From 260bde12399ccb57e07bd9f5f351297d27a5e942 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Mon, 11 Sep 2017 10:29:22 +0200 Subject: [PATCH 01/81] Add README.md. --- README.md | 8 ++++++++ 1 file changed, 8 insertions(+) create mode 100644 README.md diff --git a/README.md b/README.md new file mode 100644 index 0000000..17232af --- /dev/null +++ b/README.md @@ -0,0 +1,8 @@ +# Functional programming and type systems + +## Teachers in 2017-2018 + + * Functional Programming: Under the Hood (12h30, [http://gallium.inria.fr/~fpottier](François Pottier) + * Metatheory of Typed Programming Languages (12h30, [http://gallium.inria.fr/~remy/](Didier Rémy, *head*) + * Advanced Aspects of Type Systems (12h30, [http://www.pps.jussieu.fr/~yrg/](Yann Régis Gianas) + * Dependently-typed Functional Programming (12h30, [https://pages.lip6.fr/Pierre-Evariste.Dagand/](Pierre-Evariste Dagand) From aecd44d1a1974b5f1f8bf2237ad1041fa739e6ba Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Mon, 11 Sep 2017 10:32:16 +0200 Subject: [PATCH 02/81] Fixed links. --- README.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/README.md b/README.md index 17232af..cd8a80f 100644 --- a/README.md +++ b/README.md @@ -2,7 +2,7 @@ ## Teachers in 2017-2018 - * Functional Programming: Under the Hood (12h30, [http://gallium.inria.fr/~fpottier](François Pottier) - * Metatheory of Typed Programming Languages (12h30, [http://gallium.inria.fr/~remy/](Didier Rémy, *head*) - * Advanced Aspects of Type Systems (12h30, [http://www.pps.jussieu.fr/~yrg/](Yann Régis Gianas) - * Dependently-typed Functional Programming (12h30, [https://pages.lip6.fr/Pierre-Evariste.Dagand/](Pierre-Evariste Dagand) + * Functional Programming: Under the Hood (12h30, [François Pottier](http://gallium.inria.fr/~fpottier) + * Metatheory of Typed Programming Languages (12h30, [Didier Rémy, *head*](http://gallium.inria.fr/~remy/) + * Advanced Aspects of Type Systems (12h30, [Yann Régis Gianas](http://www.pps.jussieu.fr/~yrg/) + * Dependently-typed Functional Programming (12h30, [Pierre-Evariste Dagand](https://pages.lip6.fr/Pierre-Evariste.Dagand/) From d6f91d2412a063911939cf861e735d6061ae4660 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Mon, 11 Sep 2017 10:32:58 +0200 Subject: [PATCH 03/81] Closing parens. --- README.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/README.md b/README.md index cd8a80f..46dae3d 100644 --- a/README.md +++ b/README.md @@ -2,7 +2,7 @@ ## Teachers in 2017-2018 - * Functional Programming: Under the Hood (12h30, [François Pottier](http://gallium.inria.fr/~fpottier) - * Metatheory of Typed Programming Languages (12h30, [Didier Rémy, *head*](http://gallium.inria.fr/~remy/) - * Advanced Aspects of Type Systems (12h30, [Yann Régis Gianas](http://www.pps.jussieu.fr/~yrg/) - * Dependently-typed Functional Programming (12h30, [Pierre-Evariste Dagand](https://pages.lip6.fr/Pierre-Evariste.Dagand/) + * Functional Programming: Under the Hood (12h30, [François Pottier](http://gallium.inria.fr/~fpottier)) + * Metatheory of Typed Programming Languages (12h30, [Didier Rémy](http://gallium.inria.fr/~remy/), *head*) + * Advanced Aspects of Type Systems (12h30, [Yann Régis Gianas](http://www.pps.jussieu.fr/~yrg/)) + * Dependently-typed Functional Programming (12h30, [Pierre-Evariste Dagand](https://pages.lip6.fr/Pierre-Evariste.Dagand/)) From 0b50ea13f28e929822eb1ed6e6a4ec654e7556a0 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Mon, 11 Sep 2017 10:40:02 +0200 Subject: [PATCH 04/81] Added aims and syllabus. --- README.md | 81 +++++++++++++++++++++++++++++++++++++++++++++++++++++-- 1 file changed, 79 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index 46dae3d..4b1be86 100644 --- a/README.md +++ b/README.md @@ -1,8 +1,85 @@ -# Functional programming and type systems +# Functional programming and type systems (2017-2018) -## Teachers in 2017-2018 +## Teachers * Functional Programming: Under the Hood (12h30, [François Pottier](http://gallium.inria.fr/~fpottier)) * Metatheory of Typed Programming Languages (12h30, [Didier Rémy](http://gallium.inria.fr/~remy/), *head*) * Advanced Aspects of Type Systems (12h30, [Yann Régis Gianas](http://www.pps.jussieu.fr/~yrg/)) * Dependently-typed Functional Programming (12h30, [Pierre-Evariste Dagand](https://pages.lip6.fr/Pierre-Evariste.Dagand/)) + +## Aims + +This course presents the principles and formalisms that underlie many of +today's typed functional programming languages. + +The course is made up of four parts and can be split after the first two +parts. + +In the first part, we discuss the *operational semantics* of functional +programming languages, and we present several classic *program +transformations*, including closure conversion, defunctionalization, and the +transformation into continuation-passing style (CPS). These program +transformations are interesting from two points of view. First, they are +*useful programming techniques*, which can help write or understand +programs. Second, they are used in the *compilation* of functional +programming languages, so they help understand what happens when the machine +executes a program. We use operational semantics to *prove* that the meaning +of programs is preserved by these transformations. Finally, we suggest how +these definitions and theorems can be expressed in a form that a machine can +check. That is, although Coq is not a prerequisite of the course, we will at +least try to *read and understand Coq definitions and statements*. + +In the second part, we focus on the meta-theoretical properties of type +systems. We study parametric polymorphism (as in System F and ML), data +types and type abstraction. We show syntactic type soundness (via progress +and subject reduction) by reasoning by induction on typing derivations. We +also show how to obtain semantic properties via logical relations by +reasoning by induction on the structure of types. We also introduce +subtyping and row polymorphism and illustrate typing problems induced by +side effects (references) and the need for the value restriction. + +The third part of the course describes more advanced features of type +systems: exceptions and effect handlers, including their typechecking and +static analyses: type inference, data flow and control flow analyses. +Finally, it introduces dependent types and refinement types. + +The last part focuses on the use of dependent types for programming: +effectful programming with monads and algebraic effects; tagless +interpreters; programming with total functions; generic programming. +We also show the limits of dependently-typed functional programming. + +## Approximate syllabus + +### Functional Programming: Under the Hood + + * (22/09/2017) From a small-step operational semantics... + * (29/09/2017) ... to an efficient interpreter. (2 weeks.) + * (06/10/2017) Compiling away first-class functions: closure conversion, defunctionalization. + * (13/10/2017) Compiling away the call stack: the CPS transformation. + * (20/10/2017) Equational reasoning and program optimizations. + +### Metatheory of Typed Programming Languages + +First lesson on Sep 15, other lessons on Oct 27, Nov 02, 17, 24. + + * Metatheory of System F. (Type soundness. Erasure.) + * ADTs, existential types, GADTs. (Typed program transformations.) + * Logical relations. + * Subtyping. Rows. (Covariant arrays and covariant functions!) + * References. (Value restriction.) + +### Advanced Aspects of Type Systems + + * Exceptions and effect handlers. (Compiled away via CPS.) + * Typechecking exceptions and handlers. + * Type inference. (ML. Bidirectional. Elaboration.) + * Data/control flow analysis. + * Functional correctness. Intro to dependent/refinement types. + +### Dependently-typed Functional Programming + + * Effectful functional programming. + * Dependent functional programming. + * Total functional programming. + * Generic functional programming. + * Open problems in dependent functional programming. From 7747bcfe3b8a9c37a93375fc57e5bddcec70b01c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Mon, 11 Sep 2017 10:41:47 +0200 Subject: [PATCH 05/81] Fix lists. --- README.md | 40 ++++++++++++++++++++-------------------- 1 file changed, 20 insertions(+), 20 deletions(-) diff --git a/README.md b/README.md index 4b1be86..db4d39d 100644 --- a/README.md +++ b/README.md @@ -52,34 +52,34 @@ We also show the limits of dependently-typed functional programming. ### Functional Programming: Under the Hood - * (22/09/2017) From a small-step operational semantics... - * (29/09/2017) ... to an efficient interpreter. (2 weeks.) - * (06/10/2017) Compiling away first-class functions: closure conversion, defunctionalization. - * (13/10/2017) Compiling away the call stack: the CPS transformation. - * (20/10/2017) Equational reasoning and program optimizations. +* (22/09/2017) From a small-step operational semantics... +* (29/09/2017) ... to an efficient interpreter. (2 weeks.) +* (06/10/2017) Compiling away first-class functions: closure conversion, defunctionalization. +* (13/10/2017) Compiling away the call stack: the CPS transformation. +* (20/10/2017) Equational reasoning and program optimizations. ### Metatheory of Typed Programming Languages First lesson on Sep 15, other lessons on Oct 27, Nov 02, 17, 24. - * Metatheory of System F. (Type soundness. Erasure.) - * ADTs, existential types, GADTs. (Typed program transformations.) - * Logical relations. - * Subtyping. Rows. (Covariant arrays and covariant functions!) - * References. (Value restriction.) +* Metatheory of System F. (Type soundness. Erasure.) +* ADTs, existential types, GADTs. (Typed program transformations.) +* Logical relations. +* Subtyping. Rows. (Covariant arrays and covariant functions!) +* References. (Value restriction.) ### Advanced Aspects of Type Systems - * Exceptions and effect handlers. (Compiled away via CPS.) - * Typechecking exceptions and handlers. - * Type inference. (ML. Bidirectional. Elaboration.) - * Data/control flow analysis. - * Functional correctness. Intro to dependent/refinement types. +* Exceptions and effect handlers. (Compiled away via CPS.) +* Typechecking exceptions and handlers. +* Type inference. (ML. Bidirectional. Elaboration.) +* Data/control flow analysis. +* Functional correctness. Intro to dependent/refinement types. ### Dependently-typed Functional Programming - * Effectful functional programming. - * Dependent functional programming. - * Total functional programming. - * Generic functional programming. - * Open problems in dependent functional programming. +* Effectful functional programming. +* Dependent functional programming. +* Total functional programming. +* Generic functional programming. +* Open problems in dependent functional programming. From e1de1928c561d33fc99abb7746621da673b5d82d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Mon, 11 Sep 2017 10:43:57 +0200 Subject: [PATCH 06/81] Fixed dates. --- README.md | 12 +++++------- 1 file changed, 5 insertions(+), 7 deletions(-) diff --git a/README.md b/README.md index db4d39d..cb731b7 100644 --- a/README.md +++ b/README.md @@ -60,13 +60,11 @@ We also show the limits of dependently-typed functional programming. ### Metatheory of Typed Programming Languages -First lesson on Sep 15, other lessons on Oct 27, Nov 02, 17, 24. - -* Metatheory of System F. (Type soundness. Erasure.) -* ADTs, existential types, GADTs. (Typed program transformations.) -* Logical relations. -* Subtyping. Rows. (Covariant arrays and covariant functions!) -* References. (Value restriction.) +* (15/09/2017) Metatheory of System F. (Type soundness. Erasure.) +* (27/10/2017) ADTs, existential types, GADTs. (Typed program transformations.) +* (03/11/2017) Logical relations. +* (17/11/2017) Sub(typing. Rows. (Covariant arrays and covariant functions!) +* (24/11/2017) References. (Value restriction.) ### Advanced Aspects of Type Systems From f00f5b1b74449ccd475887f11b16fe4798f8bd22 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Mon, 11 Sep 2017 10:47:09 +0200 Subject: [PATCH 07/81] Added recommended software. --- README.md | 12 ++++++++++++ 1 file changed, 12 insertions(+) diff --git a/README.md b/README.md index cb731b7..9160de3 100644 --- a/README.md +++ b/README.md @@ -81,3 +81,15 @@ We also show the limits of dependently-typed functional programming. * Total functional programming. * Generic functional programming. * Open problems in dependent functional programming. + +## Recommended software + +OCaml 4.0x and Coq **8.5**. + +Once you have installed [opam](https://opam.ocaml.org/doc/Install.html), use the following commands: +```bash +opam init --comp=4.05 # for instance +opam repo add coq-released https://coq.inria.fr/opam/released +opam update +opam install -j4 -v coq.8.5.3 +``` From 51b07afefcf585fc384a8492dab12add36b33701 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Mon, 11 Sep 2017 10:50:20 +0200 Subject: [PATCH 08/81] Bibliography. --- README.md | 5 +++++ 1 file changed, 5 insertions(+) diff --git a/README.md b/README.md index 9160de3..6e4bbe0 100644 --- a/README.md +++ b/README.md @@ -93,3 +93,8 @@ opam repo add coq-released https://coq.inria.fr/opam/released opam update opam install -j4 -v coq.8.5.3 ``` + +## Bibliography + +[Types and Programming Languages](https://mitpress.mit.edu/books/types-and-programming-languages), +Benjamin C. Pierce, MIT Press, 2002. From 1df18e38320697ed33b60462fe6a433da6c79776 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Mon, 11 Sep 2017 10:50:45 +0200 Subject: [PATCH 09/81] .gitignore. --- .gitignore | 1 + 1 file changed, 1 insertion(+) create mode 100644 .gitignore diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..b25c15b --- /dev/null +++ b/.gitignore @@ -0,0 +1 @@ +*~ From 1a926bb89dd5d610ed11731d4b2bc728a8979e15 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Mon, 11 Sep 2017 11:07:05 +0200 Subject: [PATCH 10/81] Typo. --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 6e4bbe0..ee3e3c3 100644 --- a/README.md +++ b/README.md @@ -63,7 +63,7 @@ We also show the limits of dependently-typed functional programming. * (15/09/2017) Metatheory of System F. (Type soundness. Erasure.) * (27/10/2017) ADTs, existential types, GADTs. (Typed program transformations.) * (03/11/2017) Logical relations. -* (17/11/2017) Sub(typing. Rows. (Covariant arrays and covariant functions!) +* (17/11/2017) Subtyping. Rows. (Covariant arrays and covariant functions!) * (24/11/2017) References. (Value restriction.) ### Advanced Aspects of Type Systems From 4cceeb632a564bf1f42980c9567e001a30f03a4b Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Mon, 11 Sep 2017 11:55:20 +0200 Subject: [PATCH 11/81] Correction de dates --- README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index ee3e3c3..61da2a7 100644 --- a/README.md +++ b/README.md @@ -63,8 +63,8 @@ We also show the limits of dependently-typed functional programming. * (15/09/2017) Metatheory of System F. (Type soundness. Erasure.) * (27/10/2017) ADTs, existential types, GADTs. (Typed program transformations.) * (03/11/2017) Logical relations. -* (17/11/2017) Subtyping. Rows. (Covariant arrays and covariant functions!) -* (24/11/2017) References. (Value restriction.) +* (10/11/2017) Subtyping. Rows. (Covariant arrays and covariant functions!) +* (17/11/2017) References. (Value restriction.) ### Advanced Aspects of Type Systems From 599d85f58df67c8fa4032da7951ada4cce33d78e Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Thu, 14 Sep 2017 11:22:15 +0200 Subject: [PATCH 12/81] +biblio, course content --- README.md | 13 ++++++++----- 1 file changed, 8 insertions(+), 5 deletions(-) diff --git a/README.md b/README.md index 61da2a7..0173682 100644 --- a/README.md +++ b/README.md @@ -3,7 +3,7 @@ ## Teachers * Functional Programming: Under the Hood (12h30, [François Pottier](http://gallium.inria.fr/~fpottier)) - * Metatheory of Typed Programming Languages (12h30, [Didier Rémy](http://gallium.inria.fr/~remy/), *head*) + * [Metatheory of Typed Programming Languages](http://gallium.inria.fr/~remy/mpri/) (12h30, [Didier Rémy](http://gallium.inria.fr/~remy/), *head*) * Advanced Aspects of Type Systems (12h30, [Yann Régis Gianas](http://www.pps.jussieu.fr/~yrg/)) * Dependently-typed Functional Programming (12h30, [Pierre-Evariste Dagand](https://pages.lip6.fr/Pierre-Evariste.Dagand/)) @@ -61,10 +61,10 @@ We also show the limits of dependently-typed functional programming. ### Metatheory of Typed Programming Languages * (15/09/2017) Metatheory of System F. (Type soundness. Erasure.) -* (27/10/2017) ADTs, existential types, GADTs. (Typed program transformations.) +* (27/10/2017) ADTs, existential types, GADTs. * (03/11/2017) Logical relations. -* (10/11/2017) Subtyping. Rows. (Covariant arrays and covariant functions!) -* (17/11/2017) References. (Value restriction.) +* (10/11/2017) Subtyping. Rows. +* (17/11/2017) References, Value restriction, Side effects. ### Advanced Aspects of Type Systems @@ -96,5 +96,8 @@ opam install -j4 -v coq.8.5.3 ## Bibliography -[Types and Programming Languages](https://mitpress.mit.edu/books/types-and-programming-languages), +[Types and Programming Languages](https://mitpress.mit.edu/books/types-and-programming-languages), Benjamin C. Pierce, MIT Press, 2002. + +[Advanced Topics in Types and Programming Languages](https://www.cis.upenn.edu/~bcpierce/attapl/), +Edited by Benjamin C. Pierce, MIT Press, 2005. From 6e18628cae17d7de125e02e80498e96ec3d7d3de Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Thu, 14 Sep 2017 15:12:03 +0200 Subject: [PATCH 13/81] Previous Exams --- README.md | 27 +++++++++++++++++++++++++++ 1 file changed, 27 insertions(+) diff --git a/README.md b/README.md index 0173682..78c9715 100644 --- a/README.md +++ b/README.md @@ -82,6 +82,33 @@ We also show the limits of dependently-typed functional programming. * Generic functional programming. * Open problems in dependent functional programming. +## Evaluation of the course + +Two written exams (a partial and a final exam) and one programming project +or several programming exercises are used to evaluate the students that +follow the full course. Only the partial exam will count to grade students +who split the course. + +Although the course has changed, you may still have a look at +[http://gallium.inria.fr/~remy/mpri/index.html#evaluation](Previous exams) +available with solutions. + + +- mid-term exam 2016-2017: [partiel-2016-2017.pdf](Record concatenation) +- mid-term exam 2015-2016: [partiel-2015-2016.pdf](Type containment) +- final exam 2014-2015: [final-2014-2015.pdf {Copatterns}](subject) +- mid-term exam 2014-2015: [partiel-2014-2015.pdf](Information flow) +- final exam 2013-2014: [final-2013-2014.pdf](Operation on records) +- mid-term exam 2013-2014: [partiel-2013-2014.pdf](Typechecking Effects) +- final exam 2012-2013: [final-2012-2013.pdf](Refinement types) +- mid-term exam 2012-2013: [partiel-2012-2013.pdf](Variations on ML) +- final exam 2011-2012: [final-2011-2012.pdf](Intersection types) +- mid-term exam 2011-2012: [partiel-2011-2012.pdf](Parametricity) +- final exam 2010-2011: + [http://gallium.inria.fr/\home{xleroy}/mpri/2-4/exam-2010-2011.pdf](Compiling a language with subtyping) +- mid-term exam 2010-2011: [2010/partiel-2010-2011.pdf](Compilation of +polymorphic records) + ## Recommended software OCaml 4.0x and Coq **8.5**. From 85ceb49c7dd5d131f307008a1859e12531177c19 Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Thu, 14 Sep 2017 15:18:16 +0200 Subject: [PATCH 14/81] url --- README.md | 40 ++++++++++++++++++++++++---------------- 1 file changed, 24 insertions(+), 16 deletions(-) diff --git a/README.md b/README.md index 78c9715..79fccb3 100644 --- a/README.md +++ b/README.md @@ -90,24 +90,32 @@ follow the full course. Only the partial exam will count to grade students who split the course. Although the course has changed, you may still have a look at -[http://gallium.inria.fr/~remy/mpri/index.html#evaluation](Previous exams) -available with solutions. +Previous exams available with solutions: - -- mid-term exam 2016-2017: [partiel-2016-2017.pdf](Record concatenation) -- mid-term exam 2015-2016: [partiel-2015-2016.pdf](Type containment) -- final exam 2014-2015: [final-2014-2015.pdf {Copatterns}](subject) -- mid-term exam 2014-2015: [partiel-2014-2015.pdf](Information flow) -- final exam 2013-2014: [final-2013-2014.pdf](Operation on records) -- mid-term exam 2013-2014: [partiel-2013-2014.pdf](Typechecking Effects) -- final exam 2012-2013: [final-2012-2013.pdf](Refinement types) -- mid-term exam 2012-2013: [partiel-2012-2013.pdf](Variations on ML) -- final exam 2011-2012: [final-2011-2012.pdf](Intersection types) -- mid-term exam 2011-2012: [partiel-2011-2012.pdf](Parametricity) +- mid-term exam 2016-2017: + [http://gallium.inria.fr/~remy/mpri/exams/partiel-2016-2017.pdf](Record concatenation) +- mid-term exam 2015-2016: + [http://gallium.inria.fr/~remy/mpri/exams/partiel-2015-2016.pdf](Type containment) +- final exam 2014-2015: [http://gallium.inria.fr/~remy/mpri/exams/final-2014-2015.pdf {Copatterns}](subject) +- mid-term