Package parser-stream-thm: Properties of parse streams

Information

name	parser-stream-thm
version	1.51
description	Properties of parse streams
author	Joe Hurd <joe@gilith.com>
license	MIT
provenance	HOL Light theory extracted on 2012-03-08
requires	bool function pair natural relation option list parser-stream-def
show	Data.Bool Data.List Data.Option Data.Pair Function Number.Natural Parser.Stream Relation

Files

Package tarball parser-stream-thm-1.51.tgz
Theory file parser-stream-thm.thy (included in the package tarball)

Theorems

⊦ wellFounded isProperSuffix

⊦ ¬(error = eof)

⊦ ∀x. isSuffix x x

⊦ ∀x. ¬isProperSuffix x x

⊦ ∀l. length (fromList l) = length l

⊦ ∀l. toList (fromList l) = some l

⊦ ∀a0' a1'. ¬(eof = stream a0' a1')

⊦ ∀a0' a1'. ¬(error = stream a0' a1')

⊦ ∀x y. isProperSuffix x y ⇒ isSuffix x y

⊦ ∀s. case toList s of none → ⊤ | some l → length l = length s

⊦ ∀x y. isProperSuffix x y ⇒ length x < length y

⊦ ∀x y. isSuffix x y ⇒ length x ≤ length y

⊦ ∀l s. length (append l s) = length l + length s

⊦ ∀x y z. append (x @ y) z = append x (append y z)

⊦ ∀x y z. isProperSuffix x y ∧ isProperSuffix y z ⇒ isProperSuffix x z

⊦ ∀x y z. isSuffix x y ∧ isSuffix y z ⇒ isSuffix x z

⊦ ∀l s.
toList (append l s) =
case toList s of none → none | some ls → some (l @ ls)

⊦ ∀x. x = error ∨ x = eof ∨ ∃a0 a1. x = stream a0 a1

⊦ ∀p. (∀x. (∀y. isProperSuffix y x ⇒ p y) ⇒ p x) ⇒ ∀x. p x

⊦ ∀a0 a1 a0' a1'. stream a0 a1 = stream a0' a1' ⇔ a0 = a0' ∧ a1 = a1'

⊦ ∀h.
(∀f g s. (∀s'. isProperSuffix s' s ⇒ f s' = g s') ⇒ h f s = h g s) ⇒
∃f. ∀s. f s = h f s

Input Type Operators

→
bool
Data
- List
  - list
- Option
  - option
- Pair
  - ×
Number
- Natural
  - natural
Parser
- Stream
  - stream

Input Constants

=
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∃!
  - ∨
  - ¬
  - ⊥
  - ⊤
- List
  - ::
  - @
  - []
  - length
- Option
  - case
  - none
  - some
- Pair
  - ,
Function
- id
Number
- Natural
  - *
  - +
  - <
  - ≤
  - bit0
  - bit1
  - even
  - suc
  - zero
Parser
- Stream
  - append
  - eof
  - error
  - fromList
  - isProperSuffix
  - isSuffix
  - length
  - stream
  - toList
Relation
- irreflexive
- measure
- subrelation
- wellFounded

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ length [] = 0

⊦ bit0 0 = 0

⊦ length eof = 0

⊦ length error = 0

⊦ toList error = none

⊦ ∀t. t ⇒ t

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ ∀m. wellFounded (measure m)

⊦ ⊥ ⇔ ∀p. p

⊦ toList eof = some []

⊦ ∀x. id x = x

⊦ ∀s. ¬isProperSuffix s eof

⊦ ∀s. ¬isProperSuffix s error

⊦ (¬) = λp. p ⇒ ⊥

⊦ ∀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 ∨ ⊤ ⇔ ⊤

⊦ ∀n. ¬(suc n = 0)

