Package real: The real numbers

Information

name	real
version	1.50
description	The real numbers
author	Joe Leslie-Hurd <joe@gilith.com>
license	MIT
requires	bool function natural pair set
show	Data.Bool Data.Pair Function Number.Natural Number.Real Set

Files

Package tarball real-1.50.tgz
Theory file real.thy (included in the package tarball)

Defined Type Operator

Number
- Real
  - real

Defined Constants

Number
- Real
  - *
  - +
  - -
  - /
  - <
  - ≤
  - >
  - ≥
  - ↑
  - ~
  - abs
  - fromNatural
  - inv
  - max
  - min
  - sup

Theorems

⊦ ∀x. x ≤ x

⊦ ∀x. 0 + x = x

⊦ ∀x. x ↑ 0 = 1

⊦ ∀x. ~x + x = 0

⊦ ∀x. 1 * x = x

⊦ ∀x y. x > y ⇔ y < x

⊦ ∀x y. x ≥ y ⇔ y ≤ x

⊦ ∀x y. x * y = y * x

⊦ ∀x y. x + y = y + x

⊦ ∀x y. x ≤ y ∨ y ≤ x

⊦ ∀x y. x < y ⇔ ¬(y ≤ x)

⊦ ∀x y. x - y = x + ~y

⊦ ∀m n. fromNatural m = fromNatural n ⇔ m = n

⊦ ∀m n. fromNatural m ≤ fromNatural n ⇔ m ≤ n

⊦ ∀x. abs x = if 0 ≤ x then x else ~x

⊦ ∀m n. fromNatural m * fromNatural n = fromNatural (m * n)

⊦ ∀m n. fromNatural m + fromNatural n = fromNatural (m + n)

⊦ ∀x n. x ↑ suc n = x * x ↑ n

⊦ ∀m n. max m n = if m ≤ n then n else m

⊦ ∀m n. min m n = if m ≤ n then m else n

⊦ ∀x y. x ≤ y ∧ y ≤ x ⇔ x = y

⊦ ∀x y z. y ≤ z ⇒ x + y ≤ x + z

⊦ ∀x y z. x * (y * z) = x * y * z

⊦ ∀x y z. x + (y + z) = x + y + z

⊦ ∀x y z. x ≤ y ∧ y ≤ z ⇒ x ≤ z

⊦ ∀x. ¬(x = 0) ⇒ inv x * x = 1

⊦ ∀x y. ¬(y = 0) ⇒ x / y = x * inv y

⊦ ∀x y z. x * (y + z) = x * y + x * z

⊦ ∀x y. 0 ≤ x ∧ 0 ≤ y ⇒ 0 ≤ x * y

⊦ ∀s x. ¬(s = ∅) ∧ (∃m. ∀x. x ∈ s ⇒ x ≤ m) ∧ x ∈ s ⇒ x ≤ sup s

⊦ ∀s m.
¬(s = ∅) ∧ (∃m. ∀x. x ∈ s ⇒ x ≤ m) ∧ (∀x. x ∈ s ⇒ x ≤ m) ⇒ sup s ≤ m

⊦ ∀p.
(∃x. p x) ∧ (∃m. ∀x. p x ⇒ x ≤ m) ⇒
∃s. (∀x. p x ⇒ x ≤ s) ∧ ∀m. (∀x. p x ⇒ x ≤ m) ⇒ s ≤ m

Input Type Operators

→
bool
Data
- Pair
  - ×
Number
- Natural
  - natural
Set
- set

Input Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∃!
  - ∨
  - ¬
  - cond
  - ⊥
  - ⊤
- Pair
  - ,
  - fst
  - snd
Function
- id
Number
- Natural
  - *
  - +
  - <
  - ≤
  - bit0
  - bit1
  - distance
  - div
  - mod
  - suc
  - zero
Set
- ∅
- fromPredicate
- ∈

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ bit0 0 = 0

⊦ ∀t. t ⇒ t

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ ⊥ ⇔ ∀p. p

⊦ ∀x. id x = x

⊦ ∀t. t ∨ ¬t

⊦ ∀m. ¬(m < 0)

⊦ ∀n. ¬(n < n)

⊦ (¬) = λp. p ⇒ ⊥

⊦ (∃) = λp. p ((select) p)

⊦ ∀a. ∃x. x = a

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀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. ⊥ ∨ t ⇔ t

⊦ ∀t. ⊤ ∨ t ⇔ ⊤

⊦ ∀t. t ∨ ⊥ ⇔ t

⊦ ∀t. t ∨ ⊤ ⇔ ⊤

⊦ ∀n. ¬(suc n = 0)

⊦ ∀n. 0 * n = 0

⊦ ∀m. m * 0 = 0

⊦ ∀n. 0 + n = n

⊦ ∀m. m + 0 = m

⊦ ∀n. distance 0 n = n

⊦ ∀n. distance n 0 = n

⊦ ∀n. distance n n = 0

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀n. bit1 n = suc (bit0 n)

⊦ ∀m. m * 1 = m

⊦ ∀m. 1 * m = m

⊦ ∀x. (select y. y = x) = x

⊦ ∀m n. m ≤ m + n

⊦ ∀m n. n ≤ m + n

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀m. suc m = m + 1

⊦ ∀m. m ≤ 0 ⇔ m = 0

⊦ ∀xy. (fst xy, snd xy) = xy

⊦ ∀t₁ t₂. (if ⊥ then t₁ else t₂) = t₂

⊦ ∀t₁ t₂. (if ⊤ then t₁ else t₂) = t₁

⊦ ∀x y. fst (x, y) = x

⊦ ∀x y. snd (x, y) = y

⊦ ∀p x. p x ⇒ p ((select) p)

⊦ ∀n. bit0 (suc n) = suc (suc (bit0 n))

⊦ ∀f y. (let x ← y in f x) = f y

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀x y. x = y ⇒ y = x

⊦ ∀t₁ t₂. t₁ ∨ t₂ ⇔ t₂ ∨ t₁

⊦ ∀a b. (a ⇔ b) ⇒ a ⇒ b

⊦ ∀m n. m * n = n * m

⊦ ∀m n. m + n = n + m

⊦ ∀m n. distance m n = distance n m

⊦ ∀m n. m = n ⇒ m ≤ n

⊦ ∀m n. m < n ⇒ m ≤ n

⊦ ∀m n. m ≤ n ∨ n ≤ m

⊦ ∀m n. distance m n ≤ m + n

⊦ ∀m n. distance m (m + n) = n

⊦ ∀m n. distance (m + n) m = n

⊦ ∀n. 2 * n = n + n

⊦ ∀m n. ¬(m < n) ⇔ n ≤ m

⊦ ∀m n. ¬(m ≤ n) ⇔ n < m

⊦ ∀m n. suc m ≤ n ⇔ m < n

⊦ ∀s. (∃x. x ∈ s) ⇔ ¬(s = ∅)

⊦ (∧) = 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

⊦ ∀m n. distance m n = 0 ⇔ m = n

⊦ ∀t₁ t₂. ¬(t₁ ∨ t₂) ⇔ ¬t₁ ∧ ¬t₂

⊦ ∀m n. m * suc n = m + m * n

⊦ ∀m n. suc m * n = m * n + n

⊦ ∀m n. ¬(n = 0) ⇒ m mod n < n

⊦ ∀p. (∀xy. p xy) ⇔ ∀x y. p (x, y)

⊦ ∀m n. m ≤ n ⇔ ∃d. n = m + d

⊦ ∀a b. (∀n. a * n ≤ b) ⇔ a = 0

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g

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

⊦ ∀p. (∀x y. p x y) ⇔ ∀y x. p x y

⊦ ∀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. p ∨ (∃x. q x) ⇔ ∃x. p ∨ q x

