Package list-filter: Definitions and theorems about the list filter function
Information
| name | list-filter |
| version | 1.12 |
| description | Definitions and theorems about the list filter function |
| author | Joe Hurd <joe@gilith.com> |
| license | MIT |
| show | Data.Bool Data.List Function |
Files
- Package tarball list-filter-1.12.tgz
- Theory file list-filter.thy (included in the package tarball)
Defined Constant
- Data
- List
- filter
- List
Theorems
⊦ ∀p l. Number.Natural.≤ (length (filter p l)) (length l)
⊦ ∀p l. Set.⊆ (toSet (filter p l)) (toSet l)
⊦ ∀P l1 l2. filter P (l1 @ l2) = filter P l1 @ filter P l2
⊦ ∀P f l. filter P (map f l) = map f (filter (P ∘ f) l)
⊦ (∀P. filter P [] = []) ∧
∀P h t. filter P (h :: t) = if P h then h :: filter P t else filter P t
Input Type Operators
- →
- bool
- Data
- List
- list
- List
- Number
- Natural
- Number.Natural.natural
- Natural
- Set
- Set.set
Input Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ¬
- cond
- F
- T
- List
- ::
- @
- []
- length
- map
- toSet
- Bool
- Function
- ∘
- Number
- Natural
- Number.Natural.≤
- Number.Natural.suc
- Number.Natural.zero
- Natural
- Set
- Set.∅
- Set.insert
- Set.⊆
- Set.∪
Assumptions
⊦ T
⊦ ∀n. Number.Natural.≤ n n
⊦ ∀s. Set.⊆ s s
⊦ F ⇔ ∀p. p
⊦ ∀n. Number.Natural.≤ n (Number.Natural.suc n)
⊦ (¬) = λp. p ⇒ F
⊦ (∃) = λP. P ((select) P)
⊦ ∀t. (∀x. t) ⇔ t
⊦ (∀) = λp. p = λx. T
⊦ ∀x. x = x ⇔ T
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ (∧) = λp q. (λf. f p q) = λf. f T T
⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q
⊦ ∀x s. Set.∪ (Set.insert x Set.∅) s = Set.insert x s
⊦ ∀m n.
Number.Natural.≤ (Number.Natural.suc m) (Number.Natural.suc n) ⇔
Number.Natural.≤ m n
⊦ ∀f g x. (f ∘ g) x = f (g x)
⊦ ∀m n p.
Number.Natural.≤ m n ∧ Number.Natural.≤ n p ⇒ Number.Natural.≤ m p
⊦ ∀s t u. Set.⊆ s t ∧ Set.⊆ t u ⇒ Set.⊆ s u
⊦ ∀t1 t2. (if T then t1 else t2) = t1 ∧ (if F then t1 else t2) = t2
⊦ length [] = 0 ∧ ∀h t. length (h :: t) = Number.Natural.suc (length t)
⊦ toSet [] = Set.∅ ∧ ∀h t. toSet (h :: t) = Set.insert h (toSet t)
⊦ (∀s t. Set.⊆ s (Set.∪ s t)) ∧ ∀s t. Set.⊆ s (Set.∪ t s)
⊦ ∀s t u. Set.⊆ (Set.∪ s t) u ⇔ Set.⊆ s u ∧ Set.⊆ t u
⊦ ∀P. P [] ∧ (∀a0 a1. P a1 ⇒ P (a0 :: a1)) ⇒ ∀x. P x
⊦ ∀P c x y. P (if c then x else y) ⇔ (c ⇒ P x) ∧ (¬c ⇒ P y)
⊦ ∀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
⊦ (∀f. map f [] = []) ∧ ∀f h t. map f (h :: t) = f h :: map f t
⊦ ∀t. (T ∧ t ⇔ t) ∧ (t ∧ T ⇔ t) ∧ (F ∧ t ⇔ F) ∧ (t ∧ F ⇔ F) ∧ (t ∧ t ⇔ t)