Package pair-def: Definition of product types
Information
| name | pair-def |
| version | 1.10 |
| description | Definition of product types |
| author | Joe Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2011-11-27 |
| requires | bool |
| show | Data.Bool Data.Pair |
Files
- Package tarball pair-def-1.10.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
⊦ ∀p. ∃x y. p = (x, y)
⊦ ∀x y a b. (x, y) = (a, b) ⇔ x = a ∧ y = b
Input Type Operators
- →
- bool
Input Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- F
- T
- Bool
Assumptions
⊦ T
⊦ ¬F ⇔ T
⊦ ¬T ⇔ F
⊦ (∃) = 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. F ∧ t ⇔ F
⊦ ∀t. T ∧ t ⇔ t
⊦ ∀t. t ∧ T ⇔ t
⊦ ∀t. t ⇒ T ⇔ T
⊦ ∀t. F ∨ t ⇔ t
⊦ ∀t. T ∨ t ⇔ T
⊦ ∀t. (F ⇔ t) ⇔ ¬t
⊦ ∀t. t ⇒ F ⇔ ¬t
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)
⊦ ∀x y. x = y ⇔ y = x
⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1
⊦ (∧) = 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 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