name | function-thm |
version | 1.8 |
description | function-thm |
author | Joe Hurd <joe@gilith.com> |
license | HOLLight |
provenance | HOL Light theory extracted on 2011-07-20 |
show | Data.Bool |
⊦ ∀x. Function.id x = x
⊦ ∀f. Function.o Function.id f = f ∧ Function.o f Function.id = f
⊦ ∀f g x. Function.o f g x = f (g x)
⊦ ∀f g h. Function.o f (Function.o g h) = Function.o (Function.o f g) h
⊦ ∀f g. (∀x. ∃y. g y = f x) ⇔ ∃h. f = Function.o g h
⊦ ∀f. (∀y. ∃x. f x = y) ⇔ ∀P. (∀x. P (f x)) ⇔ ∀y. P y
⊦ ∀f. (∀y. ∃x. f x = y) ⇔ ∀P. (∃x. P (f x)) ⇔ ∃y. P y
⊦ ∀f g. (∀x y. g x = g y ⇒ f x = f y) ⇔ ∃h. f = Function.o h g
⊦ T
⊦ Function.id = λx. x
⊦ (∃) = λP. P ((select) P)
⊦ ∀t. (∀x. t) ⇔ t
⊦ (∀) = λp. p = λx. T
⊦ ∀x. x = x ⇔ T
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)
⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)
⊦ ∀f y. (let x ← y ∈ f x) = f y
⊦ ∀x y. x = y ⇔ y = x
⊦ ∀x y. x = y ⇒ y = x
⊦ ∀f g. Function.o f g = λx. f (g x)
⊦ (∧) = 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. f = g ⇔ ∀x. f x = g x
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀P. (∀x. ∃y. P x y) ⇔ ∃y. ∀x. P x (y x)
⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)
⊦ ∀t. ((T ⇔ t) ⇔ t) ∧ ((t ⇔ T) ⇔ t) ∧ ((F ⇔ t) ⇔ ¬t) ∧ ((t ⇔ F) ⇔ ¬t)
⊦ ∀t. (T ⇒ t ⇔ t) ∧ (t ⇒ T ⇔ T) ∧ (F ⇒ t ⇔ T) ∧ (t ⇒ t ⇔ T) ∧ (t ⇒ F ⇔ ¬t)