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https://github.com/lapin0t/ogs

operational game semantics, formalized in Coq
https://github.com/lapin0t/ogs

coq semantics

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operational game semantics, formalized in Coq

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README 

        
# An abstract, certified account of operational game semantics



This is the companion artifact to the paper. The main contributions of

this library are:



- an independent implementation of an indexed counterpart to the itree

  library with support for guarded and eventually guarded recursion

- an abstract OGS model of an axiomatised language proven sound w.r.t.

  substitution equivalence

- several instantiations of this abstract result to concrete

  languages: simply-typed lambda-calculus with recursive functions,

  untyped lambda calculus, Downen and Ariola\'s polarized system D and

  system L.



## Meta



- Author(s):

  - Peio Borthelle

  - Tom Hirschowitz

  - Guilhem Jaber

  - Yannick Zakowski

- License: GPLv3

- Compatible Coq versions: 8.17

- Additional dependencies:

  - dune

  - [Equations](https://github.com/mattam82/Coq-Equations)

  - [Coinduction](https://github.com/damien-pous/coinduction)

  - [Alectryon](https://github.com/cpitclaudel/alectryon)

- Coq namespace: `OGS`

- [Documentation](https://lapin0t.github.io/ogs/Readme.html)



## Building instructions



### Installing dependencies



Download the project with



``` shell

git clone https://github.com/lapin0t/ogs.git

cd ogs

```



We stress that the development has been only checked to compile against

these specific dependencies. In particular, it does not compiled at the

moment against latest version of coq-coinduction due to major changes in

the interface.



Installing the opam dependencies automatically:



``` shell

opam install --deps-only .

```



or manually:



``` shell

opam pin coq 8.17

opam pin coq-equations 1.3

opam pin coq-coinduction 1.6

```



### Building the project



``` shell

dune build

```



### Alectryon documentation



To build the html documentation, first install Alectryon:



``` shell

opam install "coq-serapi==8.17.0+0.17.1"

python3 -m pip install --user alectryon

```



Then build with:



``` shell

make doc

```



You can start a local web server to view it with:



``` shell

make serve

```



## Content



### Structure of the repository



The Coq development is contained in the theory directory, which has the

following structure, in approximate order of dependency.



- [Readme.v](https://lapin0t.github.io/ogs/Readme.html): this file

- [Prelude.v](https://lapin0t.github.io/ogs/Prelude.html): imports and setup

- Utils directory: general utilities

  - [Rel.v](https://lapin0t.github.io/ogs/Rel.html): generalities for relations over indexed types

  - [Psh.v](https://lapin0t.github.io/ogs/Psh.html): generalities for type families

- Ctx directory: general theory of contexts and variables

  - [Family.v](https://lapin0t.github.io/ogs/Family.html): definition of scoped and sorted

    families

  - [Abstract.v](https://lapin0t.github.io/ogs/Abstract.html): definition of context and variable

    structure

  - [Assignment.v](https://lapin0t.github.io/ogs/Assignment.html): generic definition of

    assignments

  - [Renaming.v](https://lapin0t.github.io/ogs/Renaming.html): generic definition of renamings

  - [Ctx.v](https://lapin0t.github.io/ogs/Ctx.html): definition of concrete contexts and DeBruijn

    indices

  - [Covering.v](https://lapin0t.github.io/ogs/Covering.html): concrete context structure for

    Ctx.v

  - [DirectSum.v](https://lapin0t.github.io/ogs/DirectSum.html): direct sum of context structures

  - [Subset.v](https://lapin0t.github.io/ogs/Subset.html): sub context structure

- ITree directory: implementation of a variant of interaction trees

  over indexed types

  - [Event.v](https://lapin0t.github.io/ogs/Event.html): indexed events parameterizing the

    interactions of an itree

  - [ITree.v](https://lapin0t.github.io/ogs/ITree.html): coinductive definition

  - [Eq.v](https://lapin0t.github.io/ogs/Eq.html): strong and weak bisimilarity over interaction

    trees

  - [Structure.v](https://lapin0t.github.io/ogs/Structure.html): combinators (definitions)

  - [Properties.v](https://lapin0t.github.io/ogs/Properties.html): general theory

  - [Guarded.v](https://lapin0t.github.io/ogs/Guarded.html): (eventually) guarded equations and

    iterations over them

  - [Delay.v](https://lapin0t.github.io/ogs/Delay.html): definition of the delay monad (as a

    special case of trees)

- OGS directory: construction of a sound OGS model for an abstract

  language

  - [Subst.v](https://lapin0t.github.io/ogs/Subst.html): axiomatization of substitution monoid,

    substitution module

  - [Obs.v](https://lapin0t.github.io/ogs/Obs.html): axiomatization of observation structure,

    normal forms

  - [Machine.v](https://lapin0t.github.io/ogs/Machine.html): axiomatization of evaluation

    structures, language machine

  - [Game.v](https://lapin0t.github.io/ogs/Game.html): abstract game and OGS game definition

  - [Strategy.v](https://lapin0t.github.io/ogs/Strategy.html): machine strategy and composition

  - [CompGuarded.v](https://lapin0t.github.io/ogs/CompGuarded.html): proof of eventual guardedness

    of the composition of strategies

  - [Adequacy.v](https://lapin0t.github.io/ogs/Adequacy.html): proof of adequacy of composition

  - [Congruence.v](https://lapin0t.github.io/ogs/Congruence.html): proof of congruence of

    composition

  - [Soundness.v](https://lapin0t.github.io/ogs/Soundness.html): proof of soundness of the OGS

- Examples directory: concrete instances of the generic construction

  - [STLC_CBV.v](https://lapin0t.github.io/ogs/STLC_CBV.html): simply typed, call by value, lambda

    calculus

  - [ULC_CBV.v](https://lapin0t.github.io/ogs/ULC_CBV.html): untyped, call by value, lambda

    calculus

  - [SystemL_CBV.v](https://lapin0t.github.io/ogs/SystemL_CBV.html): mu-mu-tilde calculus variant

    System L, in call by value

  - [SystemD.v](https://lapin0t.github.io/ogs/SystemD.html): mu-mu-tilde calculus variant System

    D, polarized



### Axioms



The whole development relies only on axiom K, a conventional and sound

axiom from Coq\'s standard library (more precisely,

[`Eq_rect_eq.rect_eq`](https://coq.inria.fr/doc/V8.19.0/stdlib/Coq.Logic.Eqdep.html#Eq_rect_eq.eq_rect_eq).



This can be double checked as follows:



-   for the abstract result of soundness of the OGS by running

    `Print Assumptions ogs_correction.` at the end of OGS/Soundness.v

-   for any particular example, for instance by running

    `Print Assumptions stlc_ciu_correct.` at the end of

    Example/CBVTyped.v