Package axiom-infinity: Axiom of Infinity
Information
| name | axiom-infinity |
| version | 1.5 |
| description | Axiom of Infinity |
| author | Joe Hurd <joe@gilith.com> |
| license | MIT |
| provenance | HOL Light theory extracted on 2011-12-18 |
| requires | bool function |
| show | Data.Bool Function |
Files
- Package tarball axiom-infinity-1.5.tgz
- Theory file axiom-infinity.thy (included in the package tarball)
Theorem
⊦ ∃f. injective f ∧ ¬surjective f
Input Type Operators
- →
- bool
- ind
Input Constants
- =
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ¬
- F
- T
- Bool
- Function
- injective
- surjective
Assumptions
⊦ T
⊦ F ⇔ ∀p. p
⊦ (¬) = λp. p ⇒ F
⊦ T ⇔ (λp. p) = λp. p
⊦ (∀) = λp. p = λx. T
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ (∧) = λp q. (λf. f p q) = λf. f T T
⊦ ∀f. surjective f ⇔ ∀y. ∃x. y = f x
⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q
⊦ ∀f. injective f ⇔ ∀x1 x2. f x1 = f x2 ⇒ x1 = x2
⊦ let a o ←
(let h ←
let p r ←
let p s ←
(let f ← o r = o s in
λg.
(let h ← f in
λi.
(λj. j h i) =
λk. k ((λd. d) = λd. d) ((λd. d) = λd. d)) g ⇔ f)
(r = s) in
p = (λq. (λd. d) = λd. d) in
p = (λq. (λd. d) = λd. d) in
λi. (λj. j h i) = λk. k ((λd. d) = λd. d) ((λd. d) = λd. d))
(let t ←
let p u ←
let v y ← u = o y in
let b w ←
(let f ←
let p x ←
(let f ← v x in
λg.
(let h ← f in
λi.
(λj. j h i) =
λk.
k ((λd. d) = λd. d)
((λd. d) = λd. d)) g ⇔ f) w in
p = (λq. (λd. d) = λd. d) in
λg.
(let h ← f in
λi.
(λj. j h i) =
λk. k ((λd. d) = λd. d) ((λd. d) = λd. d)) g ⇔
f) w in
b = (λc. (λd. d) = λd. d) in
p = (λq. (λd. d) = λd. d) in
(let f ← t in
λg.
(let h ← f in
λi. (λj. j h i) = λk. k ((λd. d) = λd. d) ((λd. d) = λd. d))
g ⇔ f) (let b d ← d in b = λc. (λd. d) = λd. d)) in
let b e ←
(let f ←
let l n ←
(let f ← a n in
λg.
(let h ← f in
λi.
(λj. j h i) =
λk. k ((λd. d) = λd. d) ((λd. d) = λd. d)) g ⇔ f) e in
l = (λm. (λd. d) = λd. d) in
λg.
(let h ← f in
λi. (λj. j h i) = λk. k ((λd. d) = λd. d) ((λd. d) = λd. d)) g ⇔
f) e in
b = λc. (λd. d) = λd. d