Package function-thm: Properties of function operators and combinators
Information
| name | function-thm |
| version | 1.29 |
| description | Properties of function operators and combinators |
| author | Joe Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2012-03-24 |
| requires | bool function-def |
| show | Data.Bool |
Files
- Package tarball function-thm-1.29.tgz
- Theory file function-thm.thy (included in the package tarball)
Theorems
⊦ ∀x. Function.id x = x
⊦ ∀x. Function.S Function.K x = Function.id
⊦ ∀f. Function.∘ f Function.id = f
⊦ ∀f. Function.∘ Function.id f = f
⊦ ∀f. Function.C (Function.C f) = f
⊦ ∀x y. Function.K x y = x
⊦ ∀f x. Function.W f x = f x x
⊦ ∀f g x. Function.∘ f g x = f (g x)
⊦ ∀f x y. Function.C f x y = f y x
⊦ ∀f g x. Function.S f g x = f x (g x)
⊦ ∀f g h. Function.∘ f (Function.∘ g h) = Function.∘ (Function.∘ f g) h
⊦ ∀f g. (∀x. ∃y. g y = f x) ⇔ ∃h. f = Function.∘ 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 = Function.∘ h g
⊦ ∀p f g. (∀x. p x ⇒ ∃y. g y = f x) ⇔ ∃h. ∀x. p x ⇒ f x = g (h x)
⊦ ∀p f g.
(∀x y. p x ∧ p y ∧ g x = g y ⇒ f x = f y) ⇔ ∃h. ∀x. p x ⇒ f x = h (g x)
Input Type Operators
- →
- bool
Input Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- ⊥
- ⊤
- Bool
- Function
- Function.id
- Function.∘
- Function.C
- Function.K
- Function.S
- Function.W
Assumptions
⊦ ⊤
⊦ Function.id = λx. x
⊦ Function.K = λx y. x
⊦ ¬⊥ ⇔ ⊤
⊦ ¬⊤ ⇔ ⊥
⊦ ∀t. t ⇒ t
⊦ (∃) = λp. p ((select) p)
⊦ ∀t. (∀x. t) ⇔ t
⊦ (∀) = λp. p = λx. ⊤
⊦ Function.W = λf x. f x x
⊦ ∀t. ¬¬t ⇔ t
⊦ ∀t. (⊤ ⇔ t) ⇔ t
⊦ ∀t. (t ⇔ ⊤) ⇔ t
⊦ ∀t. ⊤ ∧ t ⇔ t
⊦ ∀t. ⊤ ⇒ t ⇔ t
⊦ Function.∘ = λf g x. f (g x)
⊦ Function.C = λf x y. f y x
⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t
⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t
⊦ Function.S = λf g x. f x (g x)
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)
⊦ ∀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 ⊤ ⊤
⊦ ∀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 q. p ⇒ (∃x. q x) ⇔ ∃x. p ⇒ q x
⊦ ∀p. (∀x. ∃y. p x y) ⇔ ∃y. ∀x. p x (y x)