Package natural-fibonacci-exists: Existence of Fibonacci numbers
Information
| name | natural-fibonacci-exists |
| version | 1.40 |
| description | Existence of Fibonacci numbers |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | MIT |
| provenance | HOL Light theory extracted on 2015-05-05 |
| checksum | 73cea79a01bc97ade2e5905e3caaba956a06307a |
| requires | base |
| show | Data.Bool Number.Natural Relation |
Files
- Package tarball natural-fibonacci-exists-1.40.tgz
- Theory source file natural-fibonacci-exists.thy (included in the package tarball)
Theorems
⊦ ∀p. p 0 ∧ p 1 ∧ (∀n. p n ∧ p (n + 1) ⇒ p (n + 2)) ⇒ ∀n. p n
⊦ ∃f. f 0 = 0 ∧ f 1 = 1 ∧ ∀n. f (n + 2) = f (n + 1) + f n
⊦ ∀h.
(∀f g n. (∀m. m + 1 = n ∨ m + 2 = n ⇒ f m = g m) ⇒ h f n = h g n) ⇒
∃f. ∀n. f n = h f n
External Type Operators
- →
- bool
- Number
- Natural
- natural
- Natural
External Constants
- =
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- cond
- ⊥
- ⊤
- Bool
- Number
- Natural
- +
- -
- <
- ≤
- bit0
- bit1
- suc
- zero
- Natural
- Relation
- subrelation
- wellFounded
Assumptions
⊦ ⊤
⊦ wellFounded (<)
⊦ ¬⊥ ⇔ ⊤
⊦ ¬⊤ ⇔ ⊥
⊦ bit0 0 = 0
⊦ ∀n. 0 ≤ n
⊦ ∀n. n ≤ n
⊦ ∀n. n < suc n
⊦ ∀n. n ≤ suc n
⊦ ∀t. (∀x. t) ⇔ t
⊦ ∀t. (λx. t x) = 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 ⇔ t
⊦ ∀t. ⊤ ∨ t ⇔ ⊤
⊦ ∀t. t ∨ ⊥ ⇔ t
⊦ ∀n. 0 + n = n
⊦ ∀m. m + 0 = m
⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t
⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t
⊦ ∀t. t ⇒ ⊥ ⇔ ¬t
⊦ ∀n. bit1 n = suc (bit0 n)
⊦ ∀m n. n ≤ m + n
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)
⊦ ∀m. suc m = m + 1
⊦ ∀t1 t2. (if ⊥ then t1 else t2) = t2
⊦ ∀t1 t2. (if ⊤ then t1 else t2) = t1
⊦ ∀n. bit0 (suc n) = suc (suc (bit0 n))
⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1
⊦ ∀m n. m + n = n + m
⊦ ∀m n. m + n - n = m
⊦ ∀m n. ¬(m < n) ⇔ n ≤ m
⊦ ∀m n. suc m ≤ n ⇔ m < n
⊦ ∀m. m = 0 ∨ ∃n. m = suc n
⊦ ∀p. (∀b. p b) ⇔ p ⊤ ∧ p ⊥
⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤
⊦ ∀p. ¬(∀x. p x) ⇔ ∃x. ¬p x
⊦ ∀p. ¬(∃x. p x) ⇔ ∀x. ¬p x
⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q
⊦ ∀m n. m + suc n = suc (m + n)
⊦ ∀m n. suc m + n = suc (m + n)
⊦ ∀m n. suc m ≤ suc n ⇔ m ≤ n
⊦ ∀r s. subrelation r s ∧ wellFounded s ⇒ wellFounded r
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀m n. n ≤ m ⇒ m - n + n = m
⊦ ∀m n. m < n ⇔ ∃d. n = m + suc d
⊦ ∀p q. p ∧ (∃x. q x) ⇔ ∃x. p ∧ q x
⊦ ∀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 m n. m + p = n + p ⇔ m = n
⊦ ∀m n p. m + n ≤ m + p ⇔ n ≤ p
⊦ ∀m n p. m < n ∧ n < p ⇒ m < p
⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n
⊦ ∀r s. subrelation r s ⇔ ∀x y. r x y ⇒ s x y
⊦ ∀p. (∀n. (∀m. m < n ⇒ p m) ⇒ p n) ⇒ ∀n. p n
⊦ ∀p q. (∃x. p x) ∨ (∃x. q x) ⇔ ∃x. p x ∨ q x
⊦ ∀r.
wellFounded r ⇒
∀h.
(∀f g x. (∀z. r z x ⇒ f z = g z) ⇒ h f x = h g x) ⇒
∃f. ∀x. f x = h f x