Package bool-choice-cond: Theorems about the conditional relying on the axiom of choice

Information

Files

• Package tarball bool-choice-cond-1.0.tgz
• Theory file bool-choice-cond.thy (included in the package tarball)

Theorems

⊦ ∀b t. (if b then t else t) = t

⊦ ∀t1 t2. (if T then t1 else t2) = t1 ∧ (if F then t1 else t2) = t2

⊦ ∀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 c x y. P (if c then x else y) ⇔ (c ⇒ P x) ∧ (¬c ⇒ P y)

⊦ ∀b A B C D.
    (A ⇒ B) ∧ (C ⇒ D) ⇒ (if b then A else C) ⇒ (if b then B else D)

Input Type Operators

• →
• bool

Input Constants

• =
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∨
    • ¬
    • cond
    • select
    • F
    • T

Assumptions

⊦ T

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λP. P = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀x. (select y. y = x) = x

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r

⊦ cond = λt t1 t2. select x. ((t ⇔ T) ⇒ x = t1) ∧ ((t ⇔ F) ⇒ x = t2)

⊦ ∀t. ((T ⇔ t) ⇔ t) ∧ ((t ⇔ T) ⇔ t) ∧ ((F ⇔ t) ⇔ ¬t) ∧ ((t ⇔ F) ⇔ ¬t)

⊦ ∀t. (T ∧ t ⇔ t) ∧ (t ∧ T ⇔ t) ∧ (F ∧ t ⇔ F) ∧ (t ∧ F ⇔ F) ∧ (t ∧ t ⇔ t)

⊦ ∀t. (T ∨ t ⇔ T) ∧ (t ∨ T ⇔ T) ∧ (F ∨ t ⇔ t) ∧ (t ∨ F ⇔ t) ∧ (t ∨ t ⇔ t)

⊦ ∀t. (T ⇒ t ⇔ t) ∧ (t ⇒ T ⇔ T) ∧ (F ⇒ t ⇔ T) ∧ (t ⇒ t ⇔ T) ∧ (t ⇒ F ⇔ ¬t)