Package bool-class: Classical boolean theorems
Information
| name | bool-class |
| version | 1.22 |
| description | Classical boolean theorems |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | MIT |
| provenance | HOL Light theory extracted on 2014-06-12 |
| requires | axiom-choice axiom-extensionality bool-def bool-ext bool-int |
| show | Data.Bool |
Files
- Package tarball bool-class-1.22.tgz
- Theory source file bool-class.thy (included in the package tarball)
Theorems
⊦ ∀t. t ∨ ¬t
⊦ (∃) = λp. p ((select) p)
⊦ ∀t. ¬¬t ⇔ t
⊦ ∀x. (select y. y = x) = x
⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)
⊦ ∀t1 t2. (if ⊥ then t1 else t2) = t2
⊦ ∀t1 t2. (if ⊤ then t1 else t2) = t1
⊦ ∀b t. (if b then t else t) = t
⊦ ∀p. (∀b. p b) ⇔ p ⊤ ∧ p ⊥
⊦ ∀p. (∃b. p b) ⇔ p ⊤ ∨ p ⊥
⊦ ∀p. p ⊥ ∧ p ⊤ ⇒ ∀x. p x
⊦ ∀p. (∀x. ¬p x) ⇔ ¬∃x. p x
⊦ ∀p. (∃x. ¬p x) ⇔ ¬∀x. p x
⊦ ∀p. ¬(∀x. p x) ⇔ ∃x. ¬p x
⊦ ∀p. ¬(∃x. p x) ⇔ ∀x. ¬p x
⊦ ∀t1 t2. ¬(t1 ⇒ t2) ⇔ t1 ∧ ¬t2
⊦ ∀t1 t2. (¬t1 ⇔ ¬t2) ⇔ t1 ⇔ t2
⊦ ∀t1 t2. ¬t1 ⇒ ¬t2 ⇔ t2 ⇒ t1
⊦ ∀t1 t2. ¬(t1 ∧ t2) ⇔ ¬t1 ∨ ¬t2
⊦ ∀t1 t2. ¬(t1 ∨ t2) ⇔ ¬t1 ∧ ¬t2
⊦ ∀a b. ∃f. f ⊥ = a ∧ f ⊤ = b
⊦ ∀c x y. (if ¬c then x else y) = if c then y else x
⊦ ∀p q. (∀x. p ∨ q x) ⇔ p ∨ ∀x. q x
⊦ ∀p q. (∃x. p ⇒ q x) ⇔ p ⇒ ∃x. q x
⊦ ∀p q. p ⇒ (∃x. q x) ⇔ ∃x. p ⇒ q x
⊦ ∀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
⊦ ∀p q. (∀x. p x) ∨ q ⇔ ∀x. p x ∨ q
⊦ ∀p x. (∀y. p y ⇔ y = x) ⇒ (select) p = x
⊦ ∀r. (∀x. ∃y. r x y) ⇔ ∃f. ∀x. r x (f x)
⊦ ∀p. (∀x. ∃!y. p x y) ⇔ ∃!f. ∀x. p x (f x)
⊦ ∀b f g. (λx. if b then f x else g x) = if b then f else g
⊦ ∀b t1 t2. (if b then t1 else t2) ⇔ (¬b ∨ t1) ∧ (b ∨ t2)
⊦ ∀b f x y. f (if b then x else y) = if b then f x else f y
⊦ ∀b f g x. (if b then f else g) x = if b then f x else g x
⊦ ∀p. (∀x. ∃!y. p x y) ⇔ ∃f. ∀x y. p x y ⇔ f x = y
⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)
⊦ ∀p r. (∀x. p x ⇒ ∃y. r x y) ⇔ ∃f. ∀x. p x ⇒ r x (f x)
⊦ ∀b p q r s. (p ⇒ q) ∧ (r ⇒ s) ⇒ (if b then p else r) ⇒ if b then q else s
External Type Operators
- →
- bool
External Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∃!
- ∨
- ¬
- cond
- ⊥
- ⊤
- Bool
Assumptions
⊦ ⊤
⊦ ¬⊥ ⇔ ⊤
⊦ ¬⊤ ⇔ ⊥
⊦ ∀t. t ⇒ t
⊦ ⊥ ⇔ ∀p. p
⊦ (¬) = λp. p ⇒ ⊥
⊦ ∀t. (∀x. t) ⇔ t
⊦ ∀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) ⇔ ¬t
⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀p x. p x ⇒ p ((select) p)
⊦ ∀x y. x = y ⇔ y = x
⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤
⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q
⊦ ∀f g. (∀x. f x = g x) ⇔ f = g
⊦ ∀f g. (∀x. f x = g x) ⇒ f = g
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀p q. (∃x. p ∧ q x) ⇔ p ∧ ∃x. q x
⊦ ∀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. (∃!x. p x) ⇔ ∃x. ∀y. p y ⇔ x = y
⊦ ∀p q. (∀x. p x ∧ q x) ⇔ (∀x. p x) ∧ ∀x. q x
⊦ cond = λt t1 t2. select x. ((t ⇔ ⊤) ⇒ x = t1) ∧ ((t ⇔ ⊥) ⇒ x = t2)
⊦ ∀p. (∃!x. p x) ⇔ (∃x. p x) ∧ ∀x x'. p x ∧ p x' ⇒ x = x'