exam 2014-2015: + [http://gallium.inria.fr/~remy/mpri/exams/partiel-2014-2015.pdf](Information flow) +- final exam 2013-2014: + [http://gallium.inria.fr/~remy/mpri/exams/final-2013-2014.pdf](Operation on records) +- mid-term exam 2013-2014: + [http://gallium.inria.fr/~remy/mpri/exams/partiel-2013-2014.pdf](Typechecking Effects) +- final exam 2012-2013: + [http://gallium.inria.fr/~remy/mpri/exams/final-2012-2013.pdf](Refinement types) +- mid-term exam 2012-2013: + [http://gallium.inria.fr/~remy/mpri/exams/partiel-2012-2013.pdf](Variations on ML) +- final exam 2011-2012: + [http://gallium.inria.fr/~remy/mpri/exams/final-2011-2012.pdf](Intersection types) +- mid-term exam 2011-2012: + [http://gallium.inria.fr/~remy/mpri/exams/partiel-2011-2012.pdf](Parametricity) - final exam 2010-2011: - [http://gallium.inria.fr/\home{xleroy}/mpri/2-4/exam-2010-2011.pdf](Compiling a language with subtyping) -- mid-term exam 2010-2011: [2010/partiel-2010-2011.pdf](Compilation of -polymorphic records) + [http://gallium.inria.fr/~xleroy/mpri/2-4/exam-2010-2011.pdf](Compiling a language with subtyping) +- mid-term exam 2010-2011: + [http://gallium.inria.fr/~remy/mpri/exams/2010/partiel-2010-2011.pdf](Compilation + of polymorphic records) ## Recommended software From 88a9804cabaf012b5516ea20ad68eb05d932c4f6 Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Thu, 14 Sep 2017 15:20:36 +0200 Subject: [PATCH 15/81] urls --- README.md | 26 +++++++++++++------------- 1 file changed, 13 insertions(+), 13 deletions(-) diff --git a/README.md b/README.md index 79fccb3..59498ba 100644 --- a/README.md +++ b/README.md @@ -93,29 +93,29 @@ Although the course has changed, you may still have a look at Previous exams available with solutions: - mid-term exam 2016-2017: - [http://gallium.inria.fr/~remy/mpri/exams/partiel-2016-2017.pdf](Record concatenation) + [(Record concatenation](http://gallium.inria.fr/~remy/mpri/exams/partiel-2016-2017.pdf) - mid-term exam 2015-2016: - [http://gallium.inria.fr/~remy/mpri/exams/partiel-2015-2016.pdf](Type containment) -- final exam 2014-2015: [http://gallium.inria.fr/~remy/mpri/exams/final-2014-2015.pdf {Copatterns}](subject) + [(Type containment](http://gallium.inria.fr/~remy/mpri/exams/partiel-2015-2016.pdf) +- final exam 2014-2015: [(subject](http://gallium.inria.fr/~remy/mpri/exams/final-2014-2015.pdf {Copatterns}) - mid-term exam 2014-2015: - [http://gallium.inria.fr/~remy/mpri/exams/partiel-2014-2015.pdf](Information flow) + [(Information flow](http://gallium.inria.fr/~remy/mpri/exams/partiel-2014-2015.pdf) - final exam 2013-2014: - [http://gallium.inria.fr/~remy/mpri/exams/final-2013-2014.pdf](Operation on records) + [(Operation on records](http://gallium.inria.fr/~remy/mpri/exams/final-2013-2014.pdf) - mid-term exam 2013-2014: - [http://gallium.inria.fr/~remy/mpri/exams/partiel-2013-2014.pdf](Typechecking Effects) + [(Typechecking Effects](http://gallium.inria.fr/~remy/mpri/exams/partiel-2013-2014.pdf) - final exam 2012-2013: - [http://gallium.inria.fr/~remy/mpri/exams/final-2012-2013.pdf](Refinement types) + [(Refinement types](http://gallium.inria.fr/~remy/mpri/exams/final-2012-2013.pdf) - mid-term exam 2012-2013: - [http://gallium.inria.fr/~remy/mpri/exams/partiel-2012-2013.pdf](Variations on ML) + [(Variations on ML](http://gallium.inria.fr/~remy/mpri/exams/partiel-2012-2013.pdf) - final exam 2011-2012: - [http://gallium.inria.fr/~remy/mpri/exams/final-2011-2012.pdf](Intersection types) + [(Intersection types](http://gallium.inria.fr/~remy/mpri/exams/final-2011-2012.pdf) - mid-term exam 2011-2012: - [http://gallium.inria.fr/~remy/mpri/exams/partiel-2011-2012.pdf](Parametricity) + [(Parametricity](http://gallium.inria.fr/~remy/mpri/exams/partiel-2011-2012.pdf) - final exam 2010-2011: - [http://gallium.inria.fr/~xleroy/mpri/2-4/exam-2010-2011.pdf](Compiling a language with subtyping) + [(Compiling a language with subtyping](http://gallium.inria.fr/~xleroy/mpri/2-4/exam-2010-2011.pdf) - mid-term exam 2010-2011: - [http://gallium.inria.fr/~remy/mpri/exams/2010/partiel-2010-2011.pdf](Compilation - of polymorphic records) + [(Compilation + of polymorphic records](http://gallium.inria.fr/~remy/mpri/exams/2010/partiel-2010-2011.pdf) ## Recommended software From 98cf311208b46fb34d45e2db15f4e2bd96eb631b Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Thu, 14 Sep 2017 15:22:47 +0200 Subject: [PATCH 16/81] urls --- README.md | 24 ++++++++++++------------ 1 file changed, 12 insertions(+), 12 deletions(-) diff --git a/README.md b/README.md index 59498ba..a0aae54 100644 --- a/README.md +++ b/README.md @@ -93,28 +93,28 @@ Although the course has changed, you may still have a look at Previous exams available with solutions: - mid-term exam 2016-2017: - [(Record concatenation](http://gallium.inria.fr/~remy/mpri/exams/partiel-2016-2017.pdf) + [Record concatenation](http://gallium.inria.fr/~remy/mpri/exams/partiel-2016-2017.pdf) - mid-term exam 2015-2016: - [(Type containment](http://gallium.inria.fr/~remy/mpri/exams/partiel-2015-2016.pdf) -- final exam 2014-2015: [(subject](http://gallium.inria.fr/~remy/mpri/exams/final-2014-2015.pdf {Copatterns}) + [Type containment](http://gallium.inria.fr/~remy/mpri/exams/partiel-2015-2016.pdf) +- final exam 2014-2015: [Copatterns](http://gallium.inria.fr/~remy/mpri/exams/final-2014-2015.pdf) - mid-term exam 2014-2015: - [(Information flow](http://gallium.inria.fr/~remy/mpri/exams/partiel-2014-2015.pdf) + [Information flow](http://gallium.inria.fr/~remy/mpri/exams/partiel-2014-2015.pdf) - final exam 2013-2014: - [(Operation on records](http://gallium.inria.fr/~remy/mpri/exams/final-2013-2014.pdf) + [Operation on records](http://gallium.inria.fr/~remy/mpri/exams/final-2013-2014.pdf) - mid-term exam 2013-2014: - [(Typechecking Effects](http://gallium.inria.fr/~remy/mpri/exams/partiel-2013-2014.pdf) + [Typechecking Effects](http://gallium.inria.fr/~remy/mpri/exams/partiel-2013-2014.pdf) - final exam 2012-2013: - [(Refinement types](http://gallium.inria.fr/~remy/mpri/exams/final-2012-2013.pdf) + [Refinement types](http://gallium.inria.fr/~remy/mpri/exams/final-2012-2013.pdf) - mid-term exam 2012-2013: - [(Variations on ML](http://gallium.inria.fr/~remy/mpri/exams/partiel-2012-2013.pdf) + [Variations on ML](http://gallium.inria.fr/~remy/mpri/exams/partiel-2012-2013.pdf) - final exam 2011-2012: - [(Intersection types](http://gallium.inria.fr/~remy/mpri/exams/final-2011-2012.pdf) + [Intersection types](http://gallium.inria.fr/~remy/mpri/exams/final-2011-2012.pdf) - mid-term exam 2011-2012: - [(Parametricity](http://gallium.inria.fr/~remy/mpri/exams/partiel-2011-2012.pdf) + [Parametricity](http://gallium.inria.fr/~remy/mpri/exams/partiel-2011-2012.pdf) - final exam 2010-2011: - [(Compiling a language with subtyping](http://gallium.inria.fr/~xleroy/mpri/2-4/exam-2010-2011.pdf) + [Compiling a language with subtyping](http://gallium.inria.fr/~xleroy/mpri/2-4/exam-2010-2011.pdf) - mid-term exam 2010-2011: - [(Compilation + [Compilation of polymorphic records](http://gallium.inria.fr/~remy/mpri/exams/2010/partiel-2010-2011.pdf) ## Recommended software From 873f1cfde11c6336086dbea3e978c26f6b2e05eb Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Thu, 14 Sep 2017 16:35:49 +0200 Subject: [PATCH 17/81] backpointer --- README.md | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index a0aae54..2990268 100644 --- a/README.md +++ b/README.md @@ -1,5 +1,7 @@ # Functional programming and type systems (2017-2018) +[The official MPRI page](https://wikimpri.dptinfo.ens-cachan.fr/doku.php?id=cours:c-2-4-2) + ## Teachers * Functional Programming: Under the Hood (12h30, [François Pottier](http://gallium.inria.fr/~fpottier)) @@ -89,8 +91,8 @@ or several programming exercises are used to evaluate the students that follow the full course. Only the partial exam will count to grade students who split the course. -Although the course has changed, you may still have a look at -Previous exams available with solutions: +Although the course has changed, you may still have a look at previous exams +available with solutions: - mid-term exam 2016-2017: [Record concatenation](http://gallium.inria.fr/~remy/mpri/exams/partiel-2016-2017.pdf) From d235435edbbef4bcd1e56723612fe1eb2386ddef Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Thu, 14 Sep 2017 18:11:11 +0200 Subject: [PATCH 18/81] duration --- README.md | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) diff --git a/README.md b/README.md index 2990268..6a98ffc 100644 --- a/README.md +++ b/README.md @@ -1,6 +1,13 @@ # Functional programming and type systems (2017-2018) -[The official MPRI page](https://wikimpri.dptinfo.ens-cachan.fr/doku.php?id=cours:c-2-4-2) +[The official MPRI +page](https://wikimpri.dptinfo.ens-cachan.fr/doku.php?id=cours:c-2-4-2) + +## Location + +The lessons take place at University of Paris 7 - Denis Diderot, +Batiment Sophie Germain in room 2035 on Fridays at 12:45 and ends at 15:30. +There will be a break of 15' in the middle of the course. ## Teachers From 1bd9634a2ab9bd535b7ee3bb5c2986a91aeb561f Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Fri, 15 Sep 2017 08:49:14 +0200 Subject: [PATCH 19/81] slides and notes --- README.md | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) diff --git a/README.md b/README.md index 6a98ffc..04a0c3f 100644 --- a/README.md +++ b/README.md @@ -69,7 +69,14 @@ We also show the limits of dependently-typed functional programming. ### Metatheory of Typed Programming Languages -* (15/09/2017) Metatheory of System F. (Type soundness. Erasure.) +* [(15/09/2017)] + [Metatheory of System F. (Type soundness. Erasure) + ](http://gallium.inria.fr/~remy/mpri/slides1.pdf) + [Intro](http://gallium.inria.fr/~remy/mpri/slides8.pdf) + See also [chap 1](http://gallium.inria.fr/~remy/mpri/2016/cours1.pdf) + and [chap 2](http://gallium.inria.fr/~remy/mpri/2016/cours2.pdf) + of course notes. + * (27/10/2017) ADTs, existential types, GADTs. * (03/11/2017) Logical relations. * (10/11/2017) Subtyping. Rows. From 997801873bcc0cad6b71d5e1c404e3e6656f971b Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Fri, 15 Sep 2017 08:52:43 +0200 Subject: [PATCH 20/81] slides and notes --- README.md | 10 ++++------ 1 file changed, 4 insertions(+), 6 deletions(-) diff --git a/README.md b/README.md index 04a0c3f..5985b1c 100644 --- a/README.md +++ b/README.md @@ -70,13 +70,11 @@ We also show the limits of dependently-typed functional programming. ### Metatheory of Typed Programming Languages * [(15/09/2017)] - [Metatheory of System F. (Type soundness. Erasure) - ](http://gallium.inria.fr/~remy/mpri/slides1.pdf) - [Intro](http://gallium.inria.fr/~remy/mpri/slides8.pdf) - See also [chap 1](http://gallium.inria.fr/~remy/mpri/2016/cours1.pdf) - and [chap 2](http://gallium.inria.fr/~remy/mpri/2016/cours2.pdf) + [Metatheory of System F](http://gallium.inria.fr/~remy/mpri/slides1.pdf) + See also [Intro](http://gallium.inria.fr/~remy/mpri/slides8.pdf), + and chap [1](http://gallium.inria.fr/~remy/mpri/2016/cours1.pdf) + and [2](http://gallium.inria.fr/~remy/mpri/2016/cours2.pdf) of course notes. - * (27/10/2017) ADTs, existential types, GADTs. * (03/11/2017) Logical relations. * (10/11/2017) Subtyping. Rows. From 672cfcd07d82940ce2187d55f6402ceaf621fd4b Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Fri, 15 Sep 2017 08:55:27 +0200 Subject: [PATCH 21/81] slides and notes --- README.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/README.md b/README.md index 5985b1c..04234a3 100644 --- a/README.md +++ b/README.md @@ -69,12 +69,12 @@ We also show the limits of dependently-typed functional programming. ### Metatheory of Typed Programming Languages -* [(15/09/2017)] +* (15/09/2017) [Metatheory of System F](http://gallium.inria.fr/~remy/mpri/slides1.pdf) - See also [Intro](http://gallium.inria.fr/~remy/mpri/slides8.pdf), + (See also [intro](http://gallium.inria.fr/~remy/mpri/slides8.pdf), and chap [1](http://gallium.inria.fr/~remy/mpri/2016/cours1.pdf) and [2](http://gallium.inria.fr/~remy/mpri/2016/cours2.pdf) - of course notes. + of [course notes](http://gallium.inria.fr/~remy/mpri/2016/cours.pdf)) * (27/10/2017) ADTs, existential types, GADTs. * (03/11/2017) Logical relations. * (10/11/2017) Subtyping. Rows. From 20967ac6710a3e888acffa19f8449fa8fff1049b Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Wed, 20 Sep 2017 13:50:54 +0200 Subject: [PATCH 22/81] ./2016/ -> ./ --- README.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/README.md b/README.md index 04234a3..ea12d46 100644 --- a/README.md +++ b/README.md @@ -72,9 +72,9 @@ We also show the limits of dependently-typed functional programming. * (15/09/2017) [Metatheory of System F](http://gallium.inria.fr/~remy/mpri/slides1.pdf) (See also [intro](http://gallium.inria.fr/~remy/mpri/slides8.pdf), - and chap [1](http://gallium.inria.fr/~remy/mpri/2016/cours1.pdf) - and [2](http://gallium.inria.fr/~remy/mpri/2016/cours2.pdf) - of [course notes](http://gallium.inria.fr/~remy/mpri/2016/cours.pdf)) + and chap [1](http://gallium.inria.fr/~remy/mpri/cours1.pdf) + and [2](http://gallium.inria.fr/~remy/mpri/cours2.pdf) + of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)) * (27/10/2017) ADTs, existential types, GADTs. * (03/11/2017) Logical relations. * (10/11/2017) Subtyping. Rows. From ace98b8d7aaa08d3dd9f71530d17bd060d5bfd1c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Wed, 20 Sep 2017 15:48:34 +0200 Subject: [PATCH 23/81] English. (Plus spurious whitespace.) --- README.md | 48 +++++++++++++++++++++++++----------------------- 1 file changed, 25 insertions(+), 23 deletions(-) diff --git a/README.md b/README.md index ea12d46..095ac4d 100644 --- a/README.md +++ b/README.md @@ -1,13 +1,15 @@ # Functional programming and type systems (2017-2018) -[The official MPRI -page](https://wikimpri.dptinfo.ens-cachan.fr/doku.php?id=cours:c-2-4-2) +This page supplements +[the official page of MPRI 2-4](https://wikimpri.dptinfo.ens-cachan.fr/doku.php?id=cours:c-2-4-2). -## Location +## Location and duration -The lessons take place at University of Paris 7 - Denis Diderot, -Batiment Sophie Germain in room 2035 on Fridays at 12:45 and ends at 15:30. -There will be a break of 15' in the middle of the course. +The lectures take place at University Paris 7 - Denis Diderot, +Bâtiment Sophie Germain, in room 2035. + +They are scheduled on Fridays from 12:45 to 15:30. +There is a 15-minute break in the middle of each lecture. ## Teachers @@ -43,8 +45,8 @@ systems. We study parametric polymorphism (as in System F and ML), data types and type abstraction. We show syntactic type soundness (via progress and subject reduction) by reasoning by induction on typing derivations. We also show how to obtain semantic properties via logical relations by -reasoning by induction on the structure of types. We also introduce -subtyping and row polymorphism and illustrate typing problems induced by +reasoning by induction on the structure of types. Finally, we introduce +subtyping, row polymorphism, and illustrate the problems induced by side effects (references) and the need for the value restriction. The third part of the course describes more advanced features of type @@ -71,13 +73,13 @@ We also show the limits of dependently-typed functional programming. * (15/09/2017) [Metatheory of System F](http://gallium.inria.fr/~remy/mpri/slides1.pdf) - (See also [intro](http://gallium.inria.fr/~remy/mpri/slides8.pdf), - and chap [1](http://gallium.inria.fr/~remy/mpri/cours1.pdf) - and [2](http://gallium.inria.fr/~remy/mpri/cours2.pdf) + (See also [intro](http://gallium.inria.fr/~remy/mpri/slides8.pdf), + and chap [1](http://gallium.inria.fr/~remy/mpri/cours1.pdf) + and [2](http://gallium.inria.fr/~remy/mpri/cours2.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)) -* (27/10/2017) ADTs, existential types, GADTs. +* (27/10/2017) ADTs, existential types, GADTs. * (03/11/2017) Logical relations. -* (10/11/2017) Subtyping. Rows. +* (10/11/2017) Subtyping. Rows. * (17/11/2017) References, Value restriction, Side effects. ### Advanced Aspects of Type Systems @@ -106,24 +108,24 @@ who split the course. Although the course has changed, you may still have a look at previous exams available with solutions: -- mid-term exam 2016-2017: +- mid-term exam 2016-2017: [Record concatenation](http://gallium.inria.fr/~remy/mpri/exams/partiel-2016-2017.pdf) -- mid-term exam 2015-2016: +- mid-term exam 2015-2016: [Type containment](http://gallium.inria.fr/~remy/mpri/exams/partiel-2015-2016.pdf) - final exam 2014-2015: [Copatterns](http://gallium.inria.fr/~remy/mpri/exams/final-2014-2015.pdf) -- mid-term exam 2014-2015: +- mid-term exam 2014-2015: [Information flow](http://gallium.inria.fr/~remy/mpri/exams/partiel-2014-2015.pdf) -- final exam 2013-2014: +- final exam 2013-2014: [Operation on records](http://gallium.inria.fr/~remy/mpri/exams/final-2013-2014.pdf) -- mid-term exam 2013-2014: +- mid-term exam 2013-2014: [Typechecking Effects](http://gallium.inria.fr/~remy/mpri/exams/partiel-2013-2014.pdf) -- final exam 2012-2013: +- final exam 2012-2013: [Refinement types](http://gallium.inria.fr/~remy/mpri/exams/final-2012-2013.pdf) -- mid-term exam 2012-2013: +- mid-term exam 2012-2013: [Variations on ML](http://gallium.inria.fr/~remy/mpri/exams/partiel-2012-2013.pdf) -- final exam 2011-2012: +- final exam 2011-2012: [Intersection types](http://gallium.inria.fr/~remy/mpri/exams/final-2011-2012.pdf) -- mid-term exam 2011-2012: +- mid-term exam 2011-2012: [Parametricity](http://gallium.inria.fr/~remy/mpri/exams/partiel-2011-2012.pdf) - final exam 2010-2011: [Compiling a language with subtyping](http://gallium.inria.fr/~xleroy/mpri/2-4/exam-2010-2011.pdf) @@ -145,7 +147,7 @@ opam install -j4 -v coq.8.5.3 ## Bibliography -[Types and Programming Languages](https://mitpress.mit.edu/books/types-and-programming-languages), +[Types and Programming Languages](https://mitpress.mit.edu/books/types-and-programming-languages), Benjamin C. Pierce, MIT Press, 2002. [Advanced Topics in Types and Programming Languages](https://www.cis.upenn.edu/~bcpierce/attapl/), From 165e637bc5acb4ef295a1ec315f3943697d7ba28 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Wed, 20 Sep 2017 15:51:16 +0200 Subject: [PATCH 24/81] Instructions for installing AutoSubst. --- README.md | 9 +++++++++ 1 file changed, 9 insertions(+) diff --git a/README.md b/README.md index 095ac4d..178beaa 100644 --- a/README.md +++ b/README.md @@ -145,6 +145,15 @@ opam update opam install -j4 -v coq.8.5.3 ``` +Please also install François Pottier's +[variant](https://github.com/fpottier/autosubst) +of the +[AutoSubst](https://www.ps.uni-saarland.de/autosubst/) library: +```bash +git clone git@github.com:fpottier/autosubst.git +make -C autosubst install +``` + ## Bibliography [Types and Programming Languages](https://mitpress.mit.edu/books/types-and-programming-languages), From 7b95f194eae3229a6982cbf9b3e769e26e32586a Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Thu, 21 Sep 2017 11:05:14 +0200 Subject: [PATCH 25/81] Introductory slides. --- slides/fpottier-00.pdf | Bin 0 -> 289182 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 slides/fpottier-00.pdf diff --git a/slides/fpottier-00.pdf b/slides/fpottier-00.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f78bb3a25e0e52ee5dbeb4dd25e9ec72cf5b18f8 GIT binary patch literal 289182 zcmeFa2_RP67dC#HOY@|;l;%jrG^nehG$?5xG!TW5DN%9LsDaEOR7jFsQ<6%`lreLL zqB4sj8kGFk^4c74uc0ot|Mz{r+kHC2-e;e^*Lv2o);aGRGd8Uf=jNR+$Ts77TI35h zULlSJ99oKMY)hB2@vK+Y+^4sngI`dPgYh35kGSdqJ!KsZ9`OTv^psaAD{1XjW?Qz5 zO;=Axd5;F$;cOr2IL#-24SG;K@96p=yz2X=BnGL8Zn?F}EjN4erU}FPogH7@P^P&; z^Z{F-$X@552QGdpw|5U=6SrJ^wt(~Nmr{N9W%L`lR-DvCQlu(i0PvVZhSwFL$#V>URdDfncohThzMt*EE%$~P{@wR@|Y z`-@JRnH{lV#ax#=JNYZhBJV#{iRa5YBYELm`CWxA8!WyLohHw7bM%Xe(~kK*T+dq< zs%orp*L75h-mK-mv2*%wjJhVw<0G0<7kD#&-HO5L1>Y|$GT*QFmuX1Q;(T}0=^1Z@ zU9W%KVIaaI@om#-{i+FyHkOHLIRj-aN(BnoM~{5K;}(9nH2={uKk+>Ef0QJq-MrYN z{JVNiz+SP+#V1~ltsk{ZUZ?6ao3iHKCTqhUdWv@e&1Djr;GTn8=p7D~JqL7^IoNo% z7#>vS;E~y=sHUu>$HpVAuc+5}MqEor13U_UH}>FV;>s~Gpzb=)gAtd*R9aIeaorpq?%*{3Y2ub~L+AR|d= zs%WiJRnp_&r+?UW; zz`@7gc;&KXJezd1l%$pQIOI5ZfU+DsTa*v$aqviJz-?D@C?E-&`N~V9q&6Gg3U3Lw zYuqZqqH*X|T5M8mKby1hjwAxwGF{_s*?1bwgo8B)h8)jkeN9H(FrouixLtKGL=jAb zjc1Lr>c0JYI2AUYm0Ab1bfgdNQBvmM6KGr|&k9vN-A&3mE44HZYH2EK>Tw7+FN#mF zu}e#=nNt7z^FIsxpJM?VYHC9fMYWIimXkm&YX9Q)25)cRmz3!54@_>_-VI(;694`N zow-rre|ICXu31t0X8YewnBwLC6?1iSKz4h?{|7e^`F8=>9+R@DZXLXvWIUY`{5_Mj zXNqKs_tyt)Z@@@uQu_n8ck%yiZCMm|`^F$ik8ImBKzetw_9gWn^fxn5C_4 zd*-U3q`LNIIEnw=u)`9!Enl^D{(mO@pGkk&0{lp=h zx1PWL_odL|?w(_5*VX`!yCWg{Qk2LK28KSNDDI|x5mU;mrDd`y^|32U4yTqI!O)C9 zf!(w^IapC)6O3XhYAEan)6QWVtk#dBTJCom07-~u-P}F4D!`^ttsG8|YQ))F)pntnHV_3kf=~ilRE8Hg* z0hhpJ$VAMbmF^xhVy_3?VJ}aH6`JD$>~)zoHEvTXvh{S(PH4C0D^7M5cTTl-fE?Ti z9n)x~MhoFBiOtYTNN+NY=Gf{M<#1y-45#DyiQ&w~eKPl0NSwz|Fbr`zF)H4Rw39Y6o)OFqosQSi4N2BE z8Rm!kq>WtH${Enk4Rb~35eH$vM#DFqXEnYWLp#j6EBixS;4~1yFb$jqfVrE66#r4Tkl$ z)rNh{(z^0vgTkLDKqkU!o8H7@R&9;vVDfD_)Dzm!Ky6+NZK&VKNmhM=p$$3Ds;#Ya zB4B6~uD06oJjn(u`@r?h_k{b?227){zVtZ6@9{FVRT233D6CW2B9yATt_zu5eWMD zI0hc01KV0;^TL57bde$?p*suXhB$=~fle&qOgcF*FVfdS3{O9Y^TT=4?SS*eJVoXg z1^aYzlKJ6yNHn^A$UMlsnsS$Acg34)YxClgo0kInbT?t{p_>`|(p_c-$7IQvkLczV z0jZ()#XQ$)Y^GzR4gFrob&MG@KE`dUV;G-K6Pz1ycT`geL$1a3aX;_M*m&J~vdoLa z#n5?