Package parser-stream-thm: Properties of parse streams

Information

Files

• Package tarball parser-stream-thm-1.90.tgz
• Theory source 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 = cons a0' a1')

⊦ ∀a0' a1'. ¬(error = cons 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

⊦ ∀f s. toList (map f s) = map (map f) (toList 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 = cons a₀ a₁

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

⊦ ∀a₀ a₁ a0' a1'. cons a₀ a₁ = cons 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

External Type Operators

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

External Constants

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

⊦ ∀f. map f [] = []

⊦ ∀f. map f none = none

⊦ ∀f. map f eof = eof

⊦ ∀f. map f error = error

⊦ ∀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 (cons 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

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

⊦ ∀f h t. map f (h :: t) = f h :: map f t

⊦ ∀f a s. map f (cons a s) = cons (f a) (map f s)

⊦ ∀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 (cons 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 [] ∧ (∀h t. p t ⇒ p (h :: t)) ⇒ ∀l. p l

⊦ ∀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

⊦ ∀a b a' b'. (a, b) = (a', b') ⇔ a = a' ∧ b = 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 (cons 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 (cons a₀ a₁) = f₂ a₀ a₁ (fn a₁)

OpenTheory package summary generated by opentheory 1.2 (release 20131210)