Package natural-distance-thm: natural-distance-thm

Information

name	natural-distance-thm
version	1.10
description	natural-distance-thm
author	Joe Hurd <joe@gilith.com>
license	HOLLight
provenance	HOL Light theory extracted on 2011-09-21
show	Data.Bool

Files

Package tarball natural-distance-thm-1.10.tgz
Theory file natural-distance-thm.thy (included in the package tarball)

Theorems

⊦ ∀n. Number.Natural.distance 0 n = n

⊦ ∀n. Number.Natural.distance n 0 = n

⊦ ∀n. Number.Natural.distance n n = 0

⊦ ∀m n. Number.Natural.distance m n = Number.Natural.distance n m

⊦ ∀m n.
Number.Natural.≤ (Number.Natural.distance m n) (Number.Natural.+ m n)

⊦ ∀m n. Number.Natural.distance m (Number.Natural.+ m n) = n

⊦ ∀m n. Number.Natural.distance (Number.Natural.+ m n) m = n

⊦ ∀m n. Number.Natural.distance m n = 0 ⇔ m = n

⊦ ∀m n p.
    Number.Natural.≤ (Number.Natural.distance m p)
      (Number.Natural.+ (Number.Natural.distance m n)
         (Number.Natural.distance n p))

⊦ ∀m n p.
Number.Natural.distance (Number.Natural.+ m n) (Number.Natural.+ m p) =
Number.Natural.distance n p

⊦ ∀p m n.
Number.Natural.distance (Number.Natural.+ m p) (Number.Natural.+ n p) =
Number.Natural.distance m n

⊦ ∀m n p.
Number.Natural.* m (Number.Natural.distance n p) =
Number.Natural.distance (Number.Natural.* m n) (Number.Natural.* m p)

⊦ ∀p m n.
Number.Natural.* (Number.Natural.distance m n) p =
Number.Natural.distance (Number.Natural.* m p) (Number.Natural.* n p)

⊦ ∀m n p q.
    Number.Natural.≤ (Number.Natural.distance m p)
      (Number.Natural.+
         (Number.Natural.distance (Number.Natural.+ m n)
            (Number.Natural.+ p q)) (Number.Natural.distance n q))

⊦ ∀m n p q.
    Number.Natural.≤
      (Number.Natural.distance (Number.Natural.+ m n)
         (Number.Natural.+ p q))
      (Number.Natural.+ (Number.Natural.distance m p)
         (Number.Natural.distance n q))

⊦ ∀m n p q.
    Number.Natural.≤
      (Number.Natural.+ (Number.Natural.distance m n)
         (Number.Natural.distance n p)) q ⇒
    Number.Natural.≤ (Number.Natural.distance m p) q

⊦ ∀m n p.
    Number.Natural.≤ (Number.Natural.distance m n) p ⇔
    Number.Natural.≤ m (Number.Natural.+ n p) ∧
    Number.Natural.≤ n (Number.Natural.+ m p)

⊦ ∀P x y.
P (Number.Natural.distance x y) ⇔
∀d. (x = Number.Natural.+ y d ⇒ P d) ∧ (y = Number.Natural.+ x d ⇒ P d)

⊦ ∀m n p q r s.
    Number.Natural.≤ (Number.Natural.distance m n) r ∧
    Number.Natural.≤ (Number.Natural.distance p q) s ⇒
    Number.Natural.≤ (Number.Natural.distance m p)
      (Number.Natural.+ (Number.Natural.distance n q)
         (Number.Natural.+ r s))

Input Type Operators

→
bool
Number
- Natural
  - Number.Natural.natural

Input Constants

=
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - ¬
  - cond
  - F
  - T
Number
- Natural
  - Number.Natural.*
  - Number.Natural.+
  - Number.Natural.-
  - Number.Natural.<
  - Number.Natural.≤
  - Number.Natural.distance
  - Number.Natural.suc
  - Number.Natural.zero

Assumptions

⊦ T

⊦ ∀n. Number.Natural.≤ 0 n

⊦ ∀n. Number.Natural.≤ n n

⊦ F ⇔ ∀p. p

⊦ ∀t. t ∨ ¬t

⊦ (¬) = λp. p ⇒ F

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀n. ¬(Number.Natural.suc n = 0)