$LC(B*ItNhnS{=jU=>6BTFa~}6q0mk@HnW&v-lDhTeCfuFzQEKC`N)j9zrrb4OO|1tNc$7u4)$Fe1GTxyKxc zt&V{|bgp77&=-fsAcjG{;Mj=f`@ufWr`0uZOxIWBJKaIppMH!Fr{o@;5faij7+&hq z=Yi*OE$|uz|JZhyM%({vGJ05H*tK#2(=pcht=hpH%-548Mgh0m|;XYUTp-f?<3* zKgpQPTUL31eh*f0*Iw+PPSfw(k&qaH;ZaEFW8xf}YJ7+8;VINhPv9`UFF79tZfR9Bz&NCh`8nXH+`#1SR&ILLn*ZqqUB%mEnlb2;Z6|u>m#fuRb6nRXDP{`0% zhwgauYtgnjSe03=_yfymUNTG#O^>7?gQ#bCzF`Aqni=HK2$m3RqWAe8d9Mf;iTm#^Q(YVAI zg-nLWiKC&JlN#MH%#J|q#(R=QqY+vqN}3T_2?u>_ycW@%Q*d0k9?ua+p!0Bidd|Rm zlXg5$9HZ8BEnb6t5Y1yVV!^qaVu3y-OegJzDYVo3W4fWckt~gD$+E~Xz8Y)gGA3H! 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zgsgvqH_ZGN%ry0l&a!u5qLASTGKkX8m>sT|udZID%A! z1vClz*wdwh`7y7>i^bj}+2a1+G@{x+WQqeE>BR*y3Gzfn0iBgj3=q{3Tp(-T804>z z3uF&;RtDY#DxW0b!NI7LB=}?m{8xup@Co`QI)5P}ptFv?vazzhs^Lv#b?{A9Ljy2G zMFpay57AIlF;>!4SB2;T|98pB9&==n_leBBsjT^*V~_vCicyB#0ynS{NTGkw6jqY* zFo0MMYkC_gdk{45Z+BWR8f7Ke)!F1r%1!qv{3 z*d*@wg zQKrXw-Wb(P$9cjsm^n$7E$S96>N)MwE$Gsd86!M+SQ`F7f%SGtiT+QXgrnTUP*Gtg T#8A=DP*YOpl97SHjJf^|vb Date: Thu, 21 Sep 2017 11:06:57 +0200 Subject: [PATCH 26/81] Link to my introductory slides. --- README.md | 1 + 1 file changed, 1 insertion(+) diff --git a/README.md b/README.md index 178beaa..b19f007 100644 --- a/README.md +++ b/README.md @@ -22,6 +22,7 @@ There is a 15-minute break in the middle of each lecture. This course presents the principles and formalisms that underlie many of today's typed functional programming languages. +(Here are some [introductory slides](slides/fpottier-00.pdf).) The course is made up of four parts and can be split after the first two parts. From 845312be456b17defdd91e1320a2f2b11362a4a4 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Thu, 21 Sep 2017 15:40:42 +0200 Subject: [PATCH 27/81] Coq demo. --- coq/.gitignore | 7 ++ coq/DemoSyntaxReduction.v | 226 ++++++++++++++++++++++++++++++++++++++ coq/Makefile | 2 + coq/Makefile.coq | 188 +++++++++++++++++++++++++++++++ coq/MyTactics.v | 182 ++++++++++++++++++++++++++++++ 5 files changed, 605 insertions(+) create mode 100644 coq/.gitignore create mode 100644 coq/DemoSyntaxReduction.v create mode 100644 coq/Makefile create mode 100644 coq/Makefile.coq create mode 100644 coq/MyTactics.v diff --git a/coq/.gitignore b/coq/.gitignore new file mode 100644 index 0000000..a0a367a --- /dev/null +++ b/coq/.gitignore @@ -0,0 +1,7 @@ +*.vo +*.glob +*.v.d +.*.aux +.coq-native +_CoqProject +*~ diff --git a/coq/DemoSyntaxReduction.v b/coq/DemoSyntaxReduction.v new file mode 100644 index 0000000..549a787 --- /dev/null +++ b/coq/DemoSyntaxReduction.v @@ -0,0 +1,226 @@ +Require Import MyTactics. +Require Import Autosubst.Autosubst. + +(* This file is intended as a mini-demonstration of: + 1. defining the syntax of a calculus, in de Bruijn's representation; + 2. equipping it with an operational semantics; + 3. proving a basic lemma, e.g., + stability of the semantics under substitution. *) + +(* -------------------------------------------------------------------------- *) + +(* The syntax of the lambda-calculus. *) + +Inductive term := +| Var: var -> term +| Lam: {bind term} -> term +| App: term -> term -> term +. + +(* -------------------------------------------------------------------------- *) + +(* The following incantations let [AutoSubst] work its magic for us. + We obtain, for free, the operations of de Bruijn algebra: application + of a substitution to a term, composition of substitutions, etc. *) + +Instance Ids_term : Ids term. derive. Defined. +Instance Rename_term : Rename term. derive. Defined. +Instance Subst_term : Subst term. derive. Defined. +Instance SubstLemmas_term : SubstLemmas term. derive. Qed. + +(* -------------------------------------------------------------------------- *) + +(* A demo of the tactics [autosubst] and [asimpl]. *) + +Goal + forall sigma, + (Lam (App (Var 0) (Var 1))).[sigma] = + Lam (App (Var 0) (sigma 0).[ren (+1)]). +Proof. + intros. + (* The tactic [autosubst] proves this equality. *) + autosubst. +Restart. + intros. + (* If desired, we can first simplify this equality using [asimpl]. *) + asimpl. + (* [ids], the identity substitution, maps 0 to [Var 0], 1 to [Var 1], + and so on, so it is really equal to [Var] itself. As a result, the + built-in tactic [reflexivity] proves this simplified equation. *) + reflexivity. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* More demos: let us check that the laws of substitution are satisfied. *) + +Lemma subst_var: + forall x sigma, + (Var x).[sigma] = sigma x. +Proof. + autosubst. +Qed. + +Lemma subst_lam: + forall t sigma, + (Lam t).[sigma] = Lam (t.[up sigma]). +Proof. + autosubst. +Qed. + +Lemma subst_app: + forall t1 t2 sigma, + (App t1 t2).[sigma] = App t1.[sigma] t2.[sigma]. +Proof. + autosubst. +Qed. + +(* A reminder of the meaning of [up sigma]. *) + +Lemma up_def: + forall sigma, + up sigma = Var 0 .: (sigma >> ren (+1)). +Proof. + intros. autosubst. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* A small-step reduction semantics. *) + +(* This is a relation between terms: hence, its type is [term -> term -> Prop]. + It is inductively defined by three inference rules, as follows: *) + +Inductive red : term -> term -> Prop := + +(* Beta-reduction. The use of an explicit equality hypothesis is a technical + convenience. We could instead write [red (App (Lam t1) t2) t1.[t2/]] in + the conclusion, and remove the auxiliary variable [u], but that would make + it more difficult for Coq to apply the inference rule [RedBeta]. Using an + explicit equality premise makes the rule more widely applicable. Of course + the user still has to prove (after applying the rule) that the equality + holds. *) +| RedBeta: + forall t1 t2 u, + t1.[t2/] = u -> + red (App (Lam t1) t2) u + +(* Reduction in the left-hand side of an application. *) +| RedAppL: + forall t1 t2 u, + red t1 t2 -> + red (App t1 u) (App t2 u) + +(* Reduction in the right-hand side of an application. *) +| RedAppR: + forall t u1 u2, + red u1 u2 -> + red (App t u1) (App t u2) +. + +(* The following means that [eauto with red] is allowed to apply the above + three inference rules. *) + +Hint Constructors red : red. + +(* No strategy is built into this reduction relation: it is not restricted to + call-by-value or call-by-name. It is nondeterministic. Only weak reduction + is permitted here: we have not allowed reduction under a [Lam]. These choices + are arbitrary: this is just a demo anyway. *) + +(* -------------------------------------------------------------------------- *) + +(* This incantation means that [eauto with autosubst] can use the tactic + [autosubst] to prove an equality. It is used in the last "expert" proof + of the lemma [red_subst] below. *) + +Hint Extern 1 (_ = _) => autosubst : autosubst. + +(* -------------------------------------------------------------------------- *) + +(* Let us now prove that the semantics is stable under arbitrary substitutions. *) + +Lemma red_subst: + forall t1 t2, + red t1 t2 -> + forall sigma, + red t1.[sigma] t2.[sigma]. +Proof. + + (* We attack a proof by induction on the derivation of [red t1 t2]. *) + induction 1; intros. + (* Case: [RedBeta]. *) + { subst u. + eapply RedBeta. + (* Wow -- we have to prove a complicated-looking commutation property + of substitutions. Fortunately, [autosubst] is here for us! *) + autosubst. } + (* Case: [RedAppL]. The proof can be done slowly, in three steps: + 1. push the substitution into [App]; + 2. apply the rule [RedAppL]; a simpler subgoal remains to be proved; + 3. apply the induction hypothesis, which proves this subgoal. *) + { asimpl. + eapply RedAppL. + eapply IHred. } + (* Case: [RedAppR]. The proof could be done using the same three steps + as above, but one can also let the last two steps be automatically + found by [eauto]. *) + { asimpl. eauto using red. } + (* The proof is now finished. *) + + (* For the fun of it, let us do the proof again in a more "expert" style. *) +Restart. + (* The proof is still by induction. All three cases begin in the same way, + so this common pattern can be shared, as follows. We use the semicolon + which in Ltac has special meaning: when one writes [command1; command2], + [command1] can produce multiple subgoals, and [command2] is applied to + every subgoal (in parallel). Thus, here, in each of the three cases, + we perform the sequence of commands [intros; subst; asimpl; econstructor]. + The effect of [econstructor] is to apply one of [RedBeta], [RedAppL] and + [RedAppR] -- whichever is applicable. *) + induction 1; intros; subst; asimpl; econstructor. + (* Then, the three subgoals can be finished as follows: *) + { autosubst. } + { eauto. } + { eauto. } + (* The proof is now finished (again). *) + + (* For the fun of it, let us redo the proof in an even more expert style. + We remark that each of the three subgoals can be proved by [eauto with + autosubst], so we can write a fully shared command, where the subgoals + are no longer distinguished: *) +Restart. + induction 1; intros; subst; asimpl; econstructor; eauto with autosubst. + (* The proof is now finished (yet again). *) + + (* There are several lessons that one can draw from this demo: + + 1. The machine helps us by keeping track of what we may assume + and what we have to prove. + + 2. There are several ways in which a proof can be written. In the + beginning, it is advisable to write a step-by-step, simple-minded + proof; later on, when the proof is finished and well-understood, + it can be revisited for greater compactness and sharing. + + 3. The proof of this lemma *can* fit in one line. On paper, one + would say that the proof is "by induction" and "immediate". + Here, we are able to be almost as concise, yet we have much + greater confidence. + + 4. The point of the "expert" proof is not just to make the proof + more concise: the point is also to make the proof more robust + in the face of future evolution. For instance, as an EXERCISE, + extend the calculus with pairs and projections, and see how the + proof scripts must be extended. You should find that the last + "expert" proof above requires no change at all! + + *) + +Qed. + +(* -------------------------------------------------------------------------- *) + +(* As another EXERCISE, extend the operational semantics with a rule that + allows strong reduction, that is, reduction under a lambda-abstraction. + This exercise is more difficult; do not hesitate to ask for help or hints. *) diff --git a/coq/Makefile b/coq/Makefile new file mode 100644 index 0000000..2c6ec87 --- /dev/null +++ b/coq/Makefile @@ -0,0 +1,2 @@ +COQINCLUDE := -R $(shell pwd) MPRI +include Makefile.coq diff --git a/coq/Makefile.coq b/coq/Makefile.coq new file mode 100644 index 0000000..247f37c --- /dev/null +++ b/coq/Makefile.coq @@ -0,0 +1,188 @@ +############################################################################ +# Requirements. + +# We need bash. We use the pipefail option to control the exit code of a +# pipeline. + +SHELL := /usr/bin/env bash + +############################################################################ +# Configuration +# +# +# This Makefile relies on the following variables: +# COQBIN (default: empty) +# COQFLAGS (default: empty) (passed to coqc and coqide, not coqdep) +# COQINCLUDE (default: empty) +# V (default: *.v) +# V_AUX (default: undefined/empty) +# SERIOUS (default: 1) +# (if 0, we produce .vio files) +# (if 1, we produce .vo files in the old way) +# VERBOSE (default: undefined) +# (if defined, commands are displayed) + +# We usually refer to the .v files using relative paths (such as Foo.v) +# but [coqdep -R] produces dependencies that refer to absolute paths +# (such as /bar/Foo.v). This confuses [make], which does not recognize +# that these files are the same. As a result, [make] does not respect +# the dependencies. + +# We fix this by using ABSOLUTE PATHS EVERYWHERE. The paths used in targets, +# in -R options, etc., must be absolute paths. + +ifndef V + PWD := $(shell pwd) + V := $(wildcard $(PWD)/*.v) +endif + +# Typically, $(V) should list only the .v files that we are ultimately +# interested in checking. $(V_AUX) should list every other .v file in the +# project. $(VD) is obtained from $(V) and $(V_AUX), so [make] sees all +# dependencies and can rebuild files anywhere in the project, if needed, and +# only if needed. + +ifndef VD + VD := $(patsubst %.v,%.v.d,$(V) $(V_AUX)) +endif + +VIO := $(patsubst %.v,%.vio,$(V)) +VQ := $(patsubst %.v,%.vq,$(V)) +VO := $(patsubst %.v,%.vo,$(V)) + +SERIOUS := 1 + +############################################################################ +# Binaries + +COQC := $(COQBIN)coqc $(COQFLAGS) +COQDEP := $(COQBIN)coqdep +COQIDE := $(COQBIN)coqide $(COQFLAGS) + +############################################################################ +# Targets + +.PHONY: all proof depend quick proof_vo proof_vq + +all: proof +ifeq ($(SERIOUS),0) +proof: proof_vq +else +proof: proof_vo +endif +proof_vq: $(VQ) +depend: $(VD) +quick: $(VIO) +proof_vo: $(VO) + +############################################################################ +# Verbosity control. + +# Our commands are pretty long (due, among other things, to the use of +# absolute paths everywhere). So, we hide them by default, and echo a short +# message instead. However, sometimes one wants to see the command. + +# By default, VERBOSE is undefined, so the .SILENT directive is read, so no +# commands are echoed. If VERBOSE is defined by the user, then the .SILENT +# directive is ignored, so commands are echoed, unless they begin with an +# explicit @. + +ifndef VERBOSE +.SILENT: +endif + +############################################################################ +# Verbosity filter. + +# Coq is way too verbose when using one of the -schedule-* commands. +# So, we grep its output and remove any line that contains 'Checking task'. + +# We need a pipe that keeps the exit code of the *first* process. In +# bash, when the pipefail option is set, the exit code is the logical +# conjunction of the exit codes of the two processes. If we make sure +# that the second process always succeeds, then we get the exit code +# of the first process, as desired. + +############################################################################ +# Rules + +# If B uses A, then the dependencies produced by coqdep are: +# B.vo: B.v A.vo +# B.vio: B.v A.vio + +%.v.d: %.v + $(COQDEP) $(COQINCLUDE) $< > $@ + +ifeq ($(SERIOUS),0) + +%.vo: %.vio + @echo "Compiling `basename $*`..." + set -o pipefail; ( \ + $(COQC) $(COQINCLUDE) -schedule-vio2vo 1 $* \ + 2>&1 | (grep -v 'Checking task' || true)) + +# The recipe for producing %.vio destroys %.vo. In other words, we do not +# allow a young .vio file to co-exist with an old (possibly out-of-date) .vo +# file, because this seems to lead Coq into various kinds of problems +# ("inconsistent assumption" errors, "undefined universe" errors, warnings +# about the existence of both files, and so on). Destroying %.vo should be OK +# as long as the user does not try to build a mixture of .vo and .vio files in +# one invocation of make. +%.vio: %.v + @echo "Digesting `basename $*`..." + rm -f $*.vo + $(COQC) $(COQINCLUDE) -quick $< + +%.vq: %.vio + @echo "Checking `basename $*`..." + set -o pipefail; ( \ + $(COQC) $(COQINCLUDE) -schedule-vio-checking 1 $< \ + 2>&1 | (grep -v 'Checking task' || true)) + touch $@ + +endif + +ifeq ($(SERIOUS),1) + +%.vo: %.v + @echo "Compiling `basename $*`..." + $(COQC) $(COQINCLUDE) $< + +endif + +_CoqProject: .FORCE + @echo $(COQINCLUDE) > $@ + +.FORCE: + +############################################################################ +# Dependencies + +ifeq ($(findstring $(MAKECMDGOALS),depend clean),) +-include $(VD) +endif + +############################################################################ +# IDE + +.PHONY: ide + +.coqide: + @echo '$(COQIDE) $(COQINCLUDE) $$*' > .coqide + @chmod +x .coqide + +ide: _CoqProject + $(COQIDE) $(COQINCLUDE) + +############################################################################ +# Clean + +.PHONY: clean + +clean:: + rm -f *~ + rm -f $(patsubst %.v,%.v.d,$(V)) # not $(VD) + rm -f $(VIO) $(VO) $(VQ) + rm -f *.aux .*.aux *.glob *.cache *.crashcoqide + rm -rf .coq-native .coqide +# TEMPORARY *~, *.aux, etc. do not make sense in a multi-directory setting diff --git a/coq/MyTactics.v b/coq/MyTactics.v new file mode 100644 index 0000000..fec7992 --- /dev/null +++ b/coq/MyTactics.v @@ -0,0 +1,182 @@ +Require Export Omega. + +(* -------------------------------------------------------------------------- *) + +(* [false] replaces the current goal with [False]. *) + +Ltac false := + elimtype False. + +(* -------------------------------------------------------------------------- *) + +(* [tc] is a shortcut for [eauto with typeclass_instances]. For some reason, + it is often necessary to use [rewrite ... by tc]. *) + +Ltac tc := + eauto with typeclass_instances. + +(* -------------------------------------------------------------------------- *) + +(* The tactic [obvious] does nothing by default, but can be extended with hints + so as to solve relatively easy goals -- e.g., proving that a term is a value, + proving a size inequality, proving that a substitution is a renaming, etc. + These predicates are sometimes interrelated (e.g., size is preserved by + renamings; the property of being a value is preserved by renamings) so it + would be counterproductive to distinguish several hint databases. *) + +Create HintDb obvious. + +Ltac obvious := + eauto with obvious typeclass_instances. + +(* -------------------------------------------------------------------------- *) + +(* The tactic [pick R k] picks a hypothesis [h] whose statement is an + application of the inductive predicate [R], and passes [h] to the (Ltac) + continuation [k]. *) + +Ltac pick R k := + match goal with + | h: R _ |- _ => k h + | h: R _ _ |- _ => k h + | h: R _ _ _ |- _ => k h + | h: R _ _ _ _ |- _ => k h + | h: R _ _ _ _ _ |- _ => k h + end. + +(* -------------------------------------------------------------------------- *) + +(* The tactic [invert h] case-analyzes the hypothesis [h], whose statement + should be an application of an inductive predicate. *) + +Ltac invert h := + inversion h; clear h; try subst. + +(* The tactic [inv R] looks for a hypothesis [h] whose statement is an + application of the inductive predicate [R], and case-analyzes this + hypothesis. *) + +Ltac inv R := + pick R invert. + +(* -------------------------------------------------------------------------- *) + +(* The tactic [unpack] decomposes conjunctions and existential quantifiers + in the hypotheses. Then, it attempts to perform substitutions, if possible. *) + +Ltac unpack := + repeat match goal with + | h: _ /\ _ |- _ => + destruct h + | h: exists _, _ |- _ => + destruct h + end; + try subst. + +(* -------------------------------------------------------------------------- *) + +(* The tactic [forward H] introduces the term [H] as a new hypothesis, and + unpacks it (if necessary). *) + +Ltac forward H := + generalize H; intro; unpack. + +(* -------------------------------------------------------------------------- *) + +(* [push h] moves the hypothesis [h] into the goal. *) + +Ltac push h := + generalize h; clear h. + +(* [ltac_Mark] and [ltac_mark] are dummies. They are used as sentinels by + certain tactics, to mark a position in the context or in the goal. *) + +Inductive ltac_Mark : Type := +| ltac_mark : ltac_Mark. + +(* [push_until_mark] moves all hypotheses from the context into the goal, + starting from the bottom and stopping as soon as a mark (that is, a + hypothesis of type [ltac_Mark]) is reached. The mark is thrown away. The + tactic fails if no mark appears in the context. *) + +Ltac push_until_mark := + match goal with h: ?T |- _ => + match T with + | ltac_Mark => clear h + | _ => push h; push_until_mark + end end. + +(** [pop_until_mark] moves all hypotheses from the goal into the context, + until a hypothesis of type [ltac_Mark] is reached. The mark is thrown + away. The tactic fails if no mark appears in the goal. *) + +Ltac pop_until_mark := + match goal with + | |- (ltac_Mark -> _) => intros _ + | _ => intro; pop_until_mark + end. + +Ltac injections := + (* Place an initial mark, so as to not disturb the goal. *) + generalize ltac_mark; + (* Look at every equality hypothesis. *) + repeat match goal with + | h: _ = _ |- _ => + (* Try to apply the primitive tactic [injection] to this hypothesis. + If this succeeds, clear [h] and replace it with the results of + [injection]. Another mark is used for this purpose. If this fails, + just push [h] into the goal, as we do not wish to see it any more. *) + first [ + generalize ltac_mark; injection h; clear h; pop_until_mark + | push h ] + end; + (* Pop all of the hypotheses that have been set aside above. *) + pop_until_mark. + +(* -------------------------------------------------------------------------- *) + +(* The following incantation means that [eauto with omega] can solve a goal + of the form [_ < _]. The tactic [zify] is a preprocessor which increases + the number of goals that [omega] can accept; e.g., it expands away [min] + and [max]. *) + +Hint Extern 1 (le _ _) => (zify; omega) : omega. + +(* -------------------------------------------------------------------------- *) + +(* A little extra help for [eauto with omega]. *) + +Lemma arith_le_SS: forall x y, x < y -> S x < S y. +Proof. intros. omega. Qed. +Lemma arith_SS_le: forall x y, S x < S y -> x < y. +Proof. intros. omega. Qed. + +Hint Resolve arith_le_SS arith_SS_le : omega. + +(* -------------------------------------------------------------------------- *) + +(* [dblib_by_cases] simplifies goals in which a decidable integer comparison + appears. *) + +Ltac dblib_intro_case_clear := + let h := fresh in + intro h; case h; clear h. + +Ltac dblib_inspect_cases := + match goal with + | |- context [le_gt_dec ?n ?n'] => + case (le_gt_dec n n') + | h: context [le_gt_dec ?n ?n'] |- _ => + revert h; case (le_gt_dec n n'); intro h + | |- context [eq_nat_dec ?n ?n'] => + case (eq_nat_dec n n') + | h: context [eq_nat_dec ?n ?n'] |- _ => + revert h; case (eq_nat_dec n n'); intro h + | |- context [(lt_eq_lt_dec ?n ?n')] => + case (lt_eq_lt_dec n n'); [ dblib_intro_case_clear | idtac ] + | h: context [(lt_eq_lt_dec ?n ?n')] |- _ => + revert h; case (lt_eq_lt_dec n n'); [ dblib_intro_case_clear | idtac ]; intro h + end. + +Ltac dblib_by_cases := + repeat dblib_inspect_cases; try solve [ intros; elimtype False; omega ]; intros. From 86c981fb99b66dad523eda62dad6dd141dae6053 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Franc=CC=A7ois=20Pottier?