Scopes Describe Frames: the Coq Development
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This artifact consists of Coq proof scripts defining and proving type soundness for the languages described in the paper

Casper Bach Poulsen, Pierre Neron, Andrew Tolmach, and Eelco Visser, "Scopes Decribe Frames: A Uniform Model for Memory Layout in Dynamic Semantics."

"Running" these scripts, i.e. passing them successfully through the Coq type-checker, serves to verify that these proofs are correct.

Reading the scripts may give insight into the structure of the proofs, and in particular about how the scopes-as-frames framework is defined in a generic way independent of the specific languages.


To pass all scripts through Coq, run make. All proofs are known to compile with Coq 8.5.

It is also possible to step through the proofs interactively using an IDE such as CoqIDE or ProofGeneral.

Pretty-Printed Scripts

To (re-)generate pretty-printed versions of the proof scripts that may be easier to read, run make html. This runs the CoqDoc utility to generate formatted html files in the html folder. html/toc.html contains a table of contents, and html/index.html an index over the various definitions and lemmas.


The files of this Coq development are structured as follows:

  • html/ contains pretty-printed versions of the scripts.

  • framework/ contains the generic, language-independent framework for scopes-as-frames:

    • scopes.v: the formalization of resolved scope graphs, parameterized by an object language notion of types.

    • frames.v: the formalization of frames, heaps, and the good_frame and good_heap properties that capture the essence of type soundness using scopes-as-frames. Definitions and lemmas are parameterized by an object language notion of values, types, and default values.

    • GC.v: additional definitions and lemmas for sound garbage collection.

    • sub.v: a straightforward adaptation of the key lemmas from frames.v and the good_frame and good_heap properties to support subtyping.

    • maps.v: a formalization of finite maps, using lists.

    • prop_fold.v: a generic propositional fold predicate, used for sequential n-ary let-bindings in L2 (see "Differences from the paper" below).

  • langs/ contains folders for each language in the paper:

    • L1/syntax.v: the abstract syntax definition for L1.

    • L1/well_boundness.v: well-boundness rules for L1.

    • L1/well_typedness.v: well-typedness rules for L1.

    • L1/semantics.v: the dynamic semantics for L1.

    • L1/type_soundness.v: the type soundness proof for L1.

  • The files in langs/L3 are structured similarly to L1.

  • The files in langs/L2 are also structured similarly, but contains two additional files:

    • L2/GCsemantics.v: the dynamic semantics for L2 with the generic garbage collection rule.

    • L2/GCtype_soundness.v: the type soundness proof for L2 with sound garbage collection.

Differences from the paper

The languages described in our paper are simplified versions of the languages that we used to experiment with proving type soundness. There are numerous small differences of naming and terminology. In additon, the languages in this artifact differ from the paper in the following ways:

  • L1 follows the semantics in the paper.

  • L2 and L3 differ from the paper as follows:

    • Functions and function types are n-ary, and argument values are stored in call-frames immediately after they are computed (as opposed to in the big-step derivation tree).

    • The language provides three variants of n-ary let-binding: sequential lets, parallel lets, and recursive lets (following the static semantics given for these in [1, 2]). Recursive lets are restricted to bind values of function type only.

    • The language has boolean expressions and simple if-then-else branching.


[1] Pierre Neron, Andrew P. Tolmach, Eelco Visser, Guido Wachsmuth: A Theory of Name Resolution. ESOP 2015: 205-231 doi:

[2] Hendrik van Antwerpen, Pierre Néron, Andrew P. Tolmach, Eelco Visser, Guido Wachsmuth. A Constraint Language for Static Semantic Analysis based on Scope Graphs. PEPM 2016: 49–60 doi: