case class PiDefn[W <: Term with Subs[W], U <: Term with Subs[U]](variable: W, value: Typ[U]) extends GenFuncTyp[W, U] with Typ[FuncLike[W, U]] with Subs[PiDefn[W, U]] with Product with Serializable
Pi-Type, defined in terms of a variable and value, i.e.,
\Pi_{x: X}Q (latex) with x the variable and Q the value
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- GenFuncTyp
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- def !:(term: Term): FuncLike[W, U]
checks term is of this type and returns it; useful for documentation.
checks term is of this type and returns it; useful for documentation.
- Definition Classes
- Typ
- final def !=(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
- final def ##: Int
- Definition Classes
- AnyRef → Any
- def &&[UU >: FuncLike[W, U] <: Term with Subs[UU], V <: Term with Subs[V]](that: Typ[V]): ProdTyp[UU, V]
returns product type, mainly to use for "and" for structures
returns product type, mainly to use for "and" for structures
- Definition Classes
- Typ
- def &:[UU >: FuncLike[W, U] <: Term with Subs[UU], V <: Term with Subs[V]](variable: V): SigmaTyp[V, UU]
- Definition Classes
- Typ
- def +(other: String): String
- def ++[UU >: Typ[FuncLike[W, U]] <: Typ[Term] with Subs[UU], VV <: Term with Subs[VV], V <: Typ[VV] with Subs[V]](those: V): SigmaTyp[UU, VV]
returns Sigma-Type, mainly to use as "such that", for example a group type is this with product etc.
returns Sigma-Type, mainly to use as "such that", for example a group type is this with product etc. dependent on this.
- Definition Classes
- Typ
- def ->[B](y: B): (PiDefn[W, U], B)
- def ->:[W <: Term with Subs[W], UU >: FuncLike[W, U] <: Term with Subs[UU]](that: Typ[W]): FuncTyp[W, UU]
function type:
this -> thatfunction type:
this -> that- Definition Classes
- Typ
- def :->[V <: Term with Subs[V]](that: V): Func[PiDefn[W, U], V]
constructor for (pure) lambda functions, see lmbda
- def ::(name: String): FuncLike[W, U]
symbolic object with given name
symbolic object with given name
- Definition Classes
- Typ
- def :~>[V <: Term with Subs[V]](that: V): FuncLike[PiDefn[W, U], V]
constructor for (in general dependent) lambda functions, see lambda
- def =:=(rhs: PiDefn[W, U]): IdentityTyp[PiDefn[W, U]]
equality type 'term = rhs'
equality type 'term = rhs'
- Implicit
- This member is added by an implicit conversion from PiDefn[W, U] toRichTerm[PiDefn[W, U]] performed by method RichTerm in provingground.HoTT.
- Definition Classes
- RichTerm
- final def ==(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
- def Var(implicit factory: NameFactory): FuncLike[W, U]
new variable from a factory.
new variable from a factory.
- Definition Classes
- Typ
- final def asInstanceOf[T0]: T0
- Definition Classes
- Any
- def clone(): AnyRef
- Attributes
- protected[lang]
- Definition Classes
- AnyRef
- Annotations
- @throws(classOf[java.lang.CloneNotSupportedException]) @native() @HotSpotIntrinsicCandidate()
- def dependsOn(that: Term): Boolean
returns whether
thisdepends onthatreturns whether
thisdepends onthat- Definition Classes
- Term
- val domain: Typ[W]
- Definition Classes
- GenFuncTyp
- def ensuring(cond: (PiDefn[W, U]) => Boolean, msg: => Any): PiDefn[W, U]
- def ensuring(cond: (PiDefn[W, U]) => Boolean): PiDefn[W, U]
- def ensuring(cond: Boolean, msg: => Any): PiDefn[W, U]
- def ensuring(cond: Boolean): PiDefn[W, U]
- final def eq(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
- def equals(that: Any): Boolean
- Definition Classes
- GenFuncTyp → AnyRef → Any
- val fib: (W) => Typ[U]
- Definition Classes
- GenFuncTyp
- lazy val fibers: LambdaFixed[W, Typ[U]]
- def formatted(fmtstr: String): String
- final def getClass(): Class[_ <: AnyRef]
- Definition Classes
- AnyRef → Any
- Annotations
- @native() @HotSpotIntrinsicCandidate()
- lazy val hashCode: Int
- Definition Classes
- GenFuncTyp → AnyRef → Any
- def indepOf(that: Term): Boolean
returns whether
thisis independent ofthat.returns whether
thisis independent ofthat.- Definition Classes
- Term
- final def isInstanceOf[T0]: Boolean
- Definition Classes
- Any
- final def ne(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
- def newobj: PiDefn[W, U]
A new object with the same type, to be used in place of a variable to avoid name clashes.