= Date: Thu, 21 Sep 2017 21:29:24 +0200 Subject: [PATCH 28/81] Publish slides 01a and 01b. Links from README. --- README.md | 11 +++++++++-- slides/fpottier-01a.pdf | Bin 0 -> 155946 bytes slides/fpottier-01b.pdf | Bin 0 -> 414915 bytes 3 files changed, 9 insertions(+), 2 deletions(-) create mode 100644 slides/fpottier-01a.pdf create mode 100644 slides/fpottier-01b.pdf diff --git a/README.md b/README.md index b19f007..8ecdf76 100644 --- a/README.md +++ b/README.md @@ -64,8 +64,15 @@ We also show the limits of dependently-typed functional programming. ### Functional Programming: Under the Hood -* (22/09/2017) From a small-step operational semantics... -* (29/09/2017) ... to an efficient interpreter. (2 weeks.) +* (22/09/2017) + Introduction ([slides 00](slides/fpottier-00.pdf)). + Syntax and operational semantics, on paper and on a machine + ([slides 01a](slides/fpottier-01a.pdf)) + ([slides 01b](slides/fpottier-01b.pdf)) + ([Coq demo](coq/DemoSyntaxReduction.v)). +* (29/09/2017) + From a small-step semantics down to an efficient interpreter, + in several stages. * (06/10/2017) Compiling away first-class functions: closure conversion, defunctionalization. * (13/10/2017) Compiling away the call stack: the CPS transformation. * (20/10/2017) Equational reasoning and program optimizations. diff --git a/slides/fpottier-01a.pdf b/slides/fpottier-01a.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0ab092afa0ec055ed6e3d04154315088c7421f8e GIT binary patch literal 155946 zcmeFa30zKH7dC!KQHGK+k&KT_4H^w86j6pU6f!hWDGg+ZqB4buqJb1q2vMk{LM3UI zl+vV`GL#BM|Fyc?b-LXRJoWg0-}k=H?|RQU`|PvN-fLa!TI-zqb~kS6JjE%ZQzf{@ 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1 deletion(-) diff --git a/coq/DemoSyntaxReduction.v b/coq/DemoSyntaxReduction.v index 549a787..f3ffec5 100644 --- a/coq/DemoSyntaxReduction.v +++ b/coq/DemoSyntaxReduction.v @@ -1,4 +1,3 @@ -Require Import MyTactics. Require Import Autosubst.Autosubst. (* This file is intended as a mini-demonstration of: From 09761828fcab3488db8d4ac32b083193b2b0a558 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Franc=CC=A7ois=20Pottier?= Date: Thu, 21 Sep 2017 22:04:18 +0200 Subject: [PATCH 30/81] Clarified installation instructions. Added instructions for ProofGeneral. --- README.md | 27 +++++++++++++++++++++++++-- 1 file changed, 25 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index 8ecdf76..29b3f06 100644 --- a/README.md +++ b/README.md @@ -143,15 +143,19 @@ available with solutions: ## Recommended software -OCaml 4.0x and Coq **8.5**. +Please install [opam](https://opam.ocaml.org/doc/Install.html) first. -Once you have installed [opam](https://opam.ocaml.org/doc/Install.html), use the following commands: +Then, install OCaml 4.0x and Coq **8.5** via the following commands: ```bash opam init --comp=4.05 # for instance opam repo add coq-released https://coq.inria.fr/opam/released opam update opam install -j4 -v coq.8.5.3 ``` +(Do *not* install Coq 8.6. The version of AutoSubst that I am using is +not compatible with it. If for some reason you need Coq 8.6, or have +already installed Coq 8.6, note that `opam switch` can be used to let +multiple versions of Coq coexist.) Please also install François Pottier's [variant](https://github.com/fpottier/autosubst) @@ -162,6 +166,25 @@ git clone git@github.com:fpottier/autosubst.git make -C autosubst install ``` +In order to use Coq inside `emacs`, +[ProofGeneral](https://proofgeneral.github.io/) +is highly recommended. +Here is a suggested installation script: +```bash +rm -rf /tmp/PG +cd /tmp +git clone git@github.com:ProofGeneral/PG.git +cd PG +EMACS=/Applications/Aquamacs.app/Contents/MacOS/Aquamacs +if [ ! -x $EMACS ]; then + EMACS=emacs +fi +make EMACS=$EMACS compile +TARGET=/usr/local/share/emacs/site-lisp/ProofGeneral +sudo rm -rf $TARGET +sudo mv /tmp/PG $TARGET +``` + ## Bibliography [Types and Programming Languages](https://mitpress.mit.edu/books/types-and-programming-languages), From cb1a56bbda0d873fe3e9dd7ea8a3906c2ed9cbfd Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Franc=CC=A7ois=20Pottier?= Date: Thu, 21 Sep 2017 22:09:04 +0200 Subject: [PATCH 31/81] Indicate how to load ProofGeneral in .emacs. --- README.md | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/README.md b/README.md index 29b3f06..39a2abf 100644 --- a/README.md +++ b/README.md @@ -185,6 +185,13 @@ sudo rm -rf $TARGET sudo mv /tmp/PG $TARGET ``` +Enable ProofGeneral by adding the following line to your `.emacs` file: +```elisp +(load-file "/usr/local/share/emacs/site-lisp/ProofGeneral/generic/proof-site.el") +``` +If desired, ProofGeneral can be further +[customized](https://proofgeneral.github.io/doc/userman/ProofGeneral_9/). + ## Bibliography [Types and Programming Languages](https://mitpress.mit.edu/books/types-and-programming-languages), From f31a263ff7173bf42caf6587459839780c2a0311 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Franc=CC=A7ois=20Pottier?= Date: Fri, 22 Sep 2017 11:32:50 +0200 Subject: [PATCH 32/81] Updated slides, Coq demo, and OCaml exercise. --- README.md | 3 +- coq/DemoSyntaxReduction.v | 27 ++++++- ocaml/NewtonRaphson.ml | 151 ++++++++++++++++++++++++++++++++++++++ slides/fpottier-00.pdf | Bin 289182 -> 290079 bytes slides/fpottier-01a.pdf | Bin 155946 -> 162524 bytes slides/fpottier-01b.pdf | Bin 414915 -> 414915 bytes 6 files changed, 179 insertions(+), 2 deletions(-) create mode 100644 ocaml/NewtonRaphson.ml diff --git a/README.md b/README.md index 39a2abf..afd64aa 100644 --- a/README.md +++ b/README.md @@ -69,7 +69,8 @@ We also show the limits of dependently-typed functional programming. Syntax and operational semantics, on paper and on a machine ([slides 01a](slides/fpottier-01a.pdf)) ([slides 01b](slides/fpottier-01b.pdf)) - ([Coq demo](coq/DemoSyntaxReduction.v)). + ([Coq demo](coq/DemoSyntaxReduction.v)) + ([OCaml solution to Newton-Raphson exercise](ocaml/NewtonRaphson.ml)). * (29/09/2017) From a small-step semantics down to an efficient interpreter, in several stages. diff --git a/coq/DemoSyntaxReduction.v b/coq/DemoSyntaxReduction.v index f3ffec5..8e20492 100644 --- a/coq/DemoSyntaxReduction.v +++ b/coq/DemoSyntaxReduction.v @@ -137,7 +137,28 @@ Hint Extern 1 (_ = _) => autosubst : autosubst. (* -------------------------------------------------------------------------- *) -(* Let us now prove that the semantics is stable under arbitrary substitutions. *) +(* Demo: a term that reduces to itself. *) + +Definition Delta := + Lam (App (Var 0) (Var 0)). + +Definition Omega := + App Delta Delta. + +Goal + red Omega Omega. +Proof. + (* Apply the beta-reduction rule. + (This forces Coq to unfold the left-hand [Omega].) *) + eapply RedBeta. + (* Check this equality. *) + asimpl. (* optional *) + reflexivity. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* Let us prove that the semantics is stable under arbitrary substitutions. *) Lemma red_subst: forall t1 t2, @@ -223,3 +244,7 @@ Qed. (* As another EXERCISE, extend the operational semantics with a rule that allows strong reduction, that is, reduction under a lambda-abstraction. This exercise is more difficult; do not hesitate to ask for help or hints. *) + +(* Another suggested EXERCISE: define call-by-value reduction, [cbv]. Prove + that [cbv] is a subset of [red]. Prove that values do not reduce. Prove + that [cbv] is deterministic. *) diff --git a/ocaml/NewtonRaphson.ml b/ocaml/NewtonRaphson.ml new file mode 100644 index 0000000..f125997 --- /dev/null +++ b/ocaml/NewtonRaphson.ml @@ -0,0 +1,151 @@ +(* A couple abbreviations. *) + +type 'a thunk = 'a Lazy.t +let force = Lazy.force + +(* The definition of (finite or infinite) lazy lists. *) + +type 'a stream = + 'a head thunk + +and 'a head = + | Nil + | Cons of 'a * 'a stream + +(* Calling [tail xs] demands the head of the stream, that is, forces + the computation of the first element of the stream (if there is one). *) + +let tail xs = + match force xs with + | Nil -> assert false + | Cons (_, xs) -> xs + +(* Newton-Raphson approximation, following Hughes, + "Why functional programming matters", 1990. *) + +let next n x = + (x +. n /. x) /. 2. + +(* An infinite stream obtained by iterating [f]. *) + +(* The following definition, copied almost literally from Hughes' + paper, is correct but somewhat unsatisfactory; can you see why? Can + you fix it? Think about it before reading the solution below. *) + +let rec repeat (f : 'a -> 'a) (a : 'a) : 'a stream = + lazy (Cons (a, repeat f (f a))) + +(* + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + * + *) + +(* In the above definition of [repeat], the function call [f a] takes + place when the *first* element of the list is demanded by the consumer. + That's too early -- ideally, this function call should take place only + when the *second* element is demanded, since the result of [f a] is the + second element in the infinite stream [a], [f a], [f (f a)], ... *) + +(* This code exhibits the problem: *) + +let () = + let x = ref 0 in + let f () = incr x; () in + let xs = repeat f () in + let xs = tail xs in + (* This assertion fails because [x] has been incremented once: *) + assert (!x = 0); + ignore xs + +(* This can be fixed in several ways. One solution is to let [repeat] take an + argument of type ['a thunk] instead of ['a]. This approach is in fact the + standard encoding of call-by-need into call-by-value, applied to Hughes' + code. *) + +let rec repeat (f : 'a -> 'a) (a : 'a thunk) : 'a stream = + lazy ( + Cons ( + force a, + repeat f (lazy (f (force a))) + ) + ) + +(* It can also be written as follows. *) + +let rec repeat (f : 'a -> 'a) (a : 'a thunk) : 'a stream = + lazy ( + let a = force a in + Cons ( + a, + repeat f (lazy (f a)) + ) + ) + + +(* We define a wrapper so [repeat] has the desired type again: *) + +let repeat (f : 'a -> 'a) (a : 'a) : 'a stream = + repeat f (lazy a) + +(* The problematic code now behaves as desired: *) + +let () = + let x = ref 0 in + let f () = incr x; () in + let xs = repeat f () in + (* These assertions succeed: *) + let xs = tail xs in + assert (!x = 0); + let xs = tail xs in + assert (!x = 1); + let xs = tail xs in + assert (!x = 2); + ignore xs + +(* Back to Newton-Rapshon. *) + +let rec within (eps : float) (xs : float stream) : float = + match force xs with + | Nil -> assert false + | Cons (a, xs) -> + match force xs with + | Nil -> assert false + | Cons (b, _) -> + if abs_float (a /. b -. 1.) <= eps then b + else within eps xs + +let sqrt (n : float) : float = + within 0.0001 (repeat (next n) n) + +let sqrt2 = + sqrt 2. diff --git a/slides/fpottier-00.pdf b/slides/fpottier-00.pdf index f78bb3a25e0e52ee5dbeb4dd25e9ec72cf5b18f8..5210370156b418d04d32c1907e7db59291d409be 100644 GIT binary patch delta 20754 zcmY(IQ*b5#(`IAawv&l%dt%$R{l>O!+qP}n*2KIxb8=Eeu}X4me-RWwLj=7G>XaETcB3JjzG)(zg2R=LftTuH^f)f zEmzog(eSbZ{l;T>qKlF^eXPd)5aJC4oXjYY<)e%j-FUj4T!ozLbQ3~sAwkm)GW`Vw z4lMl4&H6S`cdbL6Bb`pFubRIWr?uBE-q-zxD2)IGIU<97C_JbM4=RBJHg5vBPtO5G zX7RMXGLhe@z?b@{_#}$&_0H`%s95D7MNo{L}q81gIsN<)eb#ap;tFs)6T<1LMXDt?GTdjWaT4;rRUw4 znVD(D0IDm;Md*i78RL(e&_E+`Na}pUz&Ovctvrc#{amc3Jf@8XGGv!sk#H~XM;sff zpQysk5st@`wI}>LYCsRY)|Elbbn)-nS_dwX#|En3)fEpE$$ue!+RjO-I(=}iHltG-w9@mn>?7s5Lft^zp=3u96{$yYBT|vA>y1#Hy 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the slides and OCaml/Coq files. --- README.md | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index 271a809..83aa5a1 100644 --- a/README.md +++ b/README.md @@ -72,9 +72,12 @@ We also show the limits of dependently-typed functional programming. ([OCaml solution to Newton-Raphson exercise](ocaml/NewtonRaphson.ml)) ([Even.v](coq/Even.v)). * (29/09/2017) - From a small-step semantics down to an efficient interpreter, + From a small-step semantics down to an efficient verified interpreter, in several stages - ([Coq demo](coq/DemoSyntaxReduction.v)). + ([Coq demo](coq/DemoSyntaxReduction.v)) + ([slides 02](slides/fpottier-02.pdf)) + ([OCaml code](ocaml/Lambda.ml)) + ([Coq repo](coq/)). * (06/10/2017) Compiling away first-class functions: closure conversion, defunctionalization. * (13/10/2017) Compiling away the call stack: the CPS transformation. * (20/10/2017) Equational reasoning and program optimizations. 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zo<qG5Ns(x+&>~FPVh1ZuO_aM zKe1w|iNQQ*aN9@&SP`ZjWP$JfN=~ZQYHa(*K$TqkYa5v^%%$_d@yUDvniNSKH>o{n z)Hyn-Ilu6>Rds^h<^SOaR4yQA|)w0Km*mYh7RW9(|EukZSw>u6DJo`9I2eZ~=G# zO#Q^xJ;!D!E(T9L0P0?K5lz9r`1S3-*o9!+ciNKK$*tY!>OLe6fY+nFf|RCPh#iMM>O5 zg9mov?HxY5)pCB1Y2V_%7G}Nfg+C1cc$nn%=&ox7g#DM<3G*X#J3^oZ$7E z$;<|=X#`#JziAA}Nb-6>Z2sY{6@u3a&9jBSWM0(^=CkXoAF$o1ZEGCJE$zmyAOCgWaRm2>(w9jqZZ& zpUMAYj&}=F8og^f&lV2?cq7%|B*#h@5 zq2i{SZStVB%Laov&mPqqJ_GGnPnX14#(EVA?uubayyY?~5_L03)QNoEhoyQG@})eJ n;@xh4xAsd<-J&;Wr)>Sp^z2q-vJV_=obTT;0Dy#&B+CB-;kZ%( From 083c0c18435de98e36d32de60bfe3d4bd18e39c4 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Thu, 5 Oct 2017 17:57:33 +0200 Subject: [PATCH 41/81] Expose new Coq files. --- coq/AutosubstExtra.v | 3 + coq/Autosubst_EOS.v | 329 ++++++++++++++++++++++++++ coq/Autosubst_Env.v | 130 +++++++++++ coq/Autosubst_FreeVars.v | 345 ++++++++++++++++++++++++++++ coq/ClosureConversion.v | 349 ++++++++++++++++++++++++++++ coq/LambdaCalculusFreeVars.v | 26 --- coq/LambdaCalculusSyntax.v | 5 + coq/MetaBigStep.v | 0 coq/MetalBigStep.v | 313 +++++++++++++++++++++++++ coq/MetalSyntax.v | 432 +++++++++++++++++++++++++++++++++++ coq/MyList.v | 125 ++++++++++ 11 files changed, 2031 insertions(+), 26 deletions(-) create mode 100644 coq/Autosubst_EOS.v create mode 100644 coq/Autosubst_Env.v create mode 100644 coq/Autosubst_FreeVars.v create mode 100644 coq/ClosureConversion.v create mode 100644 coq/MetaBigStep.v create mode 100644 coq/MetalBigStep.v create mode 100644 coq/MetalSyntax.v create mode 100644 coq/MyList.v diff --git a/coq/AutosubstExtra.v b/coq/AutosubstExtra.v index af2ad35..bc94dd2 100644 --- a/coq/AutosubstExtra.v +++ b/coq/AutosubstExtra.v @@ -1,5 +1,6 @@ Require Import Omega. Require Import Autosubst.Autosubst. +Require Import MyTactics. (* TEMPORARY *) (* -------------------------------------------------------------------------- *) @@ -167,3 +168,5 @@ Hint Extern 1 (_ = _) => autosubst : autosubst. metavariables, so the tactic [autosubst] fails. *) Hint Resolve scons_scomp : autosubst. + +(* -------------------------------------------------------------------------- *) diff --git a/coq/Autosubst_EOS.v b/coq/Autosubst_EOS.v new file mode 100644 index 0000000..846a79b --- /dev/null +++ b/coq/Autosubst_EOS.v @@ -0,0 +1,329 @@ +Require Import Omega. +Require Import Autosubst.Autosubst. +Require Import AutosubstExtra. (* just for [upn_ren] *) +Require Import MyTactics. (* TEMPORARY *) + +(* This file defines the construction [eos x t], which can be understood as + an end-of-scope mark for [x] in the term [t]. *) + +(* It also defines the single-variable substitution t.[u // x], which is the + substitution of [u] for [x] in [t]. *) + +(* -------------------------------------------------------------------------- *) + +Section EOS. + +Context {A} `{Ids A, Rename A, Subst A, SubstLemmas A}. + +(* The substitution [Var 0 .: Var 1 .: ... .: Var (x-1) .: Var (x+1) .: ...] + does not have [Var x] in its codomain. Thus, applying this substitution + to a term [t] can be understood as an end-of-scope construct: it means + ``end the scope of [x] in [t]''. We write [eos x t] for this construct. + It is also known as [adbmal]: see Hendriks and van Oostrom, + https://doi.org/10.1007/978-3-540-45085-6_11 *) + +(* There are at least two ways of defining the above substitution. One way + is to define it in terms of AutoSubst combinators: *) + +Definition eos_var x : var -> var := + (iterate upren x (+1)). + +Definition eos x t := + t.[ren (eos_var x)]. + +Lemma eos_eq: + forall x t, + t.[upn x (ren (+1))] = eos x t. +Proof. + intros. unfold eos, eos_var. erewrite upn_ren by tc. reflexivity. +Qed. + +(* Another way is to define directly as a function of type [var -> var]. *) + +Definition lift_var x : var -> var := + fun y => if le_gt_dec x y then 1 + y else y. + +(* The two definitions coincide: *) + +Lemma upren_lift_var: + forall x, + upren (lift_var x) = lift_var (S x). +Proof. + intros. f_ext; intros [|y]. + { reflexivity. } + { simpl. unfold lift_var, var. dblib_by_cases; omega. } +Qed. + +Lemma eos_var_eq_lift_var: + eos_var = lift_var. +Proof. + (* An uninteresting proof. *) + f_ext; intros x. + unfold eos_var. + induction x. + { reflexivity. } + { rewrite iterate_S. + rewrite IHx. + rewrite upren_lift_var. + reflexivity. } +Qed. + +(* -------------------------------------------------------------------------- *) + +(* [eos] enjoys certain commutation laws. *) + +(* Ending the scope of variable [k], then the scope of variable [s], is the + same as first ending the scope of variable [1 + s], then ending the scope + of variable [k]. This holds provided [k <= s] is true, i.e., [k] is the + most recently-introduced variable.*) + +Lemma lift_var_lift_var: + forall k s, + k <= s -> + lift_var s >>> lift_var k = lift_var k >>> lift_var (S s). +Proof. + (* By case analysis. *) + intros. f_ext; intros x. asimpl. + unfold lift_var, var. dblib_by_cases; omega. +Qed. + +Lemma eos_eos: + forall k s t, + k <= s -> + eos k (eos s t) = eos (1 + s) (eos k t). +Proof. + intros. unfold eos. asimpl. + rewrite eos_var_eq_lift_var. + rewrite lift_var_lift_var by eauto. + reflexivity. +Qed. + +(* What about the case where [k] is the least recently-introduced variable? + It is obtained by symmetry, of course. *) + +Lemma eos_eos_reversed: + forall k s t, + k >= s + 1 -> + eos k (eos s t) = eos s (eos (k - 1) t). +Proof. + intros. + replace k with (1 + (k - 1)) by omega. + rewrite <- eos_eos by omega. + replace (1 + (k - 1) - 1) with (k - 1) by omega. + reflexivity. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* Single-variable substitutions. *) + +(* [subst_var u x] is the substitution of [u] for [x]. *) + +(* We give a direct definition of it as a function of type [var -> term], + defined by cases. I don't know if it could also be nicely defined in + terms of the basic combinators of de Bruijn algebra. Note that the + candidate definition [upn x (t .: ids)] is WRONG when [x > 0]. *) + +Definition subst_var (u : A) (x y : var) : A := + match lt_eq_lt_dec y x with + | inleft (left _) => ids y + | inleft (right _) => u + | inright _ => ids (y - 1) +end. + +(* A nice notation: [t.[u // x]] is the substitution of [u] for [x] in [t]. *) + +Notation "t .[ u // x ]" := (subst (subst_var u x) t) + (at level 2, u at level 200, left associativity, + format "t .[ u // x ]") : subst_scope. + +(* The following laws serve as sanity checks: we got the definition right. *) + +Lemma subst_var_miss_1: + forall x y u, + y < x -> + (ids y).[u // x] = ids y. +Proof. + intros. asimpl. unfold subst_var. dblib_by_cases. reflexivity. +Qed. + +Lemma subst_var_match: + forall x u, + (ids x).[ u // x ] = u. +Proof. + intros. asimpl. unfold subst_var. dblib_by_cases. reflexivity. +Qed. + +Lemma subst_var_miss_2: + forall x y u, + x < y -> + (ids y).[u // x] = ids (y - 1). +Proof. + intros. asimpl. unfold subst_var. dblib_by_cases. reflexivity. +Qed. + +(* In the special case where [x] is 0, the substitution [t // 0] can also + be written [t/], which is an AutoSubst notation for [t .: ids]. *) + +Lemma subst_var_0: + forall t u, + t.[u // 0] = t.