Package function: Function operators and combinators
Information
| name | function |
| version | 1.28 |
| description | Function operators and combinators |
| author | Joe Hurd <joe@gilith.com> |
| license | MIT |
| requires | bool |
| show | Data.Bool Function |
Files
- Package tarball function-1.28.tgz
- Theory file function.thy (included in the package tarball)
Defined Constants
- Function
- c
- id
- injective
- k
- ∘
- s
- surjective
- w
Theorems
⊦ id = λx. x
⊦ k = λx y. x
⊦ ∀x. id x = x
⊦ w = λf x. f x x
⊦ ∀x. s k x = id
⊦ ∀f. f ∘ id = f
⊦ ∀f. id ∘ f = f
⊦ ∀f. c (c f) = f
⊦ c = λf x y. f y x
⊦ s = λf g x. f x (g x)
⊦ ∀x y. k x y = x
⊦ ∀f x. w f x = f x x
⊦ ∀f g. f ∘ g = λx. f (g x)
⊦ ∀f. surjective f ⇔ ∀y. ∃x. y = f x
⊦ ∀f g x. (f ∘ g) x = f (g x)
⊦ ∀f x y. c f x y = f y x
⊦ ∀f g x. s f g x = f x (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)