34,110 research outputs found
Sequent Calculus and Equational Programming
Proof assistants and programming languages based on type theories usually
come in two flavours: one is based on the standard natural deduction
presentation of type theory and involves eliminators, while the other provides
a syntax in equational style. We show here that the equational approach
corresponds to the use of a focused presentation of a type theory expressed as
a sequent calculus. A typed functional language is presented, based on a
sequent calculus, that we relate to the syntax and internal language of Agda.
In particular, we discuss the use of patterns and case splittings, as well as
rules implementing inductive reasoning and dependent products and sums.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759
On choice rules in dependent type theory
In a dependent type theory satisfying the propositions as
types correspondence together with the proofs-as-programs paradigm,
the validity of the unique choice rule or even more of the choice rule says
that the extraction of a computable witness from an existential statement
under hypothesis can be performed within the same theory.
Here we show that the unique choice rule, and hence the choice rule,
are not valid both in Coquand\u2019s Calculus of Constructions with indexed
sum types, list types and binary disjoint sums and in its predicative
version implemented in the intensional level of the Minimalist Founda-
tion. This means that in these theories the extraction of computational
witnesses from existential statements must be performed in a more ex-
pressive proofs-as-programs theory
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