Package list-length-thm: list-length-thm
Information
| name | list-length-thm |
| version | 1.12 |
| description | list-length-thm |
| author | Joe Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2011-09-21 |
| show | Data.Bool |
Files
- Package tarball list-length-thm-1.12.tgz
- Theory file list-length-thm.thy (included in the package tarball)
Theorems
⊦ ∀l. Data.List.length l = 0 ⇔ l = Data.List.[]
⊦ ∀l.
¬(l = Data.List.[]) ⇒
Data.List.length (Data.List.tail l) =
Number.Natural.- (Data.List.length l) 1
⊦ ∀l n.
Data.List.length l = Number.Natural.suc n ⇔
∃h t. l = Data.List.:: h t ∧ Data.List.length t = n
Input Type Operators
- →
- bool
- Data
- List
- Data.List.list
- List
- Number
- Natural
- Number.Natural.natural
- Natural
Input Constants
- =
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- F
- T
- List
- Data.List.::
- Data.List.[]
- Data.List.length
- Data.List.tail
- Bool
- Number
- Natural
- Number.Natural.-
- Number.Natural.bit1
- Number.Natural.suc
- Number.Natural.zero
- Natural
Assumptions
⊦ T
⊦ F ⇔ ∀p. p
⊦ (¬) = λp. p ⇒ F
⊦ ∀t. (∀x. t) ⇔ t
⊦ ∀t. (∃x. t) ⇔ t
⊦ ∀t. (λx. t x) = t
⊦ (∀) = λp. p = λx. T
⊦ ∀x. x = x ⇔ T
⊦ ∀n. ¬(Number.Natural.suc n = 0)
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)
⊦ ∀n. Number.Natural.- (Number.Natural.suc n) 1 = n
⊦ ∀h t. ¬(Data.List.:: h t = Data.List.[])
⊦ ∀h t. Data.List.tail (Data.List.:: h t) = t
⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)
⊦ (∧) = λp q. (λf. f p q) = λf. f T T
⊦ ∀P. ¬(∃x. P x) ⇔ ∀x. ¬P x
⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q
⊦ ∀m n. Number.Natural.suc m = Number.Natural.suc n ⇔ m = n
⊦ (∨) = 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
⊦ ∀t1 t2 t3. t1 ∨ t2 ∨ t3 ⇔ (t1 ∨ t2) ∨ t3
⊦ Data.List.length Data.List.[] = 0 ∧
∀h t.
Data.List.length (Data.List.:: h t) =
Number.Natural.suc (Data.List.length t)
⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)
⊦ ∀P. P Data.List.[] ∧ (∀a0 a1. P a1 ⇒ P (Data.List.:: a0 a1)) ⇒ ∀x. P x
⊦ ∀h1 h2 t1 t2. Data.List.:: h1 t1 = Data.List.:: h2 t2 ⇔ h1 = h2 ∧ t1 = 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)