Package modular: Parametric theory of modular arithmetic

Information

Files

• Package tarball modular-1.1.tgz
• Theory file modular.thy (included in the package tarball)

Defined Type Operator

• Number
  • Modular
    • modular

Defined Constants

• Number
  • Modular
    • *
    • +
    • -
    • <
    • ≤
    • ¬
    • fromNatural
    • toNatural

Theorems

⊦ ∀x. Number.Natural.< (toNatural x) modulus

⊦ ∀x. fromNatural (toNatural x) = x

⊦ ∀n. Number.Natural.< (Number.Natural.mod n modulus) modulus

⊦ ∀x. Number.Natural.div (toNatural x) modulus = 0

⊦ ∀x. toNatural (fromNatural x) = Number.Natural.mod x modulus

⊦ ∀x. ¬x = fromNatural (Number.Natural.- modulus (toNatural x))

⊦ ∀n. Number.Natural.< n modulus ⇒ Number.Natural.mod n modulus = n

⊦ ∀n.
    Number.Natural.mod (Number.Natural.mod n modulus) modulus =
    Number.Natural.mod n modulus

⊦ ∀x y. x < y ⇔ ¬(y ≤ x)

⊦ ∀x y. x - y = x + ¬y

⊦ ∀x y. x < y ⇔ Number.Natural.< (toNatural x) (toNatural y)

⊦ ∀x y. x ≤ y ⇔ Number.Natural.≤ (toNatural x) (toNatural y)

⊦ ∀x y. toNatural x = toNatural y ⇒ x = y

⊦ ∀x1 y1.
    fromNatural (Number.Natural.* x1 y1) = fromNatural x1 * fromNatural y1

⊦ ∀x1 y1.
    fromNatural (Number.Natural.+ x1 y1) = fromNatural x1 + fromNatural y1

⊦ ∀x y.
    toNatural (x + y) =
    Number.Natural.mod (Number.Natural.+ (toNatural x) (toNatural y))
      modulus

⊦ ∀x y.
    fromNatural x = fromNatural y ⇔
    Number.Natural.mod x modulus = Number.Natural.mod y modulus

⊦ ∀m n.
    Number.Natural.mod
      (Number.Natural.* (Number.Natural.mod m modulus)
         (Number.Natural.mod n modulus)) modulus =
    Number.Natural.mod (Number.Natural.* m n) modulus

⊦ ∀m n.
    Number.Natural.mod
      (Number.Natural.+ (Number.Natural.mod m modulus)
         (Number.Natural.mod n modulus)) modulus =
    Number.Natural.mod (Number.Natural.+ m n) modulus

⊦ ∀x y.
    Number.Natural.< x modulus ∧ Number.Natural.< y modulus ∧
    fromNatural x = fromNatural y ⇒ x = y

Input Type Operators

• →
• bool
• Number
  • Natural
    • Number.Natural.natural

Input Constants

• =
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • select
    • F
    • T
• Number
  • Modular
    • modulus
  • Natural
    • Number.Natural.*
    • Number.Natural.+
    • Number.Natural.-
    • Number.Natural.<
    • Number.Natural.≤
    • Number.Natural.div
    • Number.Natural.mod
  • Numeral
    • zero

Assumptions

⊦ T

⊦ ¬(modulus = 0)

⊦ F ⇔ ∀p. p

⊦ ∀t. t ∨ ¬t

⊦ (¬) = λp. p ⇒ F

⊦ (∃) = λP. P ((select) P)

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λP. P = λx. T

⊦ ∀x. x = x ⇔ T

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀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 ⇒ Number.Natural.div m n = 0

⊦ ∀m n. Number.Natural.< m n ⇒ Number.Natural.mod m n = m

⊦ ∀m n. ¬(n = 0) ⇒ Number.Natural.< (Number.Natural.mod m n) n

⊦ ∀f g. f = g ⇔ ∀x. f x = g x

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

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

⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀m n.
    ¬(n = 0) ⇒
    Number.Natural.mod (Number.Natural.mod m n) n = Number.Natural.mod m n

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

⊦ ∀m n p.
    ¬(n = 0) ⇒
    Number.Natural.mod
      (Number.Natural.* (Number.Natural.mod m n) (Number.Natural.mod p n))
      n = Number.Natural.mod (Number.Natural.* m p) n

⊦ ∀a b n.
    ¬(n = 0) ⇒
    Number.Natural.mod
      (Number.Natural.+ (Number.Natural.mod a n) (Number.Natural.mod b n))
      n = Number.Natural.mod (Number.Natural.+ a b) n

⊦ ∀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)

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