Package pair-def: Definition of product types
Information
| name | pair-def |
| version | 1.18 |
| description | Definition of product types |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2012-02-10 |
| requires | bool |
| show | Data.Bool Data.Pair |
Files
- Package tarball pair-def-1.18.tgz
- Theory file pair-def.thy (included in the package tarball)
Defined Type Operator
- Data
- Pair
- ×
- Pair
Defined Constants
- Data
- Pair
- ,
- fst
- snd
- Pair
Theorems
⊦ ∀x y. fst (x, y) = x
⊦ ∀x y. snd (x, y) = y
⊦ ∀xy. ∃x y. xy = (x, y)
⊦ ∀x y a b. (x, y) = (a, b) ⇔ x = a ∧ y = b
Input Type Operators
- →
- bool
Input Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- ⊥
- ⊤
- Bool
Assumptions
⊦ ⊤
⊦ ¬⊥ ⇔ ⊤
⊦ ¬⊤ ⇔ ⊥
⊦ (∃) = λp. p ((select) p)
⊦ ∀t. (λx. t x) = t
⊦ (∀) = λp. p = λx. ⊤
⊦ ∀t. ¬¬t ⇔ t
⊦ ∀t. (⊤ ⇔ t) ⇔ t
⊦ ∀t. (t ⇔ ⊤) ⇔ t
⊦ ∀t. ⊥ ∧ t ⇔ ⊥
⊦ ∀t. ⊤ ∧ t ⇔ t
⊦ ∀t. t ∧ ⊤ ⇔ t
⊦ ∀t. t ⇒ ⊤ ⇔ ⊤
⊦ ∀t. ⊥ ∨ t ⇔ t
⊦ ∀t. ⊤ ∨ t ⇔ ⊤
⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t
⊦ ∀t. t ⇒ ⊥ ⇔ ¬t
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)
⊦ ∀x y. x = y ⇔ y = x
⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1
⊦ (∧) = 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 q. (∃x. p x) ∧ q ⇔ ∃x. p x ∧ q
⊦ ∀p q. (∀x. p x) ∨ q ⇔ ∀x. p x ∨ q
⊦ ∀p q. (∃x. p x) ∨ q ⇔ ∃x. p x ∨ q
⊦ ∀t1 t2 t3. (t1 ∨ t2) ∨ t3 ⇔ t1 ∨ t2 ∨ t3
⊦ ∀p x. (∀y. p y ⇔ y = x) ⇒ (select) p = x
⊦ ∀p. (∀x. ∃y. p x y) ⇔ ∃y. ∀x. p x (y x)
⊦ ∀p q. (∀x. p x ∧ q x) ⇔ (∀x. p x) ∧ ∀x. q x