Package unit: Basic theory of the unit type

Information

Files

• Package tarball unit-1.2.tgz
• Theory file unit.thy (included in the package tarball)

Defined Type Operator

• Data
  • Unit
    • unit

Defined Constant

• Data
  • Unit
    • ()

Theorems

⊦ ∀v. v = ()

⊦ ∀f g. f = g

⊦ ∀e. ∃fn. fn () = e

⊦ ∀e. ∃!fn. fn () = e

⊦ ∀P. P () ⇒ ∀x. P x

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

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

⊦ ∀f g. f = g ⇔ ∀x. f x = g x

⊦ ∀P. (∃!x. P x) ⇔ (∃x. P x) ∧ ∀x x'. P x ∧ P x' ⇒ x = x'

⊦ ∀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 ⇒ t ⇔ T) ∧ (t ⇒ F ⇔ ¬t)