[u/]. +Proof. + intros. f_equal. clear t. + f_ext. intros [|x]. + { reflexivity. } + { unfold subst_var. simpl. f_equal. omega. } +Qed. + +(* -------------------------------------------------------------------------- *) + +(* A cancellation law: substituting for a variable [x] that does not occur in + [t] yields just [t]. In other words, a substitution for [x] vanishes when + it reaches [eos x _]. *) + +(* In informal syntax, this lemma would be written: + + t[u/x] = t + + under the hypothesis that x does not occur free in t. + + In de Bruijn style, the statement is just as short, and does not have a + side condition. Instead, it requires an explicit [eos x _] to appear at the + root of the term to which the substitution is applied; this may require + rewriting before this lemma can be applied. *) + +Lemma subst_eos: + forall x t u, + (eos x t).[u // x] = t. +Proof. + intros. + (* Again, let's simplify this first. *) + unfold eos. asimpl. + (* Aha! We can forget about [t], and focus on proving that two + substitutions are equal. To do so, it is sufficient that + their actions on a variable [y] are the same. *) + rewrite <- subst_id. + f_equal. clear t. + f_ext. intro y. + (* The proof is easy if we replace [eos_var] with [lift_var]. *) + rewrite eos_var_eq_lift_var. simpl. + unfold subst_var, lift_var. dblib_by_cases; f_equal; omega. +Qed. + +(* The above property allows us to prove that [eos x _] is injective. + Indeed, it has an inverse, namely [u // x], where [u] is arbitrary. *) + +Lemma eos_injective: + forall x t1 t2, + eos x t1 = eos x t2 -> + t1 = t2. +Proof. + intros. + pose (u := t1). (* dummy *) + erewrite <- (subst_eos x t1 u). + erewrite <- (subst_eos x t2 u). + congruence. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* More commutation laws. *) + +Lemma eos_subst_1: + forall k s t u, + k <= s -> + eos k (t.[u // s]) = (eos k t).[eos k u // s + 1]. +Proof. + intros. unfold eos. asimpl. f_equal. clear t. + rewrite eos_var_eq_lift_var. + f_ext. intros y. + asimpl. + unfold subst_var, lift_var. + dblib_by_cases; asimpl; dblib_by_cases; solve [ eauto | f_equal; omega ]. +Qed. + +Lemma eos_subst_2: + forall k s t u, + s <= k -> + eos k (t.[u // s]) = (eos (k + 1) t).[eos k u // s]. +Proof. + intros. unfold eos. asimpl. f_equal. clear t. + rewrite eos_var_eq_lift_var. + f_ext. intros y. + asimpl. + unfold subst_var, lift_var. + dblib_by_cases; asimpl; dblib_by_cases; solve [ eauto | f_equal; omega ]. +Qed. + +Lemma subst_subst: + forall t k v s w, + k <= s -> + t.[w // k].[v // s] = + t.[eos k v // 1 + s].[w.[v // s] // k]. +Proof. + (* First, get rid of [t]. It is sufficient to consider the action of + these substitutions at a variable [y]. *) + intros. asimpl. f_equal. clear t. f_ext. intros y. + (* Replace [eos_var] with [lift_var], whose definition is more direct. *) + unfold eos. rewrite eos_var_eq_lift_var. + (* Then, use brute force (case analysis) to prove that the goal holds. *) + unfold subst_var. simpl. + dblib_by_cases; asimpl; dblib_by_cases; + (* This case analysis yields 5 cases, of which 4 are trivial... *) + eauto. + (* ... thus, one case remains. *) + (* Now get rid of [v]. It is again sufficient to consider the action + of these substitutions at a variable [z]. *) + replace v with v.[ids] at 1 by autosubst. + f_equal. f_ext. intros z. simpl. + (* Again, use brute force. *) + unfold lift_var. dblib_by_cases; f_equal. unfold var. omega. + (* Not really proud of this proof. *) +Qed. + +Lemma pun_1: + forall t x, + (eos x t).[ ids x // x + 1 ] = t. +Proof. + (* First, get rid of [t]. It is sufficient to consider the action of + these substitutions at a variable [y]. *) + intros. unfold eos. asimpl. + replace t with t.[ids] at 2 by autosubst. + f_equal. clear t. f_ext. intros y. + (* Replace [eos_var] with [lift_var], whose definition is more direct. *) + rewrite eos_var_eq_lift_var. + (* Then, use brute force (case analysis) to prove that the goal holds. *) + simpl. unfold subst_var, lift_var. dblib_by_cases; f_equal; unfold var; omega. +Qed. + +Lemma pun_2: + forall t x, + (eos (x + 1) t).[ ids x // x ] = t. +Proof. + (* First, get rid of [t]. It is sufficient to consider the action of + these substitutions at a variable [y]. *) + intros. unfold eos. asimpl. + replace t with t.[ids] at 2 by autosubst. + f_equal. clear t. f_ext. intros y. + (* Replace [eos_var] with [lift_var], whose definition is more direct. *) + rewrite eos_var_eq_lift_var. + (* Then, use brute force (case analysis) to prove that the goal holds. *) + simpl. unfold subst_var, lift_var. dblib_by_cases; f_equal; unfold var; omega. +Qed. + +End EOS. + +(* Any notations defined in the above section must now be repeated. *) + +Notation "t .[ u // x ]" := (subst (subst_var u x) t) + (at level 2, u at level 200, left associativity, + format "t .[ u // x ]") : subst_scope. + +(* The tactic [subst_var] attempts to simplify applications of [subst_var]. *) + +Ltac subst_var := + first [ + rewrite subst_var_miss_1 by omega + | rewrite subst_var_match by omega + | rewrite subst_var_miss_2 by omega + ]. diff --git a/coq/Autosubst_Env.v b/coq/Autosubst_Env.v new file mode 100644 index 0000000..83b702f --- /dev/null +++ b/coq/Autosubst_Env.v @@ -0,0 +1,130 @@ +Require Import List. +Require Import MyTactics. (* TEMPORARY *) +Require Import Autosubst.Autosubst. +Require Import Autosubst_EOS. (* [eos_var] *) + +(* Environments are sometimes represented as finite lists. This file + provides a few notions that helps deal with this representation. *) + +Section Env. + +Context {A} `{Ids A, Rename A, Subst A, SubstLemmas A}. + +(* -------------------------------------------------------------------------- *) + +(* The function [env2subst default], where [default] is a default value, + converts an environment (a finite list) to a substitution (a total + function). *) + +Definition env2subst (default : A) (e : list A) (x : var) : A := + nth x e default. + +(* -------------------------------------------------------------------------- *) + +(* [env_ren_comp e xi e'] means (roughly) that the environment [e] is equal to + the composition of the renaming [xi] and the environment [e'], that is, + [e = xi >>> e']. We also explicitly require the environments [e] and [e'] + to have matching lengths, up to [xi], as this is *not* a consequence of + the other premise. *) + +Inductive env_ren_comp : list A -> (var -> var) -> list A -> Prop := +| EnvRenComp: + forall e xi e', + (forall x, x < length e -> xi x < length e') -> + (forall x default, nth x e default = nth (xi x) e' default) -> + env_ren_comp e xi e'. + +(* A reformulation of the second premise in the above definition. *) + +Lemma env_ren_comp_eq: + forall e xi e', + (forall default, env2subst default e = xi >>> env2subst default e') <-> + (forall x default, nth x e default = nth (xi x) e' default). +Proof. + unfold env2subst. split; intros h; intros. + { change (nth x e default) with ((fun x => nth x e default) x). + rewrite h. reflexivity. } + { f_ext; intro x. eauto. } +Qed. + +(* -------------------------------------------------------------------------- *) + +(* Initialization: [e = id >>> e]. *) + +Lemma env_ren_comp_id: + forall e, + env_ren_comp e (fun x => x) e. +Proof. + econstructor; eauto. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* The relation [e = xi >>> e'] can be taken under a binder, as follows. *) + +Lemma env_ren_comp_up: + forall e xi e' v, + env_ren_comp e xi e' -> + env_ren_comp (v :: e) (upren xi) (v :: e'). +Proof. + inversion 1; intros; subst; econstructor; + intros [|x]; intros; simpl in *; eauto with omega. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* One element can be prepended to [e'], provided [xi] is adjusted. *) + +Lemma env_ren_comp_prepend: + forall e xi e' v, + env_ren_comp e xi e' -> + env_ren_comp e (xi >>> (+1)) (v :: e'). +Proof. + inversion 1; intros; subst. + econstructor; intros; simpl; eauto with omega. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* A consequence of [env_ren_comp_id] and [env_ren_comp_prepend]. The renaming + (+1) will eat away the first entry in [v :: e]. *) + +Lemma env_ren_comp_plus1: + forall e v, + env_ren_comp e (+1) (v :: e). +Proof. + econstructor; intros; simpl; eauto with omega. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* More generally, the renaming [eos_var x], which means that [x] goes out of + scope, will eat away the entry at index [x] in [e1 ++ v :: e2]. *) + +Lemma env_ren_comp_eos_var: + forall x e1 v e2, + x = length e1 -> + env_ren_comp (e1 ++ e2) (eos_var x) (e1 ++ v :: e2). +Proof. + rewrite eos_var_eq_lift_var. unfold lift_var. + econstructor; intros y; dblib_by_cases. + { rewrite app_length in *. simpl. omega. } + { rewrite app_length in *. simpl. omega. } + { do 2 (rewrite app_nth2 by omega). + replace (1 + y - length e1) with (1 + (y - length e1)) by omega. + reflexivity. } + { do 2 (rewrite app_nth1 by omega). + reflexivity. } +Qed. + +(* -------------------------------------------------------------------------- *) + +End Env. + +Hint Resolve + env_ren_comp_id + env_ren_comp_up + env_ren_comp_prepend + env_ren_comp_plus1 + env_ren_comp_eos_var +: env_ren_comp. diff --git a/coq/Autosubst_FreeVars.v b/coq/Autosubst_FreeVars.v new file mode 100644 index 0000000..bf05011 --- /dev/null +++ b/coq/Autosubst_FreeVars.v @@ -0,0 +1,345 @@ +Require Import Omega. +Require Import Autosubst.Autosubst. +Require Import AutosubstExtra. +Require Import Autosubst_EOS. + +(* -------------------------------------------------------------------------- *) + +Class IdsLemmas (term : Type) {Ids_term : Ids term} := { + (* The identity substitution is injective. *) + ids_inj: + forall x y, + ids x = ids y -> + x = y +}. + +(* -------------------------------------------------------------------------- *) + +Section FreeVars. + +Context + {A : Type} + {Ids_A : Ids A} + {Rename_A : Rename A} + {Subst_A : Subst A} + {IdsLemmas_A : IdsLemmas A} + {SubstLemmas_A : SubstLemmas A}. + +(* -------------------------------------------------------------------------- *) + +(* A reformulation of [ids_inj]. *) + +Lemma ids_inj_False: + forall x y, + ids x = ids y -> + x <> y -> + False. +Proof. + intros. + assert (x = y). { eauto using ids_inj. } + unfold var in *. + omega. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* The predicate [fv k t] means that the free variables of the term [t] are + contained in the semi-open interval [0..k). *) + +Definition fv k t := + t.[upn k (ren (+1))] = t. + +(* -------------------------------------------------------------------------- *) + +(* The predicate [closed t] means that the term [t] is closed, that is, [t] + has no free variables. *) + +Definition closed := + fv 0. + +(* -------------------------------------------------------------------------- *) + +(* This technical lemma states that the renaming [+1] is injective. *) + +Lemma lift_inj_ids: + forall t x, + t.[ren (+1)] = ids (S x) <-> t = ids x. +Proof. + split; intros. + { eapply lift_inj. autosubst. } + { subst. autosubst. } +Qed. + +(* -------------------------------------------------------------------------- *) + +(* This lemma characterizes the meaning of [fv k] when applied to a variable. *) + +Lemma fv_ids_eq: + forall k x, + fv k (ids x) <-> x < k. +Proof. + unfold fv. induction k; intros. + (* Base case. *) + { asimpl. split; intros; elimtype False. + { eauto using ids_inj_False. } + { omega. } + } + (* Step. *) + { destruct x; asimpl. + { split; intros. { omega. } { reflexivity. } } + { rewrite lift_inj_ids. + rewrite <- id_subst. + rewrite IHk. omega. } + } +Qed. + +(* -------------------------------------------------------------------------- *) + +(* A simplification lemma. *) + +Lemma fv_lift: + forall k i t, + fv (k + i) t.[ren (+i)] <-> fv k t. +Proof. + unfold fv. intros. asimpl. + rewrite Nat.add_comm. + rewrite <- upn_upn. + erewrite plus_upn by eauto. + rewrite <- subst_comp. + split; intros. + { eauto using lift_injn. } + { f_equal. eauto. } +Qed. + +(* -------------------------------------------------------------------------- *) + +(* If [t] has at most [n - 1] free variables, + and if [x] is inserted among them, + then we get [eos x t], + which has at most [n] free variables. *) + +Lemma fv_eos: + forall x n t, + x < n -> + fv (n - 1) t -> + fv n (eos x t). +Proof. + unfold fv. intros x n t ? ht. + rewrite eos_eq in ht. + rewrite eos_eq. + rewrite eos_eos_reversed by omega. (* nice! *) + rewrite ht. + reflexivity. +Qed. + +Lemma fv_eos_eq: + forall x n t, + x < n -> + fv n (eos x t) <-> + fv (n - 1) t. +Proof. + unfold fv. intros x n t ?. + rewrite eos_eq. + rewrite eos_eq. + rewrite eos_eos_reversed by omega. (* nice! *) + split; intros h. + { eauto using eos_injective. } + { rewrite h. reflexivity. } +Qed. + +(* -------------------------------------------------------------------------- *) + +(* A substitution [sigma] is regular if and only if, for some [j], for + sufficiently large [x], [sigma x] is [x + j]. *) + +Definition regular (sigma : var -> A) := + exists i j, + ren (+i) >> sigma = ren (+j). + +Lemma regular_ids: + regular ids. +Proof. + exists 0. exists 0. autosubst. +Qed. + +Lemma regular_plus: + forall i, + regular (ren (+i)). +Proof. + intros. exists 0. exists i. autosubst. +Qed. + +Lemma regular_upn: + forall n sigma, + regular sigma -> + regular (upn n sigma). +Proof. + intros ? ? (i&j&hsigma). + exists (n + i). eexists (n + j). + replace (ren (+(n + i))) with (ren (+i) >> ren (+n)) by autosubst. + rewrite <- scompA. + rewrite up_liftn. + rewrite scompA. + rewrite hsigma. + autosubst. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* If the free variables of the term [t] are below [k], then [t] is unaffected + by a substitution of the form [upn k sigma]. *) + +(* Unfortunately, in this file, where the definition of type [A] is unknown, I + am unable to establish this result for arbitrary substitutions [sigma]. I + am able to establish it for *regular* substitutions, where The proof is somewhat interesting, so it is given here, even + though, once the definition of the type [A] is known, a more direct proof, + without a regularity hypothesis, can usually be given. *) + +(* An intermediate result states that, since [upn k (ren (+1))] does not + affect [t], then (by iteration) neither does [upn k (ren (+j))]. *) + +Lemma fv_unaffected_lift: + forall j t k, + fv k t -> + t.[upn k (ren (+j))] = t. +Proof. + induction j as [| j ]; intros t k ht. + { asimpl. rewrite up_id_n. autosubst. } + { replace (ren (+S j)) with (ren (+1) >> ren (+j)) by autosubst. + rewrite <- up_comp_n. + replace (t.[upn k (ren (+1)) >> upn k (ren (+j))]) + with (t.[upn k (ren (+1))].[upn k (ren (+j))]) by autosubst. + rewrite ht. + rewrite IHj by eauto. + eauto. } +Qed. + +(* There follows that a substitution of the form [upn k sigma], where [sigma] + is regular, does not affect [t]. The proof is slightly subtle but very + short. The previous lemma is used twice. *) + +Lemma fv_unaffected_regular: + forall k t sigma, + fv k t -> + regular sigma -> + t.[upn k sigma] = t. +Proof. + intros k t sigma ? (i&j&hsigma). + rewrite <- (fv_unaffected_lift i t k) at 1 by eauto. + asimpl. rewrite up_comp_n. + rewrite hsigma. + rewrite fv_unaffected_lift by eauto. + reflexivity. +Qed. + +(* A corollary. *) + +Lemma closed_unaffected_regular: + forall t sigma, + closed t -> + regular sigma -> + t.[sigma] = t. +Proof. + unfold closed. intros. + rewrite <- (upn0 sigma). + eauto using fv_unaffected_regular. +Qed. + +(*One might also wish to prove a result along the following lines: + + Goal + forall t k sigma1 sigma2, + fv k t -> + (forall x, x < k -> sigma1 x = sigma2 x) -> + t.[sigma1] = t.[sigma2]. + + I have not yet investigated how this could be proved. *) + +(* -------------------------------------------------------------------------- *) + +(* If some term [t] has free variables under [j], then it also has free + variables under [k], where [j <= k]. *) + +Lemma fv_monotonic: + forall j k t, + fv j t -> + j <= k -> + fv k t. +Proof. + intros. unfold fv. + replace k with (j + (k - j)) by omega. + rewrite <- upn_upn. + eauto using fv_unaffected_regular, regular_upn, regular_plus. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* These little lemmas may be occasionally useful. *) + +Lemma use_fv_length_cons: + forall A (x : A) (xs : list A) n t, + (forall x, fv (length (x :: xs)) t) -> + n = length xs -> + fv (n + 1) t. +Proof. + intros. subst. + replace (length xs + 1) with (length (x :: xs)) by (simpl; omega). + eauto. +Qed. + +Lemma prove_fv_length_cons: + forall A (x : A) (xs : list A) n t, + n = length xs -> + fv (n + 1) t -> + fv (length (x :: xs)) t. +Proof. + intros. subst. + replace (length (x :: xs)) with (length xs + 1) by (simpl; omega). + eauto. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* [closed t] is equivalent to [t.[ren (+1)] = t]. *) + +Lemma closed_eq: + forall t, + closed t <-> t.[ren (+1)] = t. +Proof. + unfold closed, fv. asimpl. tauto. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* A variable is not closed. *) + +Lemma closed_ids: + forall x, + ~ closed (ids x). +Proof. + unfold closed, fv. intros. asimpl. intro. + eauto using ids_inj_False. +Qed. + +End FreeVars. + +(* -------------------------------------------------------------------------- *) + +(* The tactic [fv] is intended to use a number of lemmas as rewriting rules. + The hint database [fv] can be extended with language-specific lemmas. *) + +Hint Rewrite @fv_ids_eq @fv_lift @fv_eos_eq : fv. + +Ltac fv := + autorewrite with fv in *; + eauto with typeclass_instances. + +(* -------------------------------------------------------------------------- *) + +(* A hint database to prove goals of the form [~ (closed _)] or [closed _]. *) + +Hint Resolve closed_ids : closed. + +(* -------------------------------------------------------------------------- *) + +Hint Resolve regular_ids regular_plus regular_upn : regular. diff --git a/coq/ClosureConversion.v b/coq/ClosureConversion.v new file mode 100644 index 0000000..208b3fd --- /dev/null +++ b/coq/ClosureConversion.v @@ -0,0 +1,349 @@ +Require Import List. +Require Import MyList. +Require Import MyTactics. +Require Import Sequences. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusFreeVars. +Require Import LambdaCalculusValues. +Require Import LambdaCalculusBigStep. +Require Import MetalSyntax. +Require Import MetalBigStep. + +(* This file contains a definition of closure conversion and a proof of + semantic preservation. Closure conversion is a transformation of the + lambda-calculus into a lower-level calculus, nicknamed Metal, where + lambda-abstractions must be closed. *) + +(* The definition is slightly painful because we are using de Bruijn indices. + (That said, it is likely that using named variables would be painful too, + just in other ways.) One important pattern that we follow is that, as soon + as a variable [x] is no longer useful, we use [eos x _] to make it go out + of scope. Following this pattern is not just convenient -- it is actually + necessary. *) + +(* The main transformation function, [cc], has type [nat -> term -> metal]. If + [t] is a source term with [n] free variables, then [cc n t] is its image + under the transformation. *) + +(* -------------------------------------------------------------------------- *) +(* -------------------------------------------------------------------------- *) + +(* The auxiliary function [cc_lam n cct] defines the transformation of + the body of a lambda-abstraction. This auxiliary function must be isolated + because it is used twice, once within [cc], and once within [cc_value]. *) + +(* The parameter [n] is supposed to be the number of free variables of the + lambda-abstraction, and the parameter [cct] is the body of the transformed + lambda-abstraction. That is, [cct] is expected to be later instantiated + with [cc (n + 1) t], where [t] is the body of the source + lambda-abstraction. *) + +(* The term [cc_lam n cct] has just one free variable, namely 0, which + represents the formal parameter of the transformed lambda-abstraction. This + parameter is expected to be a pair [(clo, x)], where [clo] is the closure + and [x] is the formal parameter of the source lambda-abstraction. *) + +(* In nominal / informal syntax, the code would be roughly: + + let (clo, x) = _ in + let x_1, ..., x_n = pi_1 clo, ..., pi_n clo in + cct + +*) + +(* We use meta-level [let] bindings, such as [let clo_x := 0 in ...]