case class IdentityTyp[U <: Term with Subs[U]](dom: Typ[U], lhs: U, rhs: U) extends Typ[Equality[U]] with Subs[IdentityTyp[U]] with Product with Serializable
The identity type.
This is the type lhs = rhs
- Alphabetic
- By Inheritance
- IdentityTyp
- Serializable
- Product
- Equals
- Typ
- Term
- Subs
- AnyRef
- Any
- by RichTerm
- by any2stringadd
- by StringFormat
- by Ensuring
- by ArrowAssoc
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- Public
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Type Members
- type Obj = Equality[U]
scala type of objects with this HoTT-type (refining
U)scala type of objects with this HoTT-type (refining
U)- Definition Classes
- IdentityTyp → Typ
Value Members
- def !:(term: Term): Equality[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
- def &&[UU >: Equality[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 >: Equality[U] <: Term with Subs[UU], V <: Term with Subs[V]](variable: V): SigmaTyp[V, UU]
- Definition Classes
- Typ
- def ++[UU >: Typ[Equality[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 ->:[W <: Term with Subs[W], UU >: Equality[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[IdentityTyp[U], V]
constructor for (pure) lambda functions, see lmbda
constructor for (pure) lambda functions, see lmbda
- Implicit
- This member is added by an implicit conversion from IdentityTyp[U] toRichTerm[IdentityTyp[U]] performed by method RichTerm in provingground.HoTT.
- Definition Classes
- RichTerm
- def ::(name: String): Equality[U]
symbolic object with given name
symbolic object with given name
- Definition Classes
- Typ
- def :~>[V <: Term with Subs[V]](that: V): FuncLike[IdentityTyp[U], V]
constructor for (in general dependent) lambda functions, see lambda
constructor for (in general dependent) lambda functions, see lambda
- Implicit
- This member is added by an implicit conversion from IdentityTyp[U] toRichTerm[IdentityTyp[U]] performed by method RichTerm in provingground.HoTT.
- Definition Classes
- RichTerm
- def =:=(rhs: IdentityTyp[U]): IdentityTyp[IdentityTyp[U]]
equality type 'term = rhs'
equality type 'term = rhs'
- Implicit
- This member is added by an implicit conversion from IdentityTyp[U] toRichTerm[IdentityTyp[U]] performed by method RichTerm in provingground.HoTT.
- Definition Classes
- RichTerm
- def Var(implicit factory: NameFactory): Equality[U]
new variable from a factory.
new variable from a factory.
- Definition Classes
- Typ
- def dependsOn(that: Term): Boolean
returns whether
thisdepends onthatreturns whether
thisdepends onthat- Definition Classes
- Term
- val dom: Typ[U]
- def indepOf(that: Term): Boolean
returns whether
thisis independent ofthat.returns whether
thisis independent ofthat.- Definition Classes
- Term
- val lhs: U
- def newobj: IdentityTyp[U]
A new object with the same type, to be used in place of a variable to avoid name clashes.
A new object with the same type, to be used in place of a variable to avoid name clashes. Should throw exception when invoked for constants.
- Definition Classes
- IdentityTyp → Subs
- def obj: Equality[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[IdentityTyp[U]]
reflexivity term
refl : term = termreflexivity term
refl : term = term- Implicit
- This member is added by an implicit conversion from IdentityTyp[U] toRichTerm[IdentityTyp[U]] performed by method RichTerm in provingground.HoTT.
- Definition Classes
- RichTerm
- def replace(x: Term, y: Term): IdentityTyp[U] with Subs[IdentityTyp[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
- val rhs: U
- def subs(x: Term, y: Term): IdentityTyp[U with Subs[U]]
substitute x by y recursively in
this.substitute x by y recursively in
this.- Definition Classes
- IdentityTyp → Subs
- def sym(implicit name: sourcecode.Name): Equality[U]
shortcut for symbolic object
shortcut for symbolic object
- Definition Classes
- Typ
- def symbObj(name: AnySym): Equality[U] with Subs[Equality[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
- def toString(): String
- Definition Classes
- IdentityTyp → AnyRef → Any
- lazy val typ: Universe
type of a type is a universe.
type of a type is a universe.
- Definition Classes
- IdentityTyp → Typ → Term
- 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): SymbEquality[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
- IdentityTyp → Typ
- def ||[UU >: Equality[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 >: Equality[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 induc[UU >: U <: Term with Subs[UU], V <: Term with Subs[V]](targetFmly: FuncLike[UU, FuncLike[UU, FuncLike[Equality[UU], Typ[V]]]]): Func[FuncLike[UU, V], FuncLike[UU, FuncLike[UU, Func[Equality[UU], V]]]]
inductive definition on the identity type family
inductive definition on the identity type family
- Annotations
- @deprecated
- Deprecated
(Since version forever) misleading since identity is an indexed family
- def rec[UU >: U <: Term with Subs[UU], V <: Term with Subs[V]](codom: Typ[V]): Func[Func[UU, V], FuncLike[UU, FuncLike[UU, Func[Equality[UU], V]]]]
recursive definition on the identity type family
recursive definition on the identity type family
- Annotations
- @deprecated
- Deprecated
(Since version forever) misleading since identity is an idexed family