ∀f g x. (f o g) x = f (g x)

⊦ ∀f g h. (f o (g o h)) = (f o g o h)

⊦ ∀f. injective f ⇔ ∀x1 x2. f x1 = f x2 ⇒ x1 = x2

Input Type Operators

• →
• bool

Input Constants

• =
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • T

Assumptions

⊦ T

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λP. P = λx. T

⊦ ∀x. x = x ⇔ T

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀f y. (λx. f x) y = f y

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ ∀f g. f = g ⇔ ∀x. f x = g x