Funcons-beta : Integers.cbs | PLAIN | PDF

OUTLINE

• Integers
  • Subtypes of integers
  • Natural numbers
  • Arithmetic
  • Comparison
  • Conversion

---

Integers

\[\begin{align*} [ \ \KEY{Type} \quad & \NAMEREF{integers} \\ \KEY{Alias} \quad & \NAMEREF{ints} \\ \KEY{Type} \quad & \NAMEREF{integers-from} \\ \KEY{Alias} \quad & \NAMEREF{from} \\ \KEY{Type} \quad & \NAMEREF{integers-up-to} \\ \KEY{Alias} \quad & \NAMEREF{up-to} \\ \KEY{Type} \quad & \NAMEREF{bounded-integers} \\ \KEY{Alias} \quad & \NAMEREF{bounded-ints} \\ \KEY{Type} \quad & \NAMEREF{positive-integers} \\ \KEY{Alias} \quad & \NAMEREF{pos-ints} \\ \KEY{Type} \quad & \NAMEREF{negative-integers} \\ \KEY{Alias} \quad & \NAMEREF{neg-ints} \\ \KEY{Type} \quad & \NAMEREF{natural-numbers} \\ \KEY{Alias} \quad & \NAMEREF{nats} \\ \KEY{Funcon} \quad & \NAMEREF{natural-successor} \\ \KEY{Alias} \quad & \NAMEREF{nat-succ} \\ \KEY{Funcon} \quad & \NAMEREF{natural-predecessor} \\ \KEY{Alias} \quad & \NAMEREF{nat-pred} \\ \KEY{Funcon} \quad & \NAMEREF{integer-add} \\ \KEY{Alias} \quad & \NAMEREF{int-add} \\ \KEY{Funcon} \quad & \NAMEREF{integer-subtract} \\ \KEY{Alias} \quad & \NAMEREF{int-sub} \\ \KEY{Funcon} \quad & \NAMEREF{integer-multiply} \\ \KEY{Alias} \quad & \NAMEREF{int-mul} \\ \KEY{Funcon} \quad & \NAMEREF{integer-divide} \\ \KEY{Alias} \quad & \NAMEREF{int-div} \\ \KEY{Funcon} \quad & \NAMEREF{integer-modulo} \\ \KEY{Alias} \quad & \NAMEREF{int-mod} \\ \KEY{Funcon} \quad & \NAMEREF{integer-power} \\ \KEY{Alias} \quad & \NAMEREF{int-pow} \\ \KEY{Funcon} \quad & \NAMEREF{integer-absolute-value} \\ \KEY{Alias} \quad & \NAMEREF{int-abs} \\ \KEY{Funcon} \quad & \NAMEREF{integer-negate} \\ \KEY{Alias} \quad & \NAMEREF{int-neg} \\ \KEY{Funcon} \quad & \NAMEREF{integer-is-less} \\ \KEY{Alias} \quad & \NAMEREF{is-less} \\ \KEY{Funcon} \quad & \NAMEREF{integer-is-less-or-equal} \\ \KEY{Alias} \quad & \NAMEREF{is-less-or-equal} \\ \KEY{Funcon} \quad & \NAMEREF{integer-is-greater} \\ \KEY{Alias} \quad & \NAMEREF{is-greater} \\ \KEY{Funcon} \quad & \NAMEREF{integer-is-greater-or-equal} \\ \KEY{Alias} \quad & \NAMEREF{is-greater-or-equal} \\ \KEY{Funcon} \quad & \NAMEREF{binary-natural} \\ \KEY{Alias} \quad & \NAMEREF{binary} \\ \KEY{Funcon} \quad & \NAMEREF{octal-natural} \\ \KEY{Alias} \quad & \NAMEREF{octal} \\ \KEY{Funcon} \quad & \NAMEREF{decimal-natural} \\ \KEY{Alias} \quad & \NAMEREF{decimal} \\ \KEY{Funcon} \quad & \NAMEREF{hexadecimal-natural} \\ \KEY{Alias} \quad & \NAMEREF{hexadecimal} \\ \KEY{Funcon} \quad & \NAMEREF{integer-sequence} \ ] \end{align*}\] \[\begin{align*} \KEY{Built-in Type} \quad & \NAMEDECL{integers} \\ \KEY{Alias} \quad & \NAMEDECL{ints} = \NAMEREF{integers} \end{align*}\]

\(\SHADE{\NAMEREF{integers}}\) is the type of unbounded integers. Decimal notation is used to express particular integer values.

Subtypes of integers

\[\begin{align*} \KEY{Built-in Type} \quad & \NAMEDECL{integers-from}( \_ : \NAMEREF{integers}) <: \NAMEREF{integers} \\ \KEY{Alias} \quad & \NAMEDECL{from} = \NAMEREF{integers-from} \end{align*}\]

\(\SHADE{\NAMEREF{integers-from} ( \VAR{M} )}\) is the subtype of integers greater than or equal to \(\SHADE{\VAR{M}}\).

\[\begin{align*} \KEY{Built-in Type} \quad & \NAMEDECL{integers-up-to}( \_ : \NAMEREF{integers}) <: \NAMEREF{integers} \\ \KEY{Alias} \quad & \NAMEDECL{up-to} = \NAMEREF{integers-up-to} \end{align*}\]

\(\SHADE{\NAMEREF{integers-up-to} ( \VAR{N} )}\) is the subtype of integers less than or equal to \(\SHADE{\VAR{N}}\).

\[\begin{align*} \KEY{Type} \quad & \NAMEDECL{bounded-integers}( \VAR{M} : \NAMEREF{integers}, \VAR{N} : \NAMEREF{integers}) \\&\quad \leadsto \NAMEREF{integers-from} ( \VAR{M} ) \mathbin{\&} \NAMEREF{integers-up-to} ( \VAR{N} ) \\ \KEY{Alias} \quad & \NAMEDECL{bounded-ints} = \NAMEREF{bounded-integers} \end{align*}\]

\(\SHADE{\NAMEREF{bounded-integers} ( \VAR{M}, \VAR{N} )}\) is the subtype of integers from \(\SHADE{\VAR{M}}\) to \(\SHADE{\VAR{N}}\), inclusive.

\[\begin{align*} \KEY{Type} \quad & \NAMEDECL{positive-integers} \leadsto \NAMEREF{integers-from} ( 1 ) \\ \KEY{Alias} \quad & \NAMEDECL{pos-ints} = \NAMEREF{positive-integers} \end{align*}\] \[\begin{align*} \KEY{Type} \quad & \NAMEDECL{negative-integers} \leadsto \NAMEREF{integers-up-to} ( -1 ) \\ \KEY{Alias} \quad & \NAMEDECL{neg-ints} = \NAMEREF{negative-integers} \end{align*}\]

Natural numbers

\[\begin{align*} \KEY{Type} \quad & \NAMEDECL{natural-numbers} \leadsto \NAMEREF{integers-from} ( 0 ) \\ \KEY{Alias} \quad & \NAMEDECL{nats} = \NAMEREF{natural-numbers} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{natural-successor}( \VAR{N} : \NAMEREF{natural-numbers}) : \TO \NAMEREF{natural-numbers} \\ \KEY{Alias} \quad & \NAMEDECL{nat-succ} = \NAMEREF{natural-successor} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{natural-predecessor}( \_ : \NAMEREF{natural-numbers}) : \TO \NAMEREF{natural-numbers}\QUERY \\ \KEY{Alias} \quad & \NAMEDECL{nat-pred} = \NAMEREF{natural-predecessor} \end{align*}\] \[\begin{align*} \KEY{Assert} \quad & \NAMEREF{natural-predecessor} ( 0 ) == ( \ ) \end{align*}\]

Arithmetic

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{integer-add}( \_ : \NAMEREF{integers}\STAR) : \TO \NAMEREF{integers} \\ \KEY{Alias} \quad & \NAMEDECL{int-add} = \NAMEREF{integer-add} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{integer-subtract}( \_ : \NAMEREF{integers}, \_ : \NAMEREF{integers}) : \TO \NAMEREF{integers} \\ \KEY{Alias} \quad & \NAMEDECL{int-sub} = \NAMEREF{integer-subtract} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{integer-multiply}( \_ : \NAMEREF{integers}\STAR) : \TO \NAMEREF{integers} \\ \KEY{Alias} \quad & \NAMEDECL{int-mul} = \NAMEREF{integer-multiply} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{integer-divide}( \_ : \NAMEREF{integers}, \_ : \NAMEREF{integers}) : \TO \NAMEREF{integers}\QUERY \\ \KEY{Alias} \quad & \NAMEDECL{int-div} = \NAMEREF{integer-divide} \end{align*}\] \[\begin{align*} \KEY{Assert} \quad & \NAMEREF{integer-divide} ( \_ : \NAMEREF{integers}, 0 ) == ( \ ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{integer-modulo}( \_ : \NAMEREF{integers}, \_ : \NAMEREF{integers}) : \TO \NAMEREF{integers}\QUERY \\ \KEY{Alias} \quad & \NAMEDECL{int-mod} = \NAMEREF{integer-modulo} \end{align*}\] \[\begin{align*} \KEY{Assert} \quad & \NAMEREF{integer-modulo} ( \_ : \NAMEREF{integers}, 0 ) == ( \ ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{integer-power}( \_ : \NAMEREF{integers}, \_ : \NAMEREF{natural-numbers}) : \TO \NAMEREF{integers} \\ \KEY{Alias} \quad & \NAMEDECL{int-pow} = \NAMEREF{integer-power} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{integer-absolute-value}( \_ : \NAMEREF{integers}) : \TO \NAMEREF{natural-numbers} \\ \KEY{Alias} \quad & \NAMEDECL{int-abs} = \NAMEREF{integer-absolute-value} \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{integer-negate}( \VAR{N} : \NAMEREF{integers}) : \TO \NAMEREF{integers} \\&\quad \leadsto \NAMEREF{integer-subtract} ( 0, \VAR{N} ) \\ \KEY{Alias} \quad & \NAMEDECL{int-neg} = \NAMEREF{integer-negate} \end{align*}\]

