case class SigmaTyp[W <: Term with Subs[W], U <: Term with Subs[U]](fibers: TypFamily[W, U]) extends Typ[AbsPair[W, U]] with Subs[SigmaTyp[W, U]] with Product with Serializable
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Type Members
- case class InducFn[V <: Term with Subs[V]](targetFmly: FuncLike[W, Func[U, Typ[V]]], data: FuncLike[W, FuncLike[U, V]]) extends InducFuncLike[AbsPair[W, U], V] with Product with Serializable
inductive definition on the Sigma-type
- type Obj = AbsPair[W, U]
scala type of objects with this HoTT-type (refining
U) - case class RecFn[V <: Term with Subs[V]](codom: Typ[V], data: FuncLike[W, Func[U, V]]) extends RecFunc[AbsPair[W, U], V] with Product with Serializable
recursively defined function on
this
Value Members
- def !:(term: Term): AbsPair[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 >: AbsPair[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 >: AbsPair[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[AbsPair[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): (SigmaTyp[W, U], B)
- def ->:[W <: Term with Subs[W], UU >: AbsPair[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[SigmaTyp[W, U], V]
constructor for (pure) lambda functions, see lmbda
- def ::(name: String): AbsPair[W, U]
symbolic object with given name
symbolic object with given name
- Definition Classes
- Typ
- def :~>[V <: Term with Subs[V]](that: V): FuncLike[SigmaTyp[W, U], V]
constructor for (in general dependent) lambda functions, see lambda
- def =:=(rhs: SigmaTyp[W, U]): IdentityTyp[SigmaTyp[W, U]]
equality type 'term = rhs'
equality type 'term = rhs'
- Implicit
- This member is added by an implicit conversion from SigmaTyp[W, U] toRichTerm[SigmaTyp[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): AbsPair[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
- def ensuring(cond: (SigmaTyp[W, U]) => Boolean, msg: => Any): SigmaTyp[W, U]
- def ensuring(cond: (SigmaTyp[W, U]) => Boolean): SigmaTyp[W, U]
- def ensuring(cond: Boolean, msg: => Any): SigmaTyp[W, U]
- def ensuring(cond: Boolean): SigmaTyp[W, U]
- final def eq(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
- lazy val fib: LambdaFixed[W, Typ[U]]
- val fibers: TypFamily[W, U]
- def formatted(fmtstr: String): String
- final def getClass(): Class[_ <: AnyRef]
- Definition Classes
- AnyRef → Any
- Annotations
- @native() @HotSpotIntrinsicCandidate()
- def indepOf(that: Term): Boolean
returns whether
thisis independent ofthat.returns whether
thisis independent ofthat.- Definition Classes
- Term
- def induc[V <: Term with Subs[V]](targetFmly: FuncLike[W, Func[U, Typ[V]]]): Func[FuncLike[W, FuncLike[U, V]], FuncLike[AbsPair[W, U], V]]
inductive definition
inductive definition
- targetFmly
the codomain
- final def isInstanceOf[T0]: Boolean
- Definition Classes
- Any
- final def ne(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
- def newobj: SigmaTyp[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: AbsPair[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
- lazy val paircons: FuncLike[W, Func[U, AbsPair[W, U]]]
introduction rule for the Sigma type
- def productElementNames: Iterator[String]
- Definition Classes
- Product
- lazy val proj1: Func[AbsPair[W, U], W]
projections from the Sigma-Type
- lazy val proj2: FuncLike[AbsPair[W, U], U]
projections from the Sigma-Type
- def rec[V <: Term with Subs[V]](target: Typ[V]): Func[FuncLike[W, Func[U, V]], FuncLike[AbsPair[W, U], V]]
recursive definition
recursive definition
- target
the codomain
- def refl: Refl[SigmaTyp[W, U]]
reflexivity term
refl : term = termreflexivity term
refl : term = term- Implicit
- This member is added by an implicit conversion from SigmaTyp[W, U] toRichTerm[SigmaTyp[W, U]] performed by method RichTerm in provingground.HoTT.
- Definition Classes
- RichTerm
- def replace(x: Term, y: Term): SigmaTyp[W, U] with Subs[SigmaTyp[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): SigmaTyp[W, U]
substitute x by y recursively in
this. - def sym(implicit name: sourcecode.Name): AbsPair[W, U]
shortcut for symbolic object
shortcut for symbolic object
- Definition Classes
- Typ
- def symbObj(name: AnySym): AbsPair[W, U] with Subs[AbsPair[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
- SigmaTyp → AnyRef → Any
- lazy val typ: Universe
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
- def variable(name: AnySym): AbsPair[W, U]
A symbolic object with this HoTT type, and with scala-type Obj
- 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 >: AbsPair[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 >: AbsPair[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): (SigmaTyp[W, U], B)
- Implicit
- This member is added by an implicit conversion from SigmaTyp[W, U] toArrowAssoc[SigmaTyp[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.