⊦ ∀m n. m < suc n ⇔ m = n ∨ m < n

⊦ ∀p q. (∀x. p x ∨ q) ⇔ (∀x. p x) ∨ q

⊦ ∀p q. (∃x. p x) ∧ q ⇔ ∃x. p x ∧ q

⊦ ∀p q. (∀x. p x) ∨ q ⇔ ∀x. p x ∨ q

⊦ ∀p q. (∃x. p x) ∨ q ⇔ ∃x. p x ∨ q

⊦ ∀p q r. p ⇒ q ⇒ r ⇔ p ∧ q ⇒ r

⊦ ∀t₁ t₂ t₃. (t₁ ∧ t₂) ∧ t₃ ⇔ t₁ ∧ t₂ ∧ t₃

⊦ ∀t₁ t₂ t₃. (t₁ ∨ t₂) ∨ t₃ ⇔ t₁ ∨ t₂ ∨ t₃

⊦ ∀m n p. distance m p ≤ distance m n + distance n p

⊦ ∀m n p. m * (n * p) = n * (m * p)

⊦ ∀m n p. m * (n * p) = m * n * p

⊦ ∀m n p. m + (n + p) = m + n + p

⊦ ∀m n p. m + n = m + p ⇔ n = p

⊦ ∀m n p. m + n ≤ m + p ⇔ n ≤ p

⊦ ∀m n p. m + p ≤ n + p ⇔ m ≤ n

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

⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p

⊦ ∀p x. (∀y. p y ⇔ y = x) ⇒ (select) p = x

⊦ ∀p. (∀x. ∃y. p x y) ⇔ ∃y. ∀x. p x (y x)

⊦ ∀m n. m ≤ suc n ⇔ m = suc n ∨ m ≤ n

⊦ ∀m n. m + n = 0 ⇔ m = 0 ∧ n = 0

⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n

⊦ ∀p x. x ∈ { y. y | p y } ⇔ p x

⊦ ∀p q r. p ⇒ q ∧ r ⇔ (p ⇒ q) ∧ (p ⇒ r)

⊦ ∀m n p. m * (n + p) = m * n + m * p

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

⊦ ∀m n p. (m + n) * p = m * p + n * p

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

⊦ (∃!) = λp. (∃) p ∧ ∀x y. p x ∧ p y ⇒ x = y

⊦ ∀p q. (∀x. p x ∧ q x) ⇔ (∀x. p x) ∧ ∀x. q x

⊦ ∀p q. (∃x. p x ∨ q x) ⇔ (∃x. p x) ∨ ∃x. q x

⊦ ∀p q. (∀x. p x) ∧ (∀x. q x) ⇔ ∀x. p x ∧ q x

⊦ ∀p q. (∃x. p x) ∨ (∃x. q x) ⇔ ∃x. p x ∨ q x

⊦ ∀e f. ∃!fn. fn 0 = e ∧ ∀n. fn (suc n) = f (fn n) n

⊦ ∀a b c. (∀n. a * n ≤ b * n + c) ⇔ a ≤ b

⊦ ∀m n. ¬(n = 0) ⇒ (m div n) * n + m mod n = m

⊦ ∀m n p. m * n ≤ m * p ⇔ m = 0 ∨ n ≤ p

⊦ ∀m n p. m * p ≤ n * p ⇔ m ≤ n ∨ p = 0

⊦ ∀x y a b. (x, y) = (a, b) ⇔ x = a ∧ y = b

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

⊦ ∀m n p q. m < p ∧ n < q ⇒ m + n < p + q

⊦ ∀m n p q. m ≤ n ∧ p ≤ q ⇒ m * p ≤ n * q

⊦ ∀m n p q. m ≤ p ∧ n ≤ q ⇒ m + n ≤ p + q

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

⊦ ∀m n p. distance m n ≤ p ⇔ m ≤ n + p ∧ n ≤ m + p

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

⊦ ∀p. (∃b. ∀n. p n ≤ b) ⇔ ∃a b. ∀n. n * p n ≤ a * n + b

⊦ ∀p. (∃n. p n) ∧ (∃m. ∀n. p n ⇒ n ≤ m) ⇔ ∃m. p m ∧ ∀n. p n ⇒ n ≤ m

⊦ ∀p q. (∃b. ∀i. p i ≤ q i + b) ⇔ ∃b n. ∀i. n ≤ i ⇒ p i ≤ q i + b

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

⊦ ∀p a b.
p 0 0 = 0 ∧ (∀m n. p m n ≤ a * (m + n) + b) ⇒
∃c. ∀m n. p m n ≤ c * (m + n)