Context
Type theory
Foundational axiom
Contents
Idea
In dependent type theory, an axiom of set truncation is a axiom or unjustified rule which implies the uniqueness of identity proofs which states that every type is an h-set. Adding an axiom of set truncation to a dependent type theory results in a set-level type theory.
Examples
TODO: show that each of the axioms below implies the uniqueness of identity proofs.
Uniqueness of identity proofs
For identification types
In dependent type theory, uniqueness of identity proofs for identification types is given by the following rule:
For heterogeneous identification types
There is uniqueness of identity proofs for heterogeneous identification types:
Uniqueness of identity proofs for identification types follows from uniqueness of identity proofs for heterogeneous identification types by taking canonical element of the unit type , identification , and elements and . Since the following two types are equivalent:
and because is always a mere proposition by uniqueness of identity proofs for heterogeneous identification types, it follows that is also always a mere proposition.
Similarly, there is also uniqueness of identity proofs for dependent heterogeneous identification types:
Uniqueness of identity proofs for identification types follows from uniqueness of identity proofs for dependent heterogeneous identification types by defining the type to be the dependent type for type family indexed by the unit type , and taking the canonical element , identification , and elements and . Since the two types are equivalent:
and because is always a mere proposition by uniqueness of identity proofs for dependent heterogeneous identification types, it follows that and thus by definition is also always a mere proposition.
Axiom K
For identification types
In dependent type theory, axiom K is given by the following rule:
For heterogeneous identification types
There is an axiom K for heterogeneous identification types:
Axiom K for identification types follows from axiom K for heterogeneous identification types by taking canonical element of the unit type , identification , and elements and . Since the following two types are equivalent:
and because is always a contractible type with center of contraction for element , and function , by axiom K for heterogeneous identification types, it follows that is also always a contractible type, with center of contraction .
Similarly, there is also an axiom K for dependent heterogeneous identification types:
Axiom K for identification types follows from axiom K for dependent heterogeneous identification types by defining the type to be the dependent type for type family indexed by the unit type , and taking the canonical element , identification , and elements and . Since the two types are equivalent:
and because is always a contractible type with center of contraction for element , and dependent function , by axiom K for heterogeneous identification types, it follows that and thus by definition is also always a contractible type, with center of contraction .
Boundary separation
In cubical type theory, boundary separation is given by the following rule:
Equality reflection
In dependent type theory, equality reflection is given by the following rule:
Strong pattern matching
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-localization
In dependent type theory, the axiom of -localization is given by the following rules:
See also