- final def notify(): Unit
- Definition Classes
- AnyRef
- Annotations
- @native() @HotSpotIntrinsicCandidate()
- final def notifyAll(): Unit
- Definition Classes
- AnyRef
- Annotations
- @native() @HotSpotIntrinsicCandidate()
- def obj: FuncLike[W, U]
factory for producing objects of the given type.
factory for producing objects of the given type. can use {{innervar}} if one wants name unchanged.
- Definition Classes
- Typ
- def productElementNames: Iterator[String]
- Definition Classes
- Product
- def refl: Refl[PiDefn[W, U]]
reflexivity term
refl : term = termreflexivity term
refl : term = term- Implicit
- This member is added by an implicit conversion from PiDefn[W, U] toRichTerm[PiDefn[W, U]] performed by method RichTerm in provingground.HoTT.
- Definition Classes
- RichTerm
- def replace(x: Term, y: Term): PiDefn[W, U] with Subs[PiDefn[W, U]]
refine substitution so if x and y are both of certain forms such as pairs or formal applications, components are substituted.
refine substitution so if x and y are both of certain forms such as pairs or formal applications, components are substituted.
- Definition Classes
- Subs
- def subs(x: Term, y: Term): PiDefn[W, U] with Subs[PiDefn[W, U]]
substitute x by y recursively in
this. - def sym(implicit name: sourcecode.Name): FuncLike[W, U]
shortcut for symbolic object
shortcut for symbolic object
- Definition Classes
- Typ
- def symbObj(name: AnySym): FuncLike[W, U] with Subs[FuncLike[W, U]]
A symbolic object with this HoTT type, and with scala-type Obj
A symbolic object with this HoTT type, and with scala-type Obj
- Definition Classes
- Typ
- final def synchronized[T0](arg0: => T0): T0
- Definition Classes
- AnyRef
- def toString(): String
- Definition Classes
- PiDefn → AnyRef → Any
- lazy val typ: Typ[Typ[Term]]
type of a type is a universe.
- lazy val typlevel: Int
- Definition Classes
- Typ
- def usesVar(t: Term): Boolean
returns whether the variable
tis used as a variable in a lambda definition.returns whether the variable
tis used as a variable in a lambda definition.- Definition Classes
- Term
- val value: Typ[U]
- def variable(name: AnySym): FuncLike[W, U]
- val variable: W
- final def wait(arg0: Long, arg1: Int): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws(classOf[java.lang.InterruptedException])
- final def wait(arg0: Long): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws(classOf[java.lang.InterruptedException]) @native()
- final def wait(): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws(classOf[java.lang.InterruptedException])
- def ||[UU >: FuncLike[W, U] <: Term with Subs[UU], V <: Term with Subs[V]](that: Typ[V]): PlusTyp[UU, V]
returns coproduct type, mainly to use for "or".
returns coproduct type, mainly to use for "or".
- Definition Classes
- Typ
- def ~>:[UU >: FuncLike[W, U] <: Term with Subs[UU], V <: Term with Subs[V]](variable: V): GenFuncTyp[V, UU]
dependent function type (Pi-Type) define by a lambda:
thisdepends on thevariable, which hence gives a type family; note that a new variable is created and substituted inthisto avoid name clashes.dependent function type (Pi-Type) define by a lambda:
thisdepends on thevariable, which hence gives a type family; note that a new variable is created and substituted inthisto avoid name clashes.- Definition Classes
- Typ
- object Elem
Pattern for element of the given type.
Pattern for element of the given type.
- Definition Classes
- Typ
Deprecated Value Members
- def finalize(): Unit
- Attributes
- protected[lang]
- Definition Classes
- AnyRef
- Annotations
- @throws(classOf[java.lang.Throwable]) @Deprecated
- Deprecated
- def →[B](y: B): (PiDefn[W, U], B)
- Implicit
- This member is added by an implicit conversion from PiDefn[W, U] toArrowAssoc[PiDefn[W, U]] performed by method ArrowAssoc in scala.Predef.
- Definition Classes
- ArrowAssoc
- Annotations
- @deprecated
- Deprecated
(Since version 2.13.0) Use
->instead. If you still wish to display it as one character, consider using a font with programming ligatures such as Fira Code.