Comparison

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{integer-is-less}( \_ : \NAMEREF{integers}, \_ : \NAMEREF{integers}) : \TO \NAMEHYPER{../.}{Booleans}{booleans} \\ \KEY{Alias} \quad & \NAMEDECL{is-less} = \NAMEREF{integer-is-less} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{integer-is-less-or-equal}( \_ : \NAMEREF{integers}, \_ : \NAMEREF{integers}) : \TO \NAMEHYPER{../.}{Booleans}{booleans} \\ \KEY{Alias} \quad & \NAMEDECL{is-less-or-equal} = \NAMEREF{integer-is-less-or-equal} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{integer-is-greater}( \_ : \NAMEREF{integers}, \_ : \NAMEREF{integers}) : \TO \NAMEHYPER{../.}{Booleans}{booleans} \\ \KEY{Alias} \quad & \NAMEDECL{is-greater} = \NAMEREF{integer-is-greater} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{integer-is-greater-or-equal}( \_ : \NAMEREF{integers}, \_ : \NAMEREF{integers}) : \TO \NAMEHYPER{../.}{Booleans}{booleans} \\ \KEY{Alias} \quad & \NAMEDECL{is-greater-or-equal} = \NAMEREF{integer-is-greater-or-equal} \end{align*}\]

Conversion

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{binary-natural}( \_ : \NAMEHYPER{../../Composite}{Strings}{strings}) : \TO \NAMEREF{natural-numbers}\QUERY \\ \KEY{Alias} \quad & \NAMEDECL{binary} = \NAMEREF{binary-natural} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{octal-natural}( \_ : \NAMEHYPER{../../Composite}{Strings}{strings}) : \TO \NAMEREF{natural-numbers}\QUERY \\ \KEY{Alias} \quad & \NAMEDECL{octal} = \NAMEREF{octal-natural} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{decimal-natural}( \_ : \NAMEHYPER{../../Composite}{Strings}{strings}) : \TO \NAMEREF{natural-numbers}\QUERY \\ \KEY{Alias} \quad & \NAMEDECL{decimal} = \NAMEREF{decimal-natural} \end{align*}\]

Literal natural numbers \(\SHADE{\VAR{N}}\) are equivalent to \(\SHADE{\NAMEREF{decimal-natural} \ \STRING{N}}\).

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{hexadecimal-natural}( \_ : \NAMEHYPER{../../Composite}{Strings}{strings}) : \TO \NAMEREF{natural-numbers}\QUERY \\ \KEY{Alias} \quad & \NAMEDECL{hexadecimal} = \NAMEREF{hexadecimal-natural} \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{integer-sequence}( \_ : \NAMEREF{integers}, \_ : \NAMEREF{integers}) : \TO \NAMEREF{integers}\STAR \end{align*}\]

\(\SHADE{\NAMEREF{integer-sequence} ( \VAR{M}, \VAR{N} )}\) is the seqeunce of integers from \(\SHADE{\VAR{M}}\) to \(\SHADE{\VAR{N}}\), except that if \(\SHADE{\VAR{M}}\) is greater than \(\SHADE{\VAR{N}}\), it is the empty sequence.

\[\begin{align*} \KEY{Rule} \quad & \RULE{ & \NAMEREF{is-greater} ( \VAR{M}, \VAR{N} ) == \NAMEHYPER{../.}{Booleans}{false} }{ & \NAMEREF{integer-sequence} ( \VAR{M} : \NAMEREF{integers}, \VAR{N} : \NAMEREF{integers} ) \leadsto \\&\quad ( \VAR{M}, \NAMEREF{integer-sequence} ( \NAMEREF{integer-add} ( \VAR{M}, 1 ), \VAR{N} ) ) } \\ \KEY{Rule} \quad & \RULE{ & \NAMEREF{is-greater} ( \VAR{M}, \VAR{N} ) == \NAMEHYPER{../.}{Booleans}{true} }{ & \NAMEREF{integer-sequence} ( \VAR{M} : \NAMEREF{integers}, \VAR{N} : \NAMEREF{integers} ) \leadsto ( \ ) } \end{align*}\]

---

---

← ↑ →

---