. + These bindings generate no Metal code -- they should not be confused + with Metal [MLet] bindings. They are just a notational convenience. *) + +Definition cc_lam (n : nat) (cct : metal) : metal := + (* Let us refer to variable 0 as [clo_x]. *) + let clo_x := 0 in + (* Initially, the variable [clo_x] is bound to a pair of [clo] and [x]. + Decompose this pair, then let [clo_x] go out of scope. *) + MLetPair (* clo, x *) (MVar clo_x) ( + let clo_x := clo_x + 2 in + eos clo_x ( + (* Two variables are now in scope, namely [clo] and [x]. *) + let clo := 1 in + let x := 0 in + (* Now, bind [n] variables, informally referred to as [x_1, ..., x_n], + to the [n] tuple projections [pi_1 clo, ..., pi_n clo]. Then, let + [clo] go out of scope, as it is no longer needed. *) + MMultiLet 0 (MProjs n (MVar clo)) ( + let clo := n + clo in + let x := n + x in + eos clo ( + (* We need [x] to correspond to de Bruijn index 0. Currently, this is + not the case. So, rebind a new variable to [x], and let the old [x] + go out of scope. (Yes, this is a bit subtle.) We use an explicit + [MLet] binding, but in principle, could also use a substitution. *) + MLet (MVar x) ( + let x := 1 + x in + eos x ( + (* We are finally ready to enter the transformed function body. *) + cct + )))))). + +(* [cc] is the main transformation function. *) + +Fixpoint cc (n : nat) (t : term) : metal := + match t with + + | Var x => + (* A variable is mapped to a variable. *) + MVar x + + | Lam t => + (* A lambda-abstraction [Lam t] is mapped to a closure, that is, a tuple + whose 0-th component is a piece of code (a closed lambda-abstraction) + and whose remaining [n] components are the [n] variables numbered 0, + 1, ..., n-1. *) + MTuple ( + MLam (cc_lam n (cc (n + 1) t)) :: + MVars n + ) + + | App t1 t2 => + (* A function application is transformed into a closure invocation. *) + (* Bind [clo] to the closure produced by the (transformed) term [t1]. *) + MLet (cc n t1) ( + let clo := 0 in + (* Bind [code] to the 0-th component of this closure. *) + MLet (MProj 0 (MVar clo)) ( + let clo := 1 + clo in + let code := 0 in + (* Apply the function [code] to a pair of the closure and the value + produced by the (transformed) term [t2]. Note that both [clo] and + [code] must go out of scope before referring to [t2]. This could + be done in one step by applying the renaming [(+2)], but we prefer + to explicitly use [eos] twice, for greater clarity. *) + MApp + (MVar code) + (MPair + (MVar clo) + (eos clo (eos code (cc n t2))) + ) + )) + + | Let t1 t2 => + (* A local definition is translated to a local definition. *) + MLet (cc n t1) (cc (n + 1) t2) + + end. + +(* -------------------------------------------------------------------------- *) +(* -------------------------------------------------------------------------- *) + +(* In order to prove that the above program transformation is sound (that is, + semantics-preserving), we must relate the values computed by the source + program with the values computed by the target program. Fortunately, the + relation is simple. (It is, in fact, a function.) The image of a source + closure [Clo t e] under the translation is a closure, represented as a + tuple, whose 0-th component is a lambda-abstraction -- the translation + of [Lam t] -- and whose remaining [n] components are the translation of + the environment [e]. *) + +Fixpoint cc_cvalue (cv : cvalue) : mvalue := + match cv with + | Clo t e => + let n := length e in + MVTuple ( + MVLam (cc_lam n (cc (n + 1) t)) :: + map cc_cvalue e + ) + end. + +(* The translation of environments is defined pointwise. *) + +Definition cc_cenv e := + map cc_cvalue e. + +(* -------------------------------------------------------------------------- *) +(* -------------------------------------------------------------------------- *) + +(* This lemma and hint increase the power of the tactic [length]. *) + +Lemma length_cc_env: + forall e, + length (cc_cenv e) = length e. +Proof. + intros. unfold cc_cenv. length. +Qed. + +Hint Rewrite length_cc_env : length. + +(* In this file, we allow [eauto with omega] to use the tactic [length]. *) + +Local Hint Extern 1 (_ = _ :> nat) => length : omega. +Local Hint Extern 1 (lt _ _) => length : omega. + +(* -------------------------------------------------------------------------- *) +(* -------------------------------------------------------------------------- *) + +(* We now check two purely syntactic properties of the transformation. First, + if the source program [t] has [n] free variables, then the transformed + program [cc n t] has [n] free variables, too. Second, in the transformed + program, every lambda-abstraction is closed. *) + +(* These proofs are "easy" and "routine", but could have been fairly lengthy + and painful if we had not set up appropriate lemmas and tactics, such as + [fv], beforehand. *) + +Lemma fv_cc_lam: + forall m n t, + fv (S n) t -> + 1 <= m -> + fv m (cc_lam n t). +Proof. + intros. unfold cc_lam. fv; length. repeat split; + eauto using fv_monotonic with omega. + { eapply fv_MProjs. fv. omega. } +Qed. + +Lemma fv_cc: + forall t n1 n2, + fv n2 t -> + n1 = n2 -> + fv n1 (cc n2 t). +Proof. + induction t; intros; subst; simpl; fv; unpack; + repeat rewrite Nat.add_1_r in *; + repeat split; eauto using fv_monotonic, fv_cc_lam with omega. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* The following lemmas help reorganize our view of the environment. They are + typically used just before applying the lemma [mbigcbv_eos], that is, when + a variable [x] is about to go out of scope. We then want to organize the + environment so that it has the shape [e1 ++ mv :: e2], where the length of + [e1] is [x], so [mv] is the value that is dropped, and the environment + afterwards is [e1 ++ e2]. *) + +Local Lemma access_1: + forall {A} xs (x0 x : A), + x0 :: x :: xs = (x0 :: nil) ++ x :: xs. +Proof. + reflexivity. +Qed. + +Local Lemma access_2: + forall {A} xs (x0 x1 x : A), + x0 :: x1 :: x :: xs = (x0 :: x1 :: nil) ++ x :: xs. +Proof. + reflexivity. +Qed. + +Local Lemma access_n_plus_1: + forall {A} xs (x : A) ys, + xs ++ x :: ys = (xs ++ x :: nil) ++ ys. +Proof. + intros. rewrite <- app_assoc. reflexivity. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* We now establish a (forward) semantic preservation statement: if under + environment [e] the source term [t] computes the value [cv], then under + the translation of [e], the translation of [t] computes the translation + of [cv]. *) + +(* The proof is fairly routine and uninteresting; it is just a matter of + checking that everything works as expected. A crucial point that helps + keep the proof manageable is that we have defined auxiliary evaluation + lemmas for tuples, projections, [MMultiLet], and so on. *) + +(* Although the proof is not "hard", it would be difficult to do by hand, as + it is quite deep and one must keep track of many details, such as the shape + of the runtime environment at every program point -- that is, which de + Bruijn index is bound to which value. An off-by-one error in a de Bruijn + index, or a list that is mistakenly reversed, would be difficult to detect + in a handwritten proof. Machine assistance is valuable in this kind of + exercise. *) + +Theorem semantic_preservation: + forall e t cv, + ebigcbv e t cv -> + forall n, + n = length e -> + mbigcbv (cc_cenv e) (cc n t) (cc_cvalue cv). +Proof. + induction 1; intros; simpl. + + (* Case: [Var]. Nothing much to say; apply the evaluation rule and + check that its side conditions hold. *) + { econstructor. + { length. } + { subst cv. + erewrite (nth_indep (cc_cenv e)) by length. + unfold cc_cenv. rewrite map_nth. eauto. } + } + + (* Case: [Lam]. We have to check that a transformed lambda-abstraction + (as per [cc]) evaluates to a transformed closure (as per [cc_value]). + This is fairly easy. *) + { subst. econstructor. + (* The code component. We must check that it is closed. *) + { econstructor. + assert (fv (length e + 1) t). + { eapply (use_fv_length_cons _ dummy_cvalue); eauto. } + unfold closed. fv. eauto using fv_cc_lam, fv_cc with omega. } + (* The remaining components. *) + { eapply MBigcbvTuple. + rewrite <- length_cc_env. eauto using MBigcbvVars. } + } + + (* Case: [App]. This is where most of the action takes place. We must + essentially "execute" the calling sequence and check that it works + as expected. *) + + { (* Evaluate the first [Let], which binds a variable [clo] to the closure. *) + econstructor. { eauto. } + (* Evaluate the second [Let], which projects the code out of the closure, + and binds the variable [code]. *) + econstructor. + { econstructor; [| eauto ]. + econstructor; simpl; eauto with omega. } + (* We are now looking at a function application. Evaluate in turn the function, + its actual argument, and the call itself. *) + econstructor. + (* The function. *) + { econstructor; simpl; eauto with omega. } + (* Its argument. *) + { econstructor. + { econstructor; simpl; eauto with omega. } + { (* Skip two [eos] constructs. *) + rewrite access_1. eapply mbigcbv_eos; [ simpl | length ]. + eapply mbigcbv_eos with (e1 := nil); [ simpl | length ]. + eauto. } + } + (* The call itself. Here, we step into a transformed lambda-abstraction, and + we begin studying how it behaves when executed. *) + unfold cc_lam at 2. + (* Evaluate [MLetPair], which binds [clo] and [x]. *) + eapply MBigcbvLetPair. + { econstructor; simpl; eauto with omega. } + (* Skip [eos]. *) + rewrite access_2. eapply mbigcbv_eos; [ simpl | eauto ]. + (* Evaluate [MMultiLet]. *) + eapply MBigcbvMultiLetProjs. + { econstructor; simpl; eauto with omega. } + { length. } + (* Skip [eos]. *) + rewrite access_n_plus_1. eapply mbigcbv_eos; [| length ]. + rewrite app_nil_r, Nat.add_0_r. + (* Evaluate the new binding of [x]. *) + econstructor. + { econstructor. + { length. } + { rewrite app_nth by (rewrite map_length; omega). simpl. eauto. } + } + (* Skip [eos]. *) + rewrite app_comm_cons. eapply mbigcbv_eos; [ rewrite app_nil_r | length ]. + (* Evaluate the function body. *) + eauto with omega. } + + (* Case: [Let]. Immediate. *) + { econstructor; eauto with omega. } + +Qed. diff --git a/coq/LambdaCalculusFreeVars.v b/coq/LambdaCalculusFreeVars.v index 1d5fdbb..423d1d6 100644 --- a/coq/LambdaCalculusFreeVars.v +++ b/coq/LambdaCalculusFreeVars.v @@ -14,12 +14,6 @@ Qed. (* -------------------------------------------------------------------------- *) -(* The predicate [fv k t] means that the free variables of the term [t] are - contained in the semi-open interval [0..k). *) - -Definition fv k t := - t.[upn k (ren (+1))] = t. - (* The predicate [fv] is characterized by the following lemmas. *) Lemma fv_Var_eq: @@ -67,28 +61,8 @@ Qed. Hint Rewrite fv_Var_eq fv_Lam_eq fv_App_eq fv_Let_eq : fv. -(* The tactic [fv] uses the above lemmas as rewriting rules. *) - -Ltac fv := - autorewrite with fv in *. - (* -------------------------------------------------------------------------- *) -(* The predicate [closed t] means that the term [t] is closed, that is, [t] - has no free variables. *) - -Definition closed := - fv 0. - -(* [closed t] is equivalent to [lift 1 t = t]. *) - -Lemma closed_eq: - forall t, - closed t <-> lift 1 t = t. -Proof. - unfold closed, fv. asimpl. tauto. -Qed. - (* The following lemmas allow decomposing a closedness hypothesis. Because [closed] is not an inductive notion, there is no lemma for [Lam] and for the right-hand side of [Let]. *) diff --git a/coq/LambdaCalculusSyntax.v b/coq/LambdaCalculusSyntax.v index 56d1898..0134bad 100644 --- a/coq/LambdaCalculusSyntax.v +++ b/coq/LambdaCalculusSyntax.v @@ -3,6 +3,8 @@ Require Import MyTactics. Require Export Autosubst.Autosubst. Require Export AutosubstExtra. Require Export Autosubst_IsRen. +(* Require Export Autosubst_EOS. *) +Require Export Autosubst_FreeVars. (* The syntax of the lambda-calculus. *) @@ -18,6 +20,9 @@ Instance Rename_term : Rename term. derive. Defined. Instance Subst_term : Subst term. derive. Defined. Instance SubstLemmas_term : SubstLemmas term. derive. Qed. +Instance IdsLemmas_term : IdsLemmas term. +Proof. econstructor. intros. injections. eauto. Qed. + (* If the image of [t] through a substitution is a variable, then [t] must itself be a variable. *) diff --git a/coq/MetaBigStep.v b/coq/MetaBigStep.v new file mode 100644 index 0000000..e69de29 diff --git a/coq/MetalBigStep.v b/coq/MetalBigStep.v new file mode 100644 index 0000000..6f0632b --- /dev/null +++ b/coq/MetalBigStep.v @@ -0,0 +1,313 @@ +Require Import List. +Require Import MyList. +Require Import MyTactics. +Require Import Sequences. +Require Import MetalSyntax. +Require Import Autosubst_Env. + +(* -------------------------------------------------------------------------- *) +(* -------------------------------------------------------------------------- *) + +(* A big-step call-by-value semantics with explicit environments. *) + +(* Because every lambda-abstraction is closed, no closures are involved: + a lambda-abstraction evaluates to itself. Thus, an mvalue [mv] must be + either a (closed) lambda-abstraction or a pair of mvalues. *) + +Inductive mvalue := +| MVLam : {bind metal} -> mvalue +| MVPair : mvalue -> mvalue -> mvalue. + +Definition dummy_mvalue : mvalue := + MVLam (MVar 0). + +(* An environment [e] is a list of mvalues. *) + +Definition menv := + list mvalue. + +(* The judgement [mbigcbv e t mv] means that, under the environment [e], the + term [t] evaluates to [mv]. *) + +Inductive mbigcbv : menv -> metal -> mvalue -> Prop := +| MBigcbvVar: + forall e x mv, + (* The variable [x] must be in the domain of [e]. *) + x < length e -> + (* This allows us to safely look up [e] at [x]. *) + mv = nth x e dummy_mvalue -> + mbigcbv e (MVar x) mv +| MBigcbvLam: + forall e t, + (* The lambda-abstraction must have no free variables. *) + closed (MLam t) -> + mbigcbv e (MLam t) (MVLam t) +| MBigcbvApp: + forall e t1 t2 u1 mv2 mv, + (* Evaluate [t1] to a lambda-abstraction, *) + mbigcbv e t1 (MVLam u1) -> + (* evaluate [t2] to a value, *) + mbigcbv e t2 mv2 -> + (* and evaluate the function body in a singleton environment. *) + mbigcbv (mv2 :: nil) u1 mv -> + mbigcbv e (MApp t1 t2) mv +| MBigcbvLet: + forall e t1 t2 mv1 mv, + (* Evaluate [t1] to a value, *) + mbigcbv e t1 mv1 -> + (* and evaluate [t2] under a suitable environment. *) + mbigcbv (mv1 :: e) t2 mv -> + mbigcbv e (MLet t1 t2) mv +| MBigcbvPair: + forall e t1 t2 mv1 mv2, + (* Evaluate each component to a value, *) + mbigcbv e t1 mv1 -> + mbigcbv e t2 mv2 -> + (* and construct a pair. *) + mbigcbv e (MPair t1 t2) (MVPair mv1 mv2) +| MBigcbvPi: + forall e i t mv1 mv2 mv, + (* Evaluate [t] to a pair value, *) + mbigcbv e t (MVPair mv1 mv2) -> + (* and project out the desired component. *) + mv = match i with 0 => mv1 | _ => mv2 end -> + mbigcbv e (MPi i t) mv +. + +Hint Constructors mbigcbv : mbigcbv. + +(* -------------------------------------------------------------------------- *) + +(* A reformulation of the evaluation rule for variables. *) + +Lemma MBigcbvVarExact: + forall e1 mv e2 x, + x = length e1 -> + mbigcbv (e1 ++ mv :: e2) (MVar x) mv. +Proof. + intros. econstructor. + { length. } + { rewrite app_nth by eauto. reflexivity. } +Qed. + +(* -------------------------------------------------------------------------- *) +(* -------------------------------------------------------------------------- *) + +(* We now examine how the big-step semantics interacts with renamings. + We prove that if (roughly) the equation [e = xi >>> e'] holds then + evaluating [t.[xi]] under [e'] is the same as evaluating [t] under + [e]. *) + +Lemma mbigcbv_ren: + forall e t mv, + mbigcbv e t mv -> + forall e' xi, + env_ren_comp e xi e' -> + mbigcbv e' t.[ren xi] mv. +Proof. + induction 1; intros; + try solve [ asimpl; eauto with env_ren_comp mbigcbv ]. + (* MVar *) + { pick @env_ren_comp invert. + econstructor; eauto. } + (* MLam *) + { rewrite closed_unaffected by eauto. + eauto with mbigcbv. } +Qed. + +(* As a special case, evaluating [eos x t] under an environment of the form + [e1 ++ mv :: e2], where length of [e1] is [x] (so [x] is mapped to [mv]) + is the same as evaluating [t] under [e1 ++ e2]. The operational effect + of the end-of-scope mark [eos x _] is to delete the value stored at index + [x] in the evaluation environment. *) + +Lemma mbigcbv_eos: + forall e1 e2 x t mv mw, + mbigcbv (e1 ++ e2) t mw -> + x = length e1 -> + mbigcbv (e1 ++ mv :: e2) (eos x t) mw. +Proof. + intros. eapply mbigcbv_ren; eauto with env_ren_comp. +Qed. + +(* -------------------------------------------------------------------------- *) +(* -------------------------------------------------------------------------- *) + +(* Evaluation rules for (simulated) tuples. *) + +Fixpoint MVTuple mvs := + match mvs with + | nil => + MVLam (MVar 0) + | mv :: mvs => + MVPair mv (MVTuple mvs) + end. + +Lemma MBigcbvTuple: + forall e ts mvs, + (* Evaluate every component to a value, *) + Forall2 (mbigcbv e) ts mvs -> + (* and construct a tuple. *) + mbigcbv e (MTuple ts) (MVTuple mvs). +Proof. + induction 1; simpl; econstructor; eauto. + { unfold closed. fv. } +Qed. + +Lemma MBigcbvProj: + forall i e t mvs mv, + i < length mvs -> + (* Evaluate [t] to a tuple value, *) + mbigcbv e t (MVTuple mvs) -> + (* and project out the desired component. *) + mv = nth i mvs dummy_mvalue -> + mbigcbv e (MProj i t) mv. +Proof. + (* By induction on [i]. In either case, [mvs] cannot be an empty list. *) + induction i; intros; (destruct mvs as [| mv0 mvs ]; [ false; simpl in *; omega |]). + (* Base case. *) + { econstructor; eauto. } + (* Step case. *) + { assert (i < length mvs). { length. } + simpl. eauto with mbigcbv. } +Qed. + +Lemma MBigcbvLetPair: + forall e t u mv mv1 mv2, + mbigcbv e t (MVPair mv1 mv2) -> + mbigcbv (mv2 :: mv1 :: e) u mv -> + mbigcbv e (MLetPair t u) mv. +Proof. + unfold MLetPair. eauto 6 using mbigcbv_ren, env_ren_comp_plus1 with mbigcbv. +Qed. + +(* -------------------------------------------------------------------------- *) +(* -------------------------------------------------------------------------- *) + +(* Evaluation rules for [MMultiLet]. *) + +Local Lemma Forall2_mbigcbv_plus1: + forall ts mvs e i mv, + Forall2 (mbigcbv e) (map (fun t : metal => lift i t) ts) mvs -> + Forall2 (mbigcbv (mv :: e)) (map (fun t : metal => lift (i + 1) t) ts) mvs. +Proof. + induction ts as [| t ts]; simpl; intros; + pick Forall2 invert; econstructor; eauto. + replace (lift (i + 1) t) with (lift 1 (lift i t)) by autosubst. + eauto using mbigcbv_ren, env_ren_comp_plus1. +Qed. + +Lemma MBigcbvMultiLet: + forall ts e i u mvs mv, + Forall2 (mbigcbv e) (map (fun t => lift i t) ts) mvs -> + mbigcbv (rev mvs ++ e) u mv -> + mbigcbv e (MMultiLet i ts u) mv. +Proof. + induction ts; simpl; intros; pick Forall2 invert. + { eauto. } + { pick mbigcbv ltac:(fun h => rewrite rev_cons_app in h). + econstructor; eauto using Forall2_mbigcbv_plus1. } +Qed. + +Lemma MBigcbvMultiLet_0: + forall ts e u mvs mv, + Forall2 (mbigcbv e) ts mvs -> + mbigcbv (rev mvs ++ e) u mv -> + mbigcbv e (MMultiLet 0 ts u) mv. +Proof. + intros. eapply MBigcbvMultiLet. + { asimpl. rewrite map_id. eauto. } + { eauto. } +Qed. + +(* -------------------------------------------------------------------------- *) +(* -------------------------------------------------------------------------- *) + +(* An evaluation rule for [MVars]. *) + +Lemma MBigcbvVars_preliminary: + forall e2 e1 e x n, + e = e1 ++ e2 -> + x = length e1 -> + n = length e2 -> + Forall2 (mbigcbv e) (map MVar (seq x n)) e2. +Proof. + induction e2 as [| mv e2 ]; intros; subst; econstructor. + { eauto using MBigcbvVarExact. } + { eapply IHe2 with (e1 := e1 ++ mv :: nil). + { rewrite <- app_assoc. reflexivity. } + { length. } + { eauto. } + } +Qed. + +Lemma MBigcbvVars: + forall e, + Forall2 (mbigcbv e) (MVars (length e)) e. +Proof. + unfold MVars, nats. intros. + eapply MBigcbvVars_preliminary with (e1 := nil); eauto. +Qed. + +(* -------------------------------------------------------------------------- *) +(* -------------------------------------------------------------------------- *) + +(* An evaluation rule for [MProjs]. If the term [t] evaluates to a tuple whose + components numbered 1, ..., n are collected in the list [mvs1], then the + family of terms [MProjs n t] evaluates to the family of values [rev mvs1]. *) + +Lemma MBigcbvProjs: + forall n e t mv mvs1 mvs2, + mbigcbv e t (MVTuple (mv :: mvs1 ++ mvs2)) -> + length mvs1 = n -> + Forall2 (mbigcbv e) (MProjs n t) (rev mvs1). +Proof. + (* This result is "obvious" but requires some low-level work. *) + unfold MProjs. induction n; intros. + (* Case: [n] is zero. Then, [mvs1] must be [nil]. The result follows. *) + { destruct mvs1; [| false; simpl in *; omega ]. + econstructor. } + (* Case: [n] is nonzero. Then, the list [mvs1] must be nonempty. *) + assert (hmvs1: mvs1 <> nil). + { intro. subst. simpl in *. omega. } + (* So, this list has one element at the end. Let us write this list in the + form [mvs1 ++ mv1]. *) + destruct (exists_last hmvs1) as (hd&mv1&?). subst mvs1. clear hmvs1. + generalize dependent hd; intro mvs1; intros. + (* Simplify. *) + rewrite rev_unit. + rewrite <- app_assoc in *. + rewrite app_length in *. + assert (length mvs1 = n). { length. } + simpl map. + econstructor. + (* The list heads. *) + { eapply MBigcbvProj; eauto. + { length. } + { replace (n + 1) with (S n) by omega. simpl. + rewrite app_nth by omega. + reflexivity. } + } + (* The list tails. *) + { eauto. } +Qed. + +(* -------------------------------------------------------------------------- *) + +(* An evaluation rule for the combination of [MMultiLet] and [MProjs]. If the + term [t] evaluates to a tuple whose components are [mv :: mvs], and if [n] + is the length of [mvs], then evaluating [MMultiLet 0 (MProjs n t) u] in