Coq Formalization of Compactness

This repo uses the Pattern Stepping Bisimulation approach to formally verify the compactness lemma for a language with control flow effects and polymorphism.

The files in the EasySubst folder represent a substitution framework similar to that of AutoSubst and are based on the formulation given in Crary's Fully Abstract Module Compilation paper.

This code provides a good starting point for utilizing the logical relation approach to formalize properties about languages.

Overview of Files

EasySubst/Sequence.v

Definitions of truncate and index operations on lists and associated lemmas.

EasySubst/Subst.v

Abstractly defines substitution based on a traverse function. It uses explicit substitutions on De Bruijn indicies. The fact it is defined abstractly makes it great as a substitution framework as an alternative to AutoSubst.

It defines a hint database with several rewrite theorems for simplifying substitutions, which for the most part works well.

EasySubst/TSubstGeneral.v

This is one of the big advantages this framework has over that of AutoSubst, namely a methodology for reasoning about substitutions $\Gamma'\vdash \sigma : \Gamma$.

This is also defined abstractly so it can be applied to any language.

SyntaX.v

Definition of the language syntax and the E[e] operation for evaluation contexts. Applies the Subst.v file substitution definitions to this language.

Rules.v

The structural operational semantics for the language, along with some lemmas about determinicity and the compositionality of running under evaluation contexts.

Substitution.v

Builds the substitution theorem for this language by utilizing the methodology given in TSubstGeneral.v. Adds several items to the hint database for substitution, which resolves most things for this language via the simp_sub tactic.

Safety.v

Progress and Preservation of the language, utilizing the substitution theorem.

Kleene.v

Definition of termination and mutual termination (Kleene equivalence). Proves a lemma that a particular term does not terminate by utilizing determinicity.

Compactness.v

Defines a pattern language for verifying the compactness lemma. Defines some helper lemmas associated with of judgment, dealing with substitution and evaluation contexts.

Then proves generalized compactness via the Pattern Stepping Bisimulation approach, and the standard definition of compactness falls out from this generalized compactness lemma.

pi314mm / compactness