Package list-quant: Definitions and theorems about list quantifiers

Information

Files

• Package tarball list-quant-1.9.tgz
• Theory file list-quant.thy (included in the package tarball)

Defined Constants

• Data
  • List
    • all
    • exists

Theorems

⊦ ∀l. all (λx. T) l

⊦ ∀P l. ¬all P l ⇔ exists (λx. ¬P x) l

⊦ ∀P l. ¬exists P l ⇔ all (λx. ¬P x) l

⊦ ∀P f l. all P (map f l) ⇔ all (P o f) l

⊦ ∀P f l. exists P (map f l) ⇔ exists (P o f) l

⊦ ∀P l. (∀x. all (P x) l) ⇔ all (λs. ∀x. P x s) l

⊦ ∀P l. (∃x. exists (P x) l) ⇔ exists (λs. ∃x. P x s) l

⊦ ∀P l1 l2. all P (l1 @ l2) ⇔ all P l1 ∧ all P l2

⊦ ∀P Q l. (∀x. P x ⇒ Q x) ⇒ all P l ⇒ all Q l

⊦ ∀f g l. all (λx. f x = g x) l ⇒ map f l = map g l

⊦ ∀P Q l. all P l ∧ all Q l ⇔ all (λx. P x ∧ Q x) l

⊦ ∀P Q l. all (λx. P x ⇒ Q x) l ∧ all P l ⇒ all Q l

⊦ (∀P. all P [] ⇔ T) ∧ ∀P h t. all P (h :: t) ⇔ P h ∧ all P t

⊦ (∀P. exists P [] ⇔ F) ∧ ∀P h t. exists P (h :: t) ⇔ P h ∨ exists P t

Input Type Operators

• →
• bool
• Data
  • List
    • list

Input Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ~
    • F
    • T
  • List
    • ::
    • @
    • []
    • map
• Function
  • id
  • o

Assumptions

⊦ T

⊦ ∀x. id x = x

⊦ (∃) = λP. P ((select) P)

⊦ ∀a. ∃x. x = a

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (∃x. t) ⇔ t

⊦ ∀t. (λx. t x) = 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

⊦ (∧) = 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 o g) x = f (g x)

⊦ ∀f g. f = g ⇔ ∀x. f x = g x

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

⊦ ∀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

⊦ ∀t1 t2 t3. t1 ∧ t2 ∧ t3 ⇔ (t1 ∧ t2) ∧ t3

⊦ ∀P. (∀x. ∃y. P x y) ⇔ ∃y. ∀x. P x (y x)

⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀P Q. (∀x. P x ∧ Q x) ⇔ (∀x. P x) ∧ ∀x. Q x

⊦ ∀P Q. (∃x. P x ∨ Q x) ⇔ (∃x. P x) ∨ ∃x. Q x

⊦ ∀P Q. (∃x. P x) ∨ (∃x. Q x) ⇔ ∃x. P x ∨ Q x

⊦ ∀P. P [] ∧ (∀a0 a1. P a1 ⇒ P (a0 :: a1)) ⇒ ∀x. P x

⊦ ∀NIL' CONS'.
    ∃fn. fn [] = NIL' ∧ ∀a0 a1. fn (a0 :: a1) = CONS' a0 a1 (fn a1)

⊦ (∀l. [] @ l = l) ∧ ∀l h t. (h :: t) @ l = h :: t @ l

⊦ ∀t1 t2. (¬(t1 ∧ t2) ⇔ ¬t1 ∨ ¬t2) ∧ (¬(t1 ∨ t2) ⇔ ¬t1 ∧ ¬t2)

⊦ (∀f. map f [] = []) ∧ ∀f h t. map f (h :: t) = f h :: map f t

⊦ ∀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)

⊦ ∀p q r.
    (p ∧ q ⇔ q ∧ p) ∧ ((p ∧ q) ∧ r ⇔ p ∧ q ∧ r) ∧ (p ∧ q ∧ r ⇔ q ∧ p ∧ r) ∧
    (p ∧ p ⇔ p) ∧ (p ∧ p ∧ q ⇔ p ∧ q)

⊦ ∀p q r.
    (p ∨ q ⇔ q ∨ p) ∧ ((p ∨ q) ∨ r ⇔ p ∨ q ∨ r) ∧ (p ∨ q ∨ r ⇔ q ∨ p ∨ r) ∧
    (p ∨ p ⇔ p) ∧ (p ∨ p ∨ q ⇔ p ∨ q)