natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
identity type, equivalence of types, definitional isomorphism
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
In logic and type theory, -equivalence is the principle that two syntactic expressions (types, terms, propositions, contexts, whatever) are equivalent for all purposes if their only difference is the renaming of bound variables.
Depending on the technicalities of how variables are managed, -equivalence may be a necessary axiom, a provable theorem, or entirely trivial. In any case, it is often (usually? always?) seen as a technicality devoid of conceptual interest. However, it can be a technically nontrivial task to implement -conversion (which is necessary to avoid capture of free variables upon substitution) when programming logic or type theory into a computer.
We work in a dependent type theory which acts as the metatheory for our object type theory.
The type is a countable set (and thus a type with decidable equality) whose terms are called names or atoms. The type of syntactic expressions is an h-set which is inductively defined by binary functions and . There are dependent types representing the occuring atoms, representing the free atoms, and representing the bound atoms, such that for each expression , there are embeddings
which may be defined by recursion on .
We define the following dependent types as
representing that occurs in at least once,
representing that freely occurs in at least once, and
representing that is fresh (does not freely occurs in at least once)
…
The distinction between -equivalence and syntactic equality of expressions is briefly discussed in:
Last revised on November 11, 2022 at 04:16:31. See the history of this page for a list of all contributions to it.