⊦ ∀m. Number.Natural.- m 0 = m

⊦ ∀n. Number.Natural.- n n = 0

⊦ ∀m n. Number.Natural.≤ m (Number.Natural.+ m n)

⊦ ∀m n. Number.Natural.≤ n (Number.Natural.+ m n)

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀m n. Number.Natural.* m n = Number.Natural.* n m

⊦ ∀m n. Number.Natural.+ m n = Number.Natural.+ n m

⊦ ∀m n. Number.Natural.≤ m n ∨ Number.Natural.≤ n m

⊦ ∀m n. Number.Natural.- (Number.Natural.+ m n) m = n

⊦ ∀m n. ¬Number.Natural.< m n ⇔ Number.Natural.≤ n m

⊦ (∧) = 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.+ m n = m ⇔ n = 0

⊦ ∀m n. Number.Natural.≤ m n ⇔ ∃d. n = Number.Natural.+ m d

⊦ ∀P a. (∀x. x = a ⇒ P x) ⇔ P a

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

⊦ ∀m n. Number.Natural.≤ m n ∧ Number.Natural.≤ n m ⇔ m = n

⊦ ∀m n.
Number.Natural.< m n ⇔
∃d. n = Number.Natural.+ m (Number.Natural.suc d)

⊦ ∀m n p.
Number.Natural.+ m (Number.Natural.+ n p) =
Number.Natural.+ (Number.Natural.+ m n) p

⊦ ∀m n p. Number.Natural.+ m n = Number.Natural.+ m p ⇔ n = p

⊦ ∀m n p.
Number.Natural.≤ (Number.Natural.+ m n) (Number.Natural.+ m p) ⇔
Number.Natural.≤ n p

⊦ ∀m n p.
Number.Natural.≤ m n ∧ Number.Natural.≤ n p ⇒ Number.Natural.≤ m p

⊦ ∀t1 t2. (if T then t1 else t2) = t1 ∧ (if F then t1 else t2) = t2

⊦ ∀m n. Number.Natural.≤ n m ⇒ (Number.Natural.- m n = 0 ⇔ m = n)

⊦ ∀m n.
    Number.Natural.distance m n =
    if Number.Natural.≤ m n then Number.Natural.- n m
    else Number.Natural.- m n

⊦ ∀m n p.
    Number.Natural.≤ p n ⇒
    Number.Natural.- (Number.Natural.+ m n) (Number.Natural.+ m p) =
    Number.Natural.- n p

⊦ ∀m n p.
Number.Natural.≤ (Number.Natural.* m n) (Number.Natural.* m p) ⇔
m = 0 ∨ Number.Natural.≤ n p

⊦ (∀n. Number.Natural.* 0 n = 0) ∧
  ∀m n.
    Number.Natural.* (Number.Natural.suc m) n =
    Number.Natural.+ (Number.Natural.* m n) n

⊦ ∀m n p q.
Number.Natural.≤ m p ∧ Number.Natural.≤ n q ⇒
Number.Natural.≤ (Number.Natural.+ m n) (Number.Natural.+ p q)

⊦ ∀m n p.
    Number.Natural.≤ p n ⇒
    Number.Natural.* m (Number.Natural.- n p) =
    Number.Natural.- (Number.Natural.* m n) (Number.Natural.* m p)

⊦ ∀P c x y. P (if c then x else y) ⇔ (c ⇒ P x) ∧ (¬c ⇒ P y)

⊦ ∀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)

⊦ ∀m n p.
    Number.Natural.+ m n = Number.Natural.+ n m ∧
    Number.Natural.+ (Number.Natural.+ m n) p =
    Number.Natural.+ m (Number.Natural.+ n p) ∧
    Number.Natural.+ m (Number.Natural.+ n p) =
    Number.Natural.+ n (Number.Natural.+ m p)

⊦ (∀n. Number.Natural.+ 0 n = n) ∧ (∀m. Number.Natural.+ m 0 = m) ∧
  (∀m n.
     Number.Natural.+ (Number.Natural.suc m) n =
     Number.Natural.suc (Number.Natural.+ m n)) ∧
  ∀m n.
    Number.Natural.+ m (Number.Natural.suc n) =
    Number.Natural.suc (Number.Natural.+ m n)