Package function: Function operators and combinators
Information
| name | function |
| version | 1.22 |
| description | Function operators and combinators |
| author | Joe Hurd <joe@gilith.com> |
| license | MIT |
| requires | bool |
| show | Data.Bool Function |
Files
- Package tarball function-1.22.tgz
- Theory file function.thy (included in the package tarball)
Defined Constants
- Function
- id
- injective
- ∘
- surjective
Theorems
⊦ id = λx. x
⊦ ∀x. id x = x
⊦ ∀f g. (f ∘ g) = λx. f (g x)
⊦ ∀f. surjective f ⇔ ∀y. ∃x. y = f x
⊦ ∀f. (id ∘ f) = f ∧ (f ∘ id) = f
⊦ ∀f g x. (f ∘ g) x = f (g x)
⊦ ∀f g h. (f ∘ (g ∘ h)) = (f ∘ g ∘ h)
⊦ ∀f. injective f ⇔ ∀x1 x2. f x1 = f x2 ⇒ x1 = x2
⊦ ∀f g. (∀x. ∃y. g y = f x) ⇔ ∃h. f = (g ∘ 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 ∘ g)
Input Type Operators
- →
- bool
Input Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- F
- T
- Bool
Assumptions
⊦ T
⊦ ¬F ⇔ T
⊦ ¬T ⇔ F
⊦ ∀t. t ⇒ t
⊦ (∃) = λP. P ((select) P)
⊦ ∀t. (∀x. t) ⇔ t
⊦ (∀) = λp. p = λx. T
⊦ ∀t. ¬¬t ⇔ t
⊦ ∀t. (T ⇔ t) ⇔ t
⊦ ∀t. (t ⇔ T) ⇔ t
⊦ ∀t. T ⇒ t ⇔ t
⊦ ∀t. (F ⇔ t) ⇔ ¬t
⊦ ∀t. (t ⇔ F) ⇔ ¬t
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)
⊦ ∀f y. (let x ← y in 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. (∀x. f x = g x) ⇔ f = g
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀P. (∀x. ∃y. P x y) ⇔ ∃y. ∀x. P x (y x)