Package list-append: Appending lists
Information
| name | list-append |
| version | 1.52 |
| description | Appending lists |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | MIT |
| requires | bool list-def list-dest list-length list-set list-thm natural set |
| show | Data.Bool Data.List Number.Natural Set |
Files
- Package tarball list-append-1.52.tgz
- Theory source file list-append.thy (included in the package tarball)
Defined Constants
- Data
- List
- @
- concat
- List
Theorems
⊦ concat [] = []
⊦ ∀l. [] @ l = l
⊦ ∀l. l @ [] = l
⊦ ∀l. null (concat l) ⇔ all null l
⊦ ∀h t. (h :: []) @ t = h :: t
⊦ ∀h t. concat (h :: t) = h @ concat t
⊦ ∀l1 l2. null (l1 @ l2) ⇔ null l1 ∧ null l2
⊦ ∀l1 l2. length (l1 @ l2) = length l1 + length l2
⊦ ∀l1 l2. toSet (l1 @ l2) = toSet l1 ∪ toSet l2
⊦ ∀l h t. (h :: t) @ l = h :: t @ l
⊦ ∀l1 l2 l3. l1 @ l2 @ l3 = (l1 @ l2) @ l3
⊦ ∀l l1 l2. l @ l1 = l @ l2 ⇔ l1 = l2
⊦ ∀l l1 l2. l1 @ l = l2 @ l ⇔ l1 = l2
⊦ ∀l l1 l2. l @ l1 = l @ l2 ⇒ l1 = l2
⊦ ∀l l1 l2. l1 @ l = l2 @ l ⇒ l1 = l2
⊦ ∀l1 l2. l1 @ l2 = [] ⇔ l1 = [] ∧ l2 = []
⊦ ∀l1 l2 x. member x (l1 @ l2) ⇔ member x l1 ∨ member x l2
⊦ ∀p l1 l2. all p (l1 @ l2) ⇔ all p l1 ∧ all p l2
⊦ ∀p l1 l2. any p (l1 @ l2) ⇔ any p l1 ∨ any p l2
⊦ ∀x l. member x l ⇔ ∃l1 l2. ¬member x l1 ∧ l = l1 @ x :: l2
⊦ ∀l1 l1' l2 l2'. length l1 = length l1' ∧ l1 @ l2 = l1' @ l2' ⇒ l1 = l1'
⊦ ∀l1 l1' l2 l2'. length l2 = length l2' ∧ l1 @ l2 = l1' @ l2' ⇒ l2 = l2'
External Type Operators
- →
- bool
- Data
- List
- list
- List
- Number
- Natural
- natural
- Natural
- Set
- set
External Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- ⊥
- ⊤
- List
- ::
- []
- all
- any
- length
- member
- null
- toSet
- Bool
- Number
- Natural
- +
- suc
- zero
- Natural
- Set
- ∅
- insert
- ∈
- ∪
Assumptions
⊦ ⊤
⊦ null []
⊦ ¬⊥ ⇔ ⊤
⊦ length [] = 0
⊦ toSet [] = ∅
⊦ ∀t. t ⇒ t
⊦ ∀p. all p []
⊦ ⊥ ⇔ ∀p. p
⊦ ∀x. ¬member x []
⊦ ∀t. t ∨ ¬t
⊦ (¬) = λp. p ⇒ ⊥
⊦ (∃) = λp. p ((select) p)
⊦ ∀t. (∀x. t) ⇔ 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 ⇔ ⊤
⊦ ∀n. ¬(suc n = 0)
⊦ ∀n. 0 + n = n
⊦ ∀s. ∅ ∪ s = s
⊦ ∀l. null l ⇔ l = []
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀l. length l = 0 ⇔ null l
⊦ ∀h t. length (h :: t) = suc (length t)
⊦ ∀l x. member x l ⇔ x ∈ toSet l
⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤
⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q
⊦ ∀h t. toSet (h :: t) = insert h (toSet t)
⊦ ∀x s. insert x ∅ ∪ s = insert x s
⊦ ∀m n. suc m + n = suc (m + n)
⊦ ∀m n. suc m = suc n ⇔ m = n
⊦ ∀m n. m + n = n ⇔ m = 0
⊦ ∀t1 t2. ¬(t1 ∨ t2) ⇔ ¬t1 ∧ ¬t2
⊦ ∀p l. ¬any (λx. ¬p x) l ⇔ all p l
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀p. (∀x y. p x y) ⇔ ∀y x. p x y
⊦ ∀p m n. m + p = n + p ⇔ m = n
⊦ ∀s t u. s ∪ t ∪ u = s ∪ (t ∪ u)
⊦ ∀p l. any p l ⇔ ∃x. x ∈ toSet l ∧ p x
⊦ ∀r. (∀x. ∃y. r x y) ⇔ ∃f. ∀x. r x (f x)
⊦ ∀m n. m + n = 0 ⇔ m = 0 ∧ n = 0
⊦ ∀p h t. all p (h :: t) ⇔ p h ∧ all p t
⊦ ∀x h t. member x (h :: t) ⇔ x = h ∨ member x t
⊦ ∀p q r. (p ∨ q) ∧ r ⇔ p ∧ r ∨ q ∧ r
⊦ ∀s t x. x ∈ s ∪ t ⇔ x ∈ s ∨ x ∈ t
⊦ ∀p q. (∃x. p x ∨ q x) ⇔ (∃x. p x) ∨ ∃x. q x
⊦ ∀p. p [] ∧ (∀h t. p t ⇒ p (h :: t)) ⇒ ∀l. p l
⊦ ∀h1 h2 t1 t2. h1 :: t1 = h2 :: t2 ⇔ h1 = h2 ∧ t1 = t2
⊦ ∀b f. ∃fn. fn [] = b ∧ ∀h t. fn (h :: t) = f h t (fn t)