sweirich / equations-ln

An interactive tutorial on specifying and implementing the simply-typed lambda calculus in Coq using a locally nameless representation.

This tutorial is based on the Stlc tutorial found in the metalib repository. However, it experiments with making two major changes:

1. This version uses the Equations Coq package to index the expression type with the number of bound variables. This definition means that the local closure predicate is not needed --- instead the type "exp 0" only contains locally closed terms.

2. This version uses type classes to simplify the names of operations and their lemmas. See the Classes module for the definition of those type classes. Some of the instances are defined in the middle of Definitions and others at the end of Lemmas.

INSTALLATION

The code has been tested with Coq 8.15.0. Other versions of Coq may not be compatible.

This tutorial depends on the Metalib.Metatheory library, available from https://github.com/plclub/metalib. Make sure that you compile and install this library first.

You also need to install the equations library, version 1.3+8.15 available from opam.

After you have done that, you can use make

`make`             - Compile all Coq files

CONTENTS

 _CoqProject         - list of modules

 Fin.v               - Finite numbers, defined using equations

 DefinitionsSyntax.v - Specification of STLC using locally nameless
                       representation (LN)

 Lemmas.v            - infrastructure lemmas about binding, similar 
                       to the form generated by LNgen. Proofs differ 
                       due to the use of equations.

 Classes.v           - type class definitions for operations on 
                       syntax (open/close/etc) and their properties
 
 ClassInstances.v    - Use Definitions and Lemmas to instantiate 
                       general type classes.
                     
 ClassRewrites.v     - Rewrite lemmas (equations) for the operations in
                       terms of the class names.
                     
 DefinitionsTyping.v - Typing rules for STLC
                     
 Lec2.v              - type soundness for STLC

WORKING GROUP PLAN (equations)

• Fin.v try proof: decrease_fin_inj

• DefinitionsSyntax.v compare definitions with equations and Fixpoint

• Lemmas.v pick a proof and try to recreate/simplify it

QUESTIONS (general)

• Should "open" and "subst" call weaken u with every recursion under a binder? Or should it be one call where u actually appears. (i.e. should u have type exp n or exp 0?)

• Is there a (confluent) rewriting theory for the lemmas? Have created a hint db "syntax" for this rewriting theory. But how to use it?

• Is this version any better than the previous version? Not clear.

  • lc predicates are gone, but we now have "weaken" to reason about.
  • The use of dependent types places additional constraints (depelim/dependent induction instead of case analysis, inversion, regular induction).