an + environment [e] leads to evaluating [u] in environment [mvs ++ e]. *) + +Lemma MBigcbvMultiLetProjs: + forall e t mvs u mv mw n, + mbigcbv e t (MVTuple (mv :: mvs)) -> + length mvs = n -> + mbigcbv (mvs ++ e) u mw -> + mbigcbv e (MMultiLet 0 (MProjs n t) u) mw. +Proof. + intros. + eapply MBigcbvMultiLet_0. + { eapply MBigcbvProjs with (mvs2 := nil); [ rewrite app_nil_r |]; eauto. } + { rewrite rev_involutive. eauto. } +Qed. diff --git a/coq/MetalSyntax.v b/coq/MetalSyntax.v new file mode 100644 index 0000000..f846e20 --- /dev/null +++ b/coq/MetalSyntax.v @@ -0,0 +1,432 @@ +Require Import List. +Require Import MyList. +Require Import MyTactics. +Require Export Autosubst.Autosubst. +Require Export AutosubstExtra. +Require Export Autosubst_EOS. +Require Export Autosubst_FreeVars. + +(* -------------------------------------------------------------------------- *) + +(* Metal is a lambda-calculus where every lambda-abstractions must be closed. + It is equipped with pairs and projections. It is intended to serve as the + target of closure conversion. *) + +(* -------------------------------------------------------------------------- *) + +(* Syntax. *) + +Inductive metal := +| MVar : var -> metal +| MLam : {bind metal} -> metal +| MApp : metal -> metal -> metal +| MLet : metal -> {bind metal} -> metal +| MPair : metal -> metal -> metal +| MPi : nat -> metal -> metal +. + +Instance Ids_metal : Ids metal. derive. Defined. +Instance Rename_metal : Rename metal. derive. Defined. +Instance Subst_metal : Subst metal. derive. Defined. +Instance SubstLemmas_metal : SubstLemmas metal. derive. Qed. + +Instance IdsLemmas_metal : IdsLemmas metal. +Proof. econstructor. intros. injections. eauto. Qed. + +(* -------------------------------------------------------------------------- *) + +(* We equip the calculus with pairs, instead of general tuples, because + this makes life in Coq simpler -- we would otherwise have an occurrence + of [list metal] in the definition of [metal], and would need to ask for + a custom induction scheme. *) + +(* Instead, we simulate tuples using nested pairs. This would of course be + inefficient in real life, but fur our purposes, should be fine. *) + +Fixpoint MTuple ts := + match ts with + | nil => + (* A dummy value serves as the tail. *) + MLam (MVar 0) + | t :: ts => + MPair t (MTuple ts) + end. + +Fixpoint MProj i t := + match i with + | 0 => + (* Take the left pair component. *) + MPi 0 t + | S i => + (* Take the right pair component and continue. *) + MProj i (MPi 1 t) + end. + +Definition MLetPair t u := + MLet (* x_0 = *) (MPi 0 t) ( + MLet (* x_1 = *) (MPi 1 (lift 1 t)) + u + ). + +(* -------------------------------------------------------------------------- *) + +(* [MMultiLet 0 ts u] is a sequence of [MLet] bindings. [n] variables, where + [n] is [length ts], are bound to the terms [ts] in the term [u]. The + recursive structure of the definition is such that the *last* term in the + list [ts] is bound *last*, hence is referred to as variable 0 in [u]. *) + +Fixpoint MMultiLet i ts u := + match ts with + | nil => + u + | t :: ts => + MLet (lift i t) (MMultiLet (i + 1) ts u) + end. + +(* The auxiliary parameter [i] in [MMultiLet i ts u] is required for the + recursive definition to go through, but, as far as the end user is + concerned, is not very useful. It can be eliminated as follows. *) + +Lemma MMultiLet_ij: + forall ts delta i j u, + i + delta = j -> + MMultiLet j ts u = MMultiLet i (map (fun t => lift delta t) ts) u. +Proof. + induction ts; intros; simpl; eauto. + f_equal. + { asimpl. congruence. } + { erewrite IHts. eauto. omega. } +Qed. + +Lemma MMultiLet_i0: + forall i ts u, + MMultiLet i ts u = MMultiLet 0 (map (fun t => lift i t) ts) u. +Proof. + eauto using MMultiLet_ij with omega. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* [MVars n] is the list [MVar 0, ..., MVar (n-1)]. *) + +Definition MVars (n : nat) : list metal := + map MVar (nats n). + +(* -------------------------------------------------------------------------- *) + +(* [MProjs n t] is the list [MProj n t, ..., MProj 1 t]. *) + +Definition MProjs (n : nat) (t : metal) := + map (fun i => MProj (i + 1) t) (rev_nats n). + +(* This list has length [n]. *) + +Lemma length_MProjs: + forall n t, + length (MProjs n t) = n. +Proof. + unfold MProjs. intros. rewrite map_length, length_rev_nats. eauto. +Qed. + +Hint Rewrite length_MProjs : length. + +(* -------------------------------------------------------------------------- *) + +(* Substitution (boilerplate lemmas). *) + +Lemma subst_MVar: + forall x sigma, + (MVar x).[sigma] = sigma x. +Proof. + intros. autosubst. +Qed. + +Lemma subst_MLam: + forall t sigma, + (MLam t).[sigma] = MLam t.[up sigma]. +Proof. + intros. autosubst. +Qed. + +Lemma subst_MApp: + forall t1 t2 sigma, + (MApp t1 t2).[sigma] = MApp t1.[sigma] t2.[sigma]. +Proof. + intros. autosubst. +Qed. + +Lemma subst_MLet: + forall t1 t2 sigma, + (MLet t1 t2).[sigma] = MLet t1.[sigma] t2.[up sigma]. +Proof. + intros. autosubst. +Qed. + +Lemma subst_MPair: + forall t1 t2 sigma, + (MPair t1 t2).[sigma] = MPair t1.[sigma] t2.[sigma]. +Proof. + intros. autosubst. +Qed. + +Lemma subst_MPi: + forall i t sigma, + (MPi i t).[sigma] = MPi i t.[sigma]. +Proof. + intros. autosubst. +Qed. + +Lemma subst_MTuple: + forall ts sigma, + (MTuple ts).[sigma] = MTuple (map (subst sigma) ts). +Proof. + induction ts; intros; asimpl; [ | rewrite IHts ]; reflexivity. +Qed. + +Lemma subst_MProj: + forall n t sigma, + (MProj n t).[sigma] = MProj n t.[sigma]. +Proof. + induction n; intros; asimpl; [ | rewrite IHn ]; autosubst. +Qed. + +Lemma subst_MLetPair: + forall t u sigma, + (MLetPair t u).[sigma] = MLetPair t.[sigma] u.[upn 2 sigma]. +Proof. + unfold MLetPair. intros. autosubst. +Qed. + +Lemma subst_MMultiLet: + forall ts i u sigma, + (MMultiLet i ts u).[upn i sigma] = + MMultiLet i (map (subst sigma) ts) u.[upn (length ts) (upn i sigma)]. +Proof. + induction ts; intros; asimpl; f_equal. + { erewrite plus_upn by tc. eauto. } + { rewrite IHts. + repeat erewrite upn_upn by tc. + do 3 f_equal. omega. } +Qed. + +Lemma subst_MMultiLet_0: + forall ts u sigma, + (MMultiLet 0 ts u).[sigma] = + MMultiLet 0 (map (subst sigma) ts) u.[upn (length ts) sigma]. +Proof. + intros. + change sigma with (upn 0 sigma) at 1. + eapply subst_MMultiLet. +Qed. + +Lemma subst_MVars: + forall n sigma, + map (subst sigma) (MVars n) = map sigma (nats n). +Proof. + intros. unfold MVars. rewrite map_map. reflexivity. +Qed. + +Lemma subst_MProjs: + forall n t sigma, + map (subst sigma) (MProjs n t) = MProjs n t.[sigma]. +Proof. + unfold MProjs. induction n; intros; simpl; eauto. + rewrite subst_MProj, IHn. eauto. +Qed. + +Hint Rewrite + subst_MVar subst_MLam subst_MApp subst_MLet subst_MPair subst_MPi + subst_MTuple subst_MProj subst_MLetPair + subst_MMultiLet subst_MMultiLet_0 + subst_MVars subst_MProjs +: subst. + +(* -------------------------------------------------------------------------- *) + +(* Free variables (boilerplate lemmas). *) + +Lemma fv_MVar: + forall x k, + fv k (MVar x) <-> x < k. +Proof. + eauto using fv_ids_eq. +Qed. + +Ltac prove_fv := + unfold fv; intros; asimpl; + split; intros; unpack; [ injections | f_equal ]; eauto. + +Lemma fv_MLam: + forall t k, + fv k (MLam t) <-> fv (k + 1) t. +Proof. + prove_fv. +Qed. + +Lemma fv_MApp: + forall t1 t2 k, + fv k (MApp t1 t2) <-> fv k t1 /\ fv k t2. +Proof. + prove_fv. +Qed. + +Lemma fv_MLet: + forall t1 t2 k, + fv k (MLet t1 t2) <-> fv k t1 /\ fv (k + 1) t2. +Proof. + prove_fv. +Qed. + +Lemma fv_MPair: + forall t1 t2 k, + fv k (MPair t1 t2) <-> fv k t1 /\ fv k t2. +Proof. + prove_fv. +Qed. + +Lemma fv_MPi: + forall i t k, + fv k (MPi i t) <-> fv k t. +Proof. + prove_fv. +Qed. + +Hint Rewrite fv_MVar fv_MLam fv_MApp fv_MLet fv_MPair fv_MPi : fv. + +Lemma fv_MTuple: + forall k ts, + fv k (MTuple ts) <-> Forall (fv k) ts. +Proof. + induction ts; simpl; intros; fv; split; intros; unpack. + { econstructor. } + { omega. } + { rewrite IHts in *. econstructor; eauto. } + { rewrite IHts in *. pick Forall invert. eauto. } +Qed. + +Lemma fv_MProj: + forall k n t, + fv k (MProj n t) <-> fv k t. +Proof. + induction n; intros; simpl; [ | rewrite IHn ]; fv; tauto. +Qed. + +Lemma fv_MLetPair: + forall k t u, + fv k (MLetPair t u) <-> fv k t /\ fv (k + 2) u. +Proof. + intros. unfold MLetPair. fv; tc. + replace (k + 1 + 1) with (k + 2) by omega. + tauto. +Qed. + +Local Lemma MLet_inj: + forall t1 u1 t2 u2, + MLet t1 u1 = MLet t2 u2 <-> t1 = t2 /\ u1 = u2. +Proof. + split; intros; injections; unpack; subst; eauto. +Qed. + +Local Lemma cons_cons_eq: + forall {A} (x1 x2 : A) xs1 xs2, + x1 :: xs1 = x2 :: xs2 <-> + x1 = x2 /\ xs1 = xs2. +Proof. + split; intros; injections; unpack; subst; eauto. +Qed. + +Local Lemma MMultiLet_inj: + forall ts1 u1 ts2 u2 i, + length ts1 = length ts2 -> + MMultiLet i ts1 u1 = MMultiLet i ts2 u2 <-> + ts1 = ts2 /\ u1 = u2. +Proof. + induction ts1; destruct ts2; intros; simpl; + try solve [ false; simpl in *; omega ]. + { tauto. } + { rewrite MLet_inj. + rewrite lift_injn_eq. + rewrite IHts1 by (simpl in *; omega). + rewrite cons_cons_eq. + tauto. } +Qed. + +Local Lemma map_inj: + forall {A} (f : A -> A) xs, + map f xs = xs <-> + Forall (fun x => f x = x) xs. +Proof. + induction xs; simpl; intros; split; intros; eauto using Forall; + injections. + { econstructor; tauto. } + { pick Forall invert. f_equal; tauto. } +Qed. + +Lemma fv_MMultiLet_0: + forall ts u k, + fv k (MMultiLet 0 ts u) <-> + Forall (fv k) ts /\ fv (k + length ts) u. +Proof. + intros. unfold fv. + autorewrite with subst. + rewrite MMultiLet_inj by eauto using map_length. + rewrite upn_upn, Nat.add_comm. + rewrite map_inj. + tauto. +Qed. + +Lemma fv_MVars: + forall n, + Forall (fv n) (MVars n) <-> + True. +Proof. + split; [ eauto | intros _ ]. + unfold MVars, nats. + eapply Forall_map. + eapply Forall_seq; intros. + fv. omega. +Qed. + +Lemma fv_MProjs: + forall k n t, + fv k t -> (* not an equivalence, as [k] could be zero *) + Forall (fv k) (MProjs n t). +Proof. + unfold MProjs. induction n; simpl; intros; + econstructor; [ rewrite fv_MProj |]; eauto. +Qed. + +Hint Rewrite + fv_MTuple fv_MProj fv_MLetPair fv_MMultiLet_0 + fv_MVars + (* not fv_MProjs *) +: fv. + +(* -------------------------------------------------------------------------- *) + +(* The following lemma is analogous to [fv_unaffected_regular], except it does + not have a regularity hypothesis, which makes it more pleasant to use. It is + proved by induction on terms, which is why we are unable to prove it in a + generic setting. *) + +Lemma fv_unaffected: + forall t k sigma, + fv k t -> + t.[upn k sigma] = t. +Proof. + induction t; intros; fv; unpack; asimpl; repeat rewrite Nat.add_1_r in *; + try solve [ eauto using upn_k_sigma_x with typeclass_instances + | f_equal; eauto ]. +Qed. + +(* A corollary. *) + +Lemma closed_unaffected: + forall t sigma, + closed t -> + t.[sigma] = t. +Proof. + unfold closed. intros. + rewrite <- (upn0 sigma). + eauto using fv_unaffected. +Qed. diff --git a/coq/MyList.v b/coq/MyList.v new file mode 100644 index 0000000..f56127a --- /dev/null +++ b/coq/MyList.v @@ -0,0 +1,125 @@ +Require Import List. +Require Import MyTactics. + +(* A few random additions to the [List] module, which is woefully incomplete. *) + +(* -------------------------------------------------------------------------- *) + +Lemma rev_cons_app: + forall {A} (x : A) xs ys, + rev (x :: xs) ++ ys = rev xs ++ x :: ys. +Proof. + intros. simpl. rewrite <- app_assoc. reflexivity. +Qed. + +(* -------------------------------------------------------------------------- *) + +Lemma length_nil: + forall A, + length (@nil A) = 0. +Proof. + reflexivity. +Qed. + +Lemma length_cons: + forall A (x : A) xs, + length (x :: xs) = 1 + length xs. +Proof. + reflexivity. +Qed. + +Hint Rewrite length_nil length_cons app_length map_length : length. + +Ltac length := + autorewrite with length in *; + try omega. + +(* -------------------------------------------------------------------------- *) + +(* We have [app_nth1] and [app_nth2], but the following lemma, which can be + viewed as a special case of [app_nth2], is missing. *) + +Lemma app_nth: + forall {A} (xs ys : list A) x n, + n = length xs -> + nth n (xs ++ ys) x = nth 0 ys x. +Proof. + intros. + rewrite app_nth2 by omega. + replace (n - length xs) with 0 by omega. + reflexivity. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* [rev_nats n] is the semi-open interval (n, 0], counted down. *) + +(* It could also be defined as [rev (seq 0 n)], but a direct definition + is easier to work with, as it is immediately amenable to proofs by + induction. *) + +Fixpoint rev_nats (n : nat) : list nat := + match n with + | 0 => + nil + | S n => + n :: rev_nats n + end. + +(* [nats n] is the semi-open interval [0, n), counted up. *) + +Definition nats (n : nat) : list nat := + seq 0 n. + +(* These sequences have length [n]. *) + +Lemma length_rev_nats: + forall n, + length (rev_nats n) = n. +Proof. + induction n; intros; simpl; [| rewrite IHn ]; eauto. +Qed. + +Lemma length_nats: + forall n, + length (nats n) = n. +Proof. + unfold nats. intros. eauto using seq_length. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* A few basic lemmas about [Forall]. *) + +Lemma Forall_map: + forall A B (f : A -> B) (P : B -> Prop) xs, + Forall (fun x => P (f x)) xs -> + Forall P (map f xs). +Proof. + induction 1; intros; subst; simpl; econstructor; eauto. +Qed. + +Lemma Forall_app: + forall A (P : A -> Prop) xs ys, + Forall P xs -> + Forall P ys -> + Forall P (xs ++ ys). +Proof. + induction 1; intros; subst; simpl; eauto. +Qed. + +Lemma Forall_rev: + forall A (P : A -> Prop) xs, + Forall P xs -> + Forall P (rev xs). +Proof. + induction 1; intros; subst; simpl; eauto using Forall_app. +Qed. + +Lemma Forall_seq: + forall (P : nat -> Prop) len start, + (forall i, start <= i < start + len -> P i) -> + Forall P (seq start len). +Proof. + induction len; intros; simpl; econstructor; eauto with omega. +Qed. 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([slides 02](slides/fpottier-02.pdf)) ([OCaml code](ocaml/Lambda.ml)) ([Coq repo](coq/)). -* (06/10/2017) Compiling away first-class functions: closure conversion, defunctionalization. +* (06/10/2017) Compiling away first-class functions: + closure conversion, defunctionalization + ([slides 03](slides/fpottier-03.pdf)) + ([Coq repo](coq/)). * (13/10/2017) Compiling away the call stack: the CPS transformation. * (20/10/2017) Equational reasoning and program optimizations. 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zHQvtAIPZ*#Ijq}KAY2Fk8ayROvS0Skax*F+UOfDn{I<1ty#le@C z+l>n-=XEdLO7yGweIQZJUE0Xzr?T8!N{H7EJX$ov@3iVhzh3flMK??q!Q9f;BfAV7edARQ|wSM`!4H2^wRDw(M iv^%i=3r@~7ZSrp`F2!yd9tV*d#DPdnEvY1h`2PTquf(7L From 79af3f8d1ca6c475f340d13c26a58782e7131b39 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Wed, 11 Oct 2017 15:27:50 +0200 Subject: [PATCH 46/81] Removed MetaBigStep.v -- a typo in a file name. --- coq/MetaBigStep.v | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 coq/MetaBigStep.v diff --git a/coq/MetaBigStep.v b/coq/MetaBigStep.v deleted file mode 100644 index e69de29..0000000 From 555822838f0267bdf7da1a7c4650266906544e84 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Wed, 11 Oct 2017 15:28:20 +0200 Subject: [PATCH 47/81] Add the Coq formalization of the CPS transformation. --- coq/CPSContextSubstitution.v | 77 +++ coq/CPSCorrectness.v | 138 ++++++ coq/CPSCounterExample.v | 107 ++++ coq/CPSDefinition.v | 446 +++++++++++++++++ coq/CPSIndifference.v | 262 ++++++++++ coq/CPSKubstitution.v | 120 +++++ coq/CPSRenaming.v | 92 ++++ coq/CPSSimulation.v | 177 +++++++ coq/CPSSimulationWithoutLet.v | 137 ++++++ coq/CPSSpecialCases.v | 48 ++ coq/CPSSubstitution.v | 149 ++++++ coq/FixExtra.v | 20 + coq/LambdaCalculusParallelReduction.v | 174 +++++++ coq/LambdaCalculusStandardization.v | 679 ++++++++++++++++++++++++++ coq/Relations.v | 498 +++++++++++++++++++ 15 files changed, 3124 insertions(+) create mode 100644 coq/CPSContextSubstitution.v create mode 100644 coq/CPSCorrectness.v create mode 100644 coq/CPSCounterExample.v create mode 100644 coq/CPSDefinition.v create mode 100644 coq/CPSIndifference.v create mode 100644 coq/CPSKubstitution.v create mode 100644 coq/CPSRenaming.v create mode 100644 coq/CPSSimulation.v create mode 100644 coq/CPSSimulationWithoutLet.v create mode 100644 coq/CPSSpecialCases.v create mode 100644 coq/CPSSubstitution.v create mode 100644 coq/FixExtra.v create mode 100644 coq/LambdaCalculusParallelReduction.v create mode 100644 coq/LambdaCalculusStandardization.v create mode 100644 coq/Relations.v diff --git a/coq/CPSContextSubstitution.v b/coq/CPSContextSubstitution.v new file mode 100644 index 0000000..11864a8 --- /dev/null +++ b/coq/CPSContextSubstitution.v @@ -0,0 +1,77 @@ +Require Import MyTactics. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusValues. +Require Import CPSDefinition. + +(* This file contains a few lemmas about [substc]. *) + +(* Two successive applications of [substc] can be fused. *) + +Lemma substc_substc: + forall sigma1 sigma2 c, + substc sigma2 (substc sigma1 c) = substc (sigma1 >> sigma2) c. +Proof. + intros. destruct c; autosubst. +Qed. + +(* Two successive applications of [liftc] can be fused. *) + +Lemma liftc_liftc: + forall i j c, + liftc i (liftc j c) = liftc (i + j) c. +Proof. + intros i j c. destruct c; autosubst. +Qed. + +(* [apply] commutes with substitutions. *) + +Lemma apply_substitution: + forall c sigma c' v, + substc sigma c = c' -> + (apply c v).[sigma] = apply c' v.[sigma]. +Proof. + intros. subst. destruct c; autosubst. +Qed. + +(* [reify] commutes with substitutions. *) + +Lemma reify_substitution: + forall c sigma c', + substc sigma c = c' -> + (reify c).[sigma] = reify c'. +Proof. + intros. subst. destruct c; reflexivity. +Qed. + +(* As a special case, [reify] commutes with lifting. *) + +Lemma lift_reify: + forall i c, + lift i (reify c) = reify (liftc i c). +Proof. + intros. destruct c; reflexivity. +Qed. + +(* [substc] is preserved by [liftc]. *) + +Lemma substc_liftc_liftc: + forall i c sigma c', + substc sigma c = c' -> + substc (upn i sigma) (liftc i c) = liftc i c'. +Proof. + intros. subst. destruct c; simpl. + { rewrite lift_upn by tc. reflexivity. } + { asimpl. erewrite plus_upn by tc. autosubst. } +Qed. + +Hint Resolve substc_liftc_liftc : obvious. + +(* As is the case for terms, lifting [c] by 1, then applying a substitution + of the form [v .: ids], yields [c] again. *) + +Lemma substc_liftc_single: + forall c v, + substc (v .: ids) (liftc 1 c) = c. +Proof. + intros. destruct c; autosubst. +Qed. diff --git a/coq/CPSCorrectness.v b/coq/CPSCorrectness.v new file mode 100644 index 0000000..a8bedb7 --- /dev/null +++ b/coq/CPSCorrectness.v @@ -0,0 +1,138 @@ +Require Import MyTactics. +Require Import Sequences. +Require Import Relations. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusValues. +Require Import LambdaCalculusReduction. +Require Import LambdaCalculusStandardization. +Require Import CPSDefinition. +Require Import CPSSpecialCases. +Require Import CPSSimulation. + +(* [cbv+ . pcbv] implies [pcbv*]. *) + +Lemma technical_inclusion_0: + inclusion plus_cbv_pcbv (star pcbv). +Proof. + intros t1 t2. unfold composition. intros. unpack. + eauto 6 using cbv_subset_pcbv, plus_covariant with sequences. +Qed. + +(* [(cbv+ . pcbv)*] implies [pcbv*]. *) + +Lemma technical_inclusion_1: + inclusion (star plus_cbv_pcbv) (star pcbv). +Proof. + eapply inclusion_transitive; [| eapply inclusion_star_star ]. + eapply star_covariant_inclusion. + eapply technical_inclusion_0. +Qed. + +(* A simplified simulation diagram. *) + +Lemma simulation_cbv_pcbv: + forall t t', + star cbv t t' -> + star pcbv (cps t init) (cps t' init). +Proof. + intros t t' Hred. + (* According to the simulation diagram (iterated), [cps t c] reduces to + [cps v c] via a series of [cbv] and [pcbv] steps. *) + destruct (star_diamond_left _ _ _ cps_init_simulation _ _ Hred _ eq_refl) + as (?&?