Package function: Basic theory of functions

Information

Files

• Package tarball function-1.8.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

⊦ ∀f g. (∀x. ∃y. g y = f x) ⇔ ∃h. f = (g o h)

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

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

⊦ ∀f g. (∀x y. g x = g y ⇒ f x = f y) ⇔ ∃h. f = (h o g)

Input Type Operators

• →
• bool

Input Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ~
    • F
    • T

Assumptions

⊦ T

⊦ (∃) = λP. P ((select) P)

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀f y. (let x ← y ∈ f x) = f y

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀x y. x = y ⇒ y = x

⊦ (∧) = 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. f = g ⇔ ∀x. f x = g x

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

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

⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)

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

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