pleiad / GradualizingCIC

Companion code of the paper Gradualizing the Calculus of Inductive Constructions

This artefact contains the Agda source code accompanying the paper Gradualizing the Calculus of Inductive Constructions

Requirements

The development relies on a recent version of Agda (compiles with master#cae986ea1cd51ac00faca65cd7c9dad53d0debdb ; expected to work with Agda 2.6.1 and 2.6.2).

See https://agda.readthedocs.io/en/v2.6.1/getting-started/index.html for installation instructions.

The Agda-stdlib is also required, see https://github.com/agda/agda-stdlib

Organisation of the files

The directory src/ contains the following files:

• prelude.agda: pervasives definitions

• Poset.agda: definition of posets, distributors (aka ep-pairs)

• top.agda: order structure on top

• nat.agda: order structure on the extended natural numbers

• bool.agda: order structure on the extended booleans

• pi.agda: order structure on monotone depdendent products

• DiscreteModelPartial.agda: partial discrete model

• MaybeProp.agda: utilities on Prop and Maybe datatypes

• GroundTypes.agda: head constructors of types for the monotone partial universe hierarchy

• UnivPartial.agda: monotone partial universe hierarchy and order on it

• FiniteOrdinal.agda: representation of ordinals for strong induction

• DiscreteModelTotal.agda: total discrete model (capturing GCIC-shift)

The subdirectory src/Unknown contains the development of the monotone unknown type:

• Quotient.agda: axiomatization of co-equalizers
• Interface.agda: interface for the unknown type
• Core.agda: definition of the quotiented unknown type
• Rel.agda: order on the unknown type
• Into.agda : definition of the heterogeneous relation into the unknown type
• IntoRel.agda: definition of distributors into the unknown type

Axioms and assumptions

We use the following axioms:

• functional extensionality in prelude.agda (funext, funext-impl, ∀impl-ext)
• various standard lemmas from HoTT wrt homotopy levels in prelude.agda; these are provable (∀-hSet, ∀impl-hSet, Σ-hset)
• Propositional extensionality for h-propositions in prelude.agda (hpropext, hProp-hSet)
• the existence of coequalizers in Unknown/Quotient.agda
• the proof that ep-pairs for Pi-types enjoy the section-rectraction property in pi.agda (troubles with dependent rewrite)

We also use the following pragmas:

• --prop: to obtain the definitionally irrelevant hierarchy of propositions Prop
• --without-K: to disable UIP in pattern-matching, in order to be agnostic wrt UIP
• --rewriting: to add a definitional reduction rule for quotients
• NO_POSITIVITY_CHECK and TERMINATING in DiscreteModelPartial.agda and UnivPartial.agda reflecting the non-terminating nature of these models