⊦ ∀n. 0 + n = n

⊦ ∀m. m + 0 = m

⊦ ∀l. [] @ l = l

⊦ ∀s. append [] s = s

⊦ ∀r. wellFounded r ⇒ irreflexive r

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀n. bit1 n = suc (bit0 n)

⊦ ∀l. fromList l = append l eof

⊦ ∀x. case none some x = x

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀n. even (suc n) ⇔ ¬even n

⊦ ∀m. m ≤ 0 ⇔ m = 0

⊦ ∀b f. case b f none = b

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

⊦ ∀f y. (let x ← y in f x) = f y

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

⊦ ∀m n. m < n ⇒ m ≤ n

⊦ ∀r x. irreflexive r ⇒ ¬r x x

⊦ ∀n. 2 * n = n + n

⊦ ∀h t. length (h :: t) = suc (length t)

⊦ ∀a s. length (stream a s) = suc (length s)

⊦ ∀m n. ¬(m < n ∧ n ≤ m)

⊦ ∀m n. ¬(m ≤ n ∧ n < m)

⊦ ∀m n. m < suc n ⇔ m ≤ n

⊦ ∀m n. suc m ≤ n ⇔ m < n

⊦ ∀x. x = none ∨ ∃a. x = some a

⊦ (∧) = 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

⊦ ∀t1 t2. ¬(t1 ⇒ t2) ⇔ t1 ∧ ¬t2

⊦ ∀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

⊦ ∀b f a. case b f (some a) = f a

⊦ ∀t1 t2. ¬(t1 ∧ t2) ⇔ ¬t1 ∨ ¬t2

⊦ ∀m n. even (m * n) ⇔ even m ∨ even n

⊦ ∀m n. even (m + n) ⇔ even m ⇔ even n

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

⊦ ∀m n. m ≤ n ⇔ m < n ∨ m = n

⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n

⊦ ∀s s'. isSuffix s s' ⇔ s = s' ∨ isProperSuffix s s'

⊦ ∀m x y. measure m x y ⇔ m x < m y

⊦ ∀p q. (∀x. p ∨ q x) ⇔ p ∨ ∀x. q x

⊦ ∀p q. (∃x. p ∧ q x) ⇔ p ∧ ∃x. 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

⊦ ∀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

⊦ ∀h t s. append (h :: t) s = stream h (append t s)

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

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

⊦ ∀l h t. (h :: t) @ l = h :: t @ l

⊦ ∀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

⊦ ∀a s.
toList (stream a s) =
case toList s of none → none | some l → some (a :: l)

⊦ ∀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

⊦ ∀s a s'. isProperSuffix s (stream a s') ⇔ s = s' ∨ isProperSuffix s s'

⊦ (∃!) = 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 ⇒ q x) ⇒ (∃x. p x) ⇒ ∃x. q x

⊦ ∀P. P [] ∧ (∀a0 a1. P a1 ⇒ P (a0 :: a1)) ⇒ ∀x. P x

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

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

⊦ ∀m n p. m * n < m * p ⇔ ¬(m = 0) ∧ n < p

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

⊦ ∀p1 p2 q1 q2. (p1 ⇒ p2) ∧ (q1 ⇒ q2) ⇒ p1 ∧ q1 ⇒ p2 ∧ q2

⊦ ∀p1 p2 q1 q2. (p2 ⇒ p1) ∧ (q1 ⇒ q2) ⇒ (p1 ⇒ q1) ⇒ p2 ⇒ q2

⊦ ∀P. P error ∧ P eof ∧ (∀a0 a1. P a1 ⇒ P (stream a0 a1)) ⇒ ∀x. P x

⊦ ∀r. wellFounded r ⇔ ∀p. (∀x. (∀y. r y x ⇒ p y) ⇒ p x) ⇒ ∀x. p x

⊦ ∀f0 f1 f2.
    ∃fn.
      fn error = f0 ∧ fn eof = f1 ∧
      ∀a0 a1. fn (stream a0 a1) = f2 a0 a1 (fn a1)