&?). subst. + (* Thus, [cps t c] reduces to [cps t' c] via [pcbv*]. *) + eapply technical_inclusion_1. eauto. +Qed. + +(* If [t] diverges, then [cps t init] diverges, too. *) + +Lemma cps_preserves_divergence: + forall t, + infseq cbv t -> + infseq cbv (cps t init). +Proof. + intros. + eapply pcbv_preserves_divergence. + eapply infseq_simulation. + { eapply cps_init_simulation. } + { eauto. } + { tauto. } +Qed. + +(* If [t] converges to a value [v], then [cps t init] converges to a value [w]. + Furthermore, [w] reduces to [cpsv v] via a number of parallel reduction + steps. *) + +Lemma cps_preserves_convergence: + forall t v, + star cbv t v -> + is_value v -> + exists w, + star cbv (cps t init) w /\ + is_value w /\ + star pcbv w (cpsv v). +Proof. + intros ? ? Htv Hv. + (* [cps t init] reduces to [cps v init] via [pcbv*]. *) + generalize (simulation_cbv_pcbv _ _ Htv); intro Hred. + (* [cps v init] is [cpsv v]. *) + assert (Heq: cps v init = cpsv v). + { cps. reflexivity. } + (* Thus, [cps t init] reduces to [cpsv v] via [pcbv*]. *) + rewrite Heq in Hred. + (* Bifurcate this reduction sequence. *) + forward1 crarys_lemma9. clear Hred. + (* This gives us the value [w] that we are looking for. *) + eexists. split. eauto. split. + { eauto using + (star_implication_reversed _ ipcbv_preserves_values_reversed) + with obvious. } + { eauto using star_covariant, ipcbv_subset_pcbv. } +Qed. + +(* If [t] is stuck, then [cps t c] is stuck. Not a really interesting + property, but we prove it, just so that no stone is left unturned. *) + +Lemma cps_preserves_stuck: + forall t, + stuck t -> + forall c, + stuck (cps t c). +Proof. + induction 1; intros. + (* StuckApp *) + { rewrite cps_app_value_value by eauto. + eapply StuckAppL. + eapply StuckApp; [ obvious | obvious |]. + (* Only [Lam] is translated to [Lam]. *) + intros. destruct v1. + { cpsv. congruence. } + { cpsv. false. congruence. } + { obvious. } + { obvious. } + } + (* StuckAppL *) + { cps. eauto. } + (* StuckAppR *) + { rewrite cps_app_value by eauto. eauto. } + (* StuckLetL *) + { cps. eauto. } +Qed. + +(* As a corollary, the property of going wrong is preserved by the CPS + transformation. *) + +Lemma cps_preserves_going_wrong: + forall t, + goes_wrong t -> + goes_wrong (cps t init). +Proof. + intros ? [ t' [ Htt' ? ]]. + (* [cps t init] reduces to [cps t' init] via [pcbv*]. *) + generalize (simulation_cbv_pcbv _ _ Htt'); intro Hred. + (* Bifurcate this reduction sequence. *) + forward1 crarys_lemma9. clear Hred. + (* This gives us the stuck term we are looking for. *) + eexists. split. eauto. + eauto using cps_preserves_stuck, reverse_star_ipcbv_preserves_stuck. +Qed. diff --git a/coq/CPSCounterExample.v b/coq/CPSCounterExample.v new file mode 100644 index 0000000..5db8181 --- /dev/null +++ b/coq/CPSCounterExample.v @@ -0,0 +1,107 @@ +Require Import MyTactics. +Require Import Sequences. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusValues. +Require Import LambdaCalculusReduction. +Require Import CPSDefinition. + +(* The single-step simulation lemma in Danvy and Filinski's paper states that + if [t1] reduces to [t2], then [cps t1 c] reduces (in one or more steps) to + [cps t2 c]. Although this lemma is true in the pure lambda calculus, it + fails when the calculus is extended with [Let]. This file provides two + counter-examples. *) + +(* Although Danvy and Filinski's paper does not claim that this lemma holds + when the calculus is extended with [Let], it does not indicate that the + lemma fails, either. *) + +(* -------------------------------------------------------------------------- *) + +(* The tactic [analyze] assumes that there is a hypothesis [star cbv t1 t2]. + It checks that [t1] and [t2] are distinct and, if [t1] reduces to [t'1], + updates this hypothesis to [star cbv t'1 t2]. Repeating this tactic allows + proving that [t1] does *not* reduce to [t2]. *) + +Ltac analyze := + invert_star_cbv; repeat invert_cbv; compute in *; fold cbv_mask in *; + repeat match goal with h: True |- _ => clear h end. + +Transparent cps cpsv. (* required by [compute] *) + +(* -------------------------------------------------------------------------- *) + +(* Consider the term [t1], defined as follows. In informal syntax, [t1] + is written (\z.let w = z in w) (\x.x). *) + +Definition t1 := + App (Lam (Let (Var 0) (Var 0))) (Lam (Var 0)). + +(* The term [t1] reduces to [t2], which in informal syntax is written + let w = \x.x in w. *) + +Definition t2 := + Let (Lam (Var 0)) (Var 0). + +Goal + cbv t1 t2. +Proof. + unfold t1, t2. obvious. +Qed. + +(* The single-step simulation diagram is violated: [cps t1 init] does + *not* reduce (in any number of steps) to [cps t2 init]. *) + +Goal + ~ (star cbv (cps t1 init) (cps t2 init)). +Proof. + compute; fold cbv_mask. intro. + analyze. + analyze. + (* This point is the near miss: + [cps t1 init] has now reduced to a [Let] construct, whereas + [cps t2 init] is a similar-looking [Let] construct. + Both have the same value on the left-hand side of the [Let]. + But the right-hand sides of the [Let] construct differ. *) + analyze. + analyze. + analyze. +Qed. + +(* Let us summarize. + + The term [t1] reduces in one step to [t2] as follows: + + (\z.let w = z in w) (\x.x) + -> + let w = \x.x in w + + The term [cps t1 init], in informal notation, is as follows: + + (\z.\k.let w = z in k w) + (\x.\k. k x) + (\w.w) + + This term reduces in two steps to: + + let w = \x.\k. k x in + (\w.w) w + + But the term [cps t2 init], in informal notation, is: + + let w = \x.\k. k x in + w + + This is our near miss. Both terms are [let] constructs and both have + the same left-hand side, but the right-hand sides differ by a beta-v + reduction. Thus, [cps t1 init] does not reduce *in call-by-value* to + [cps t2 init]. In order to allow [cps u1 init] to join [cps u2 init], + we must allow beta-v reductions in the right-hand side of [let] + constructs (and, it turns out, under lambda-abstractions, too.) + This is visible in the proof of the [simulation] lemma in the file + CPSSimulation: there, we use the reduction strategy [pcbv], which + allows (parallel) beta-v reductions under arbitrary contexts. *) + +(* This counter-example is one of two closed counter-examples of minimal size. + It has size 4 (counting [Lam], [App], and [Let] nodes) and involves only + one [Let] construct. There are no smaller counter-examples. An exhaustive + search procedure, coded in OCaml, was used to find it. *) diff --git a/coq/CPSDefinition.v b/coq/CPSDefinition.v new file mode 100644 index 0000000..4757cd1 --- /dev/null +++ b/coq/CPSDefinition.v @@ -0,0 +1,446 @@ +Require Import MyTactics. +Require Import FixExtra. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusValues. + +(* This is a definition of the CPS transformation. *) + +(* This CPS transformation is "one-pass" in the sense that it does not produce + any administrative redexes. (In other words, there is no need for a second + pass, whose purpose would be to remove administrative redexes.) + + To achieve this, instead of defining [cps t k], where the continuation [k] + is a term, we define [cps t c], where the continuation [c] is either a term + (also known as an object-level continuation) or a term-with-a-hole [K] + (also known as a meta-level continuation). + + This formulation of the CPS transformation is analogous to Danvy and + Filinski's higher-order formulation. Yet, it is still technically + first-order, because we represent a term-with-a-hole [K] as a term, + where the variable 0 denotes the hole. *) + +(* This CPS transformation is defined by recursion on the size of terms. This + allows recursive calls of the form [cps (lift 1 t)], which would be illegal + if [cps] was defined by structural induction. Definitions by well-founded + recursion in Coq are somewhat tricky, requiring the use of the fixed point + combinator [Fix] and the tactic [refine]. For explanations, see the chapter + on general recursion in Chlipala's book at + http://adam.chlipala.net/cpdt/html/GeneralRec.html *) + +(* The situation could be complicated by the fact that we wish to define two + functions simultaneously: + + [cpsv v] is the translation of a value [v]. + + [cps t c] is the translation of a term [t] with continuation [c]. + + To avoid this problem, we follow Danvy and Filinski's approach, which is to + first define [cps] directly (as this does not cause much duplication), then + define [cpsv] in terms of [cps]. In the latter step, no case analysis is + required: [cpsv] is obtained by invoking [cps] with an identity meta-level + continuation. + + Regardless of how [cps] and [cpsv] are defined, we prove that the they + satisfy the desired recurrence equations, so, in the end, everything is + just as if they had been defined in a mutually recursive manner. *) + +(* -------------------------------------------------------------------------- *) + +(* As explained above, a continuation [c] is + + either [O k], where [k] is a term (in fact, a value) + (an object-level continuation) + + or [M K], where [K] is term-with-a-hole + (a meta-level continuation). + + A term-with-a-hole [K] is represented as a term where the variable 0 denotes + the hole (and, of course, all other variables are shifted up). *) + +Inductive continuation := +| O (k : term) +| M (K : term). + +(* The term [apply c v] is the application of the continuation [c] to the + value [v]. If [c] is an object-level continuation [k] (that is, a term), + then it is just the object-level application [App k v]. If [c] is a + meta-level continuation [K], then it is the meta-level operation of filling + the hole with the value [v], which in fact is just a substitution, + [K.[v/]]. *) + +Definition apply (c : continuation) (v : term) : term := + match c with + | O k => + App k v + | M K => + K.[v/] + end. + +(* The term [reify c] is the reification of the continuation [c] as an + object-level continuation (that is, a term). If [c] is an object-level + continuation [k], then it is just [k]. If [c] is a meta-level continuation + [K], then [reify c] is the term [\x.K x]. (This is usually known as a + two-level eta-expansion.) Because the hole is already represented by the + variable 0, filling the hole with the variable [x] is a no-op. Therefore, + it suffices to write [Lam K] to obtain the desired lambda-abstraction. *) + +Definition reify (c : continuation) : term := + match c with + | O k => + k + | M K => + Lam K + end. + +(* The continuation [substc sigma c] is the result of applying the + substitution [sigma] to the continuation [c]. *) + +Definition substc sigma (c : continuation) : continuation := + match c with + | O k => + O k.[sigma] + | M K => + M K.[up sigma] + end. + +(* [liftc i c] is the result of lifting the free names of the continuation [c] + up by [i]. The function [liftc] can be defined in terms of [substc]. *) + +Notation liftc i c := + (substc (ren (+i)) c). + +(* -------------------------------------------------------------------------- *) + +(* Here is the definition of [cps]. Because we must keep track of sizes and + prove that the recursive calls cause a decrease in the size, this + definition cannot be easily written as Coq terms. Instead, we switch to + proof mode and use the tactic [refine]. This allows us to write some of the + code, with holes [_] in it, and use proof mode to fill the holes. *) + +(* [cps t c] is the CPS-translation of the term [t] with continuation [c]. *) + +Definition cps : term -> continuation -> term. +Proof. + (* The definition is by well-founded recursion on the size of [t]. *) + refine (Fix smaller_wf_transparent (fun _ => continuation -> term) _). + (* We receive the arguments [t] and [c] as well as a function [cps_] + which we use for recursive calls. At every call to [cps_], there + is an obligation to prove that the size of the argument is less + than the size of [t]. *) + intros t cps_ c. + (* The definition is by cases on the term [t]. *) + destruct t as [ x | t | t1 t2 | t1 t2 ]. + (* When [t] is a value, we return an application of the continuation [c] + to a value which will later be known as [cpsv t]. *) + (* Case: [Var x]. *) + { refine (apply c (Var x)). } + (* Case: [Lam t]. *) + (* In informal notation, the term [\x.t] is transformed to an application of + [c] to [\x.\k.[cps t k]], where [k] is a fresh variable. Here, [k] is + represented by the de Bruijn index 0, and the term [t] must be lifted + because it is brought into the scope of [k]. *) + { refine (apply c + (Lam (* x *) (Lam (* k *) (cps_ (lift 1 t) _ (O (Var 0))))) + ); abstract size. } + (* Case: [App t1 t2]. *) + (* The idea is, roughly, to first obtain the value [v1] of [t1], then obtain + the value [v2] of [t2], then perform the application [v1 v2 c]. + + Two successive calls to [cps] are used to obtain [v1] and [v2]. In each + case, we avoid administrative redexes by using an [M] continuation. + Thus, [v1] and [v2] are represented by two holes, denoted by the + variables [Var 1] and [Var 0]. + + If [t1] is a value, then the first hole will be filled directly with (the + translation of) [t1]; otherwise, it will be filled with a fresh variable, + bound to the result of evaluating (the translation of) [t1]. The same + goes for [t2]. + + The application [v1 v2 c] does not directly make sense, since [c] is a + continuation, not a term. Instead of [c], we must use [reify c]. The + continuation [c] must turned into a term, so it can be used in this + term-level application. + + A little de Bruijn wizardry is involved. The term [t2] must be lifted up + by 1 because it is brought into the scope of the first meta-level + continuation. Similarly, the first hole must be lifted by 1 because it is + brought into the scope of the second meta-level continuation: thus, it + becomes Var 1. Finally, the continuation [c] must be lifted up by [2] + because it is brought into the scope of both. Here, we have a choice + between [reify (liftc _ c)] and [lift _ (reify c)], which mean the same + thing. *) + { refine ( + cps_ t1 _ (M ( + cps_ (lift 1 t2) _ (M ( + App (App (Var 1) (Var 0)) (lift 2 (reify c)) + )) + )) + ); + abstract size. + } + (* Case: [Let x = t1 in t2]. *) + (* The idea is to first obtain the value [v1] of [t1]. This value is + represented by the hole [Var 0] in the [M] continuation. We bind + this value via a [Let] construct to the variable [x] (represented by the + index 0 in [t2]), then execute [t2], under the original continuation [c]. + All variables in [t2] except [x] must lifted up by one because they are + brought in the scope of the meta-level continuation. The continuation [c] + must be lifted up by 2 because it is brought in the scope of the + meta-level continuation and in the scope of the [Let] construct. *) + { refine ( + cps_ t1 _ (M ( + Let (Var 0) ( + cps_ t2.[up (ren (+1))] _ (liftc 2 c) + ) + )) + ); + abstract size. + } +Defined. + +(* -------------------------------------------------------------------------- *) + +(* The above definition can be used inside Coq to compute the image of a term + through the transformation. For instance, the image under [cps] of [\x.x] + with object-level continuation [k] (a variable) is [k (\x.\k.k x)]. *) + +Goal + cps (Lam (Var 0)) (O (Var 0)) = + App (Var 0) (Lam (Lam (App (Var 0) (Var 1)))). +Proof. + compute. (* optional *) + reflexivity. +Qed. + +(* The image of [(\x.x) y] with continuation [k] is [(\x.\k.k x) y k]. *) + +Goal + cps (App (Lam (Var 0)) (Var 0)) (O (Var 1)) = + App (App (Lam (Lam (App (Var 0) (Var 1)))) (Var 0)) (Var 1). +Proof. + compute. (* optional *) + reflexivity. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* The initial continuation is used when invoking [cps] at the top level. *) + +(* We could use [O (Lam (Var 0))] -- that is, the identity function -- as + the initial continuation. Instead, we use [M (Var 0)], that is, the + empty context. This sometimes saves one beta-redex. *) + +Definition init := + M (Var 0). + +Definition cpsinit t := + cps t init. + +(* -------------------------------------------------------------------------- *) + +(* Now that [cps] is defined, we can define [cpsv] in terms of it. *) + +(* We explicitly check whether [v] is a value, and if it is not, we return a + dummy closed value. [cpsv] is supposed to be applied only to values, + anyway. Using a dummy value allows us to prove that [cpsv v] is always a + value, without requiring that [v] itself be a value. *) + +Definition cpsv (v : term) := + if_value v + (cpsinit v) + (Lam (Var 0)). + +(* -------------------------------------------------------------------------- *) + +(* The function [cps] satisfies the expected recurrence equations. *) + +(* The lemmas [cps_var] and [cps_lam] are not used outside this file, as we + use [cps_value] instead, followed with [cpsv_var] or [cpsv_lam]. *) + +Lemma cps_var: + forall x c, + cps (Var x) c = apply c (Var x). +Proof. + reflexivity. +Qed. + +Lemma cps_lam: + forall t c, + cps (Lam t) c = + apply c (Lam (* x *) (Lam (* k *) (cps (lift 1 t) (O (Var 0))))). +Proof. + intros. erewrite Fix_eq_simplified with (f := cps) by reflexivity. + reflexivity. +Qed. + +Lemma cps_app: + forall t1 t2 c, + cps (App t1 t2) c = + cps t1 (M ( + cps (lift 1 t2) (M ( + App (App (Var 1) (Var 0)) (lift 2 (reify c)) + )) + )). +Proof. + intros. erewrite Fix_eq_simplified with (f := cps) by reflexivity. + reflexivity. +Qed. + +Lemma cps_let: + forall t1 t2 c, + cps (Let t1 t2) c = + cps t1 (M ( + Let (Var 0) ( + cps t2.[up (ren (+1))] (liftc 2 c) + ) + )). +Proof. + intros. erewrite Fix_eq_simplified with (f := cps) by reflexivity. + reflexivity. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* The translation of values is uniform: we also have the following equation. *) + +Lemma cps_value: + forall v c, + is_value v -> + cps v c = apply c (cpsv v). +Proof. + destruct v; simpl; intros; try not_a_value; unfold cpsv, cpsinit. + { rewrite !cps_var. reflexivity. } + { rewrite !cps_lam. reflexivity. } +Qed. + +(* -------------------------------------------------------------------------- *) + +(* The function [cpsv] satisfies the expected recurrence equations. *) + +Lemma cpsv_var: + forall x, + cpsv (Var x) = Var x. +Proof. + reflexivity. +Qed. + +Lemma cpsv_lam: + forall t, + cpsv (Lam t) = Lam (Lam (cps (lift 1 t) (O (Var 0)))). +Proof. + intros. unfold cpsv, cpsinit. rewrite cps_lam. reflexivity. +Qed. + +(* If [theta] is a renaming, then [theta x] is a variable, so [cpsv (theta x)] + is [theta x], which can also be written [(Var x).[theta]]. *) + +Lemma cpsv_var_theta: + forall theta x, + is_ren theta -> + cpsv (theta x) = (Var x).[theta]. +Proof. + inversion 1. subst. asimpl. reflexivity. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* The tactic [cps] applies the rewriting rules [cps_value] and [cps_app] as + many times as possible, therefore expanding applications of the function + [cps] to values and to applications. *) + +Ltac cps := + repeat first [ rewrite cps_value by obvious + | rewrite cps_app | rewrite cps_let ]. + +(* -------------------------------------------------------------------------- *) + +(* The translation of a value is a value. *) + +(* In fact, thanks to the manner in which we have defined [cpsv], the image of + every term through [cpsv] is a value. This turns out to be quite pleasant, + as it allows removing nasty side conditions in several lemmas. *) + +Lemma is_value_cpsv: + forall v, + (* is_value v -> *) + is_value (cpsv v). +Proof. + intros. destruct v; simpl; tauto. +Qed. + +Hint Resolve is_value_cpsv : is_value obvious. + +Hint Rewrite cpsv_var cpsv_lam : cpsv. +Ltac cpsv := autorewrite with cpsv. + +(* -------------------------------------------------------------------------- *) + +(* As a final step, we prove an induction principle that helps work with the + functions [cpsv] and [cps]. When trying to establish a property of the + function [cps], we often need to prove this property by induction on the + size of terms. Furthermore, we usually need to state an auxiliary property + of the function [cpsv] and to prove the two statements simultaneously, by + induction on the size of terms. The following lemma is tailored for this + purpose. It proves the properties [Pcpsv] and [Pcps] simultaneously. The + manner in which the index [n] is used reflects precisely the manner in + which each function depends on the other, with or without a decrease in + [n]. The dependencies are as follows: + + [cpsv] invokes [cps] with a size decrease. + + [cps] invokes [cpsv] without a size decrease and + [cps] with a size decrease. + + It is worth noting that this proof has nothing to do with the definitions + of [cpsv] and [cps]. It happens to reflect just the right dependencies + between them. *) + +Lemma mutual_induction: + forall + (Pcpsv : term -> Prop) + (Pcps : term -> continuation -> Prop), + (forall n, + (forall t c, size t < n -> Pcps t c) -> + (forall v, size v < S n -> Pcpsv v) + ) -> + (forall n, + (forall v, size v < S n -> Pcpsv v) -> + (forall t c, size t < n -> Pcps t c) -> + (forall t c, size t < S n -> Pcps t c) + ) -> + ( + (forall v, Pcpsv v) /\ + (forall t c, Pcps t c) + ). +Proof. + intros Pcpsv Pcps IHcpsv IHcps. + assert (fact: + forall n, + (forall v, size v < n -> Pcpsv v) /\ + (forall t k, size t < n -> Pcps t k) + ). + { induction n; intros; split; intros; + try solve [ elimtype False; omega ]; + destruct IHn as (?&?); eauto. } + split; intros; eapply fact; eauto. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* In the proofs that follow, we never expand the definition of [cpsv] or + [cps]: we use the tactics [cpsv] and [cps] instead. We actually forbid + unfolding [cpsv] and [cps], so our proofs cannot depend on the details of + the above definitions. + + In general, when working with complex objects, it is good practice anyway + to characterize an object and forget how it was defined. Here, the + functions [cpsv] and [cps] are characterized by the equations that they + satisfy; the rest does not matter. + + Attempting to work with transparent [cpsv] and [cps] would be problematic + for several reasons. The tactics [simpl] and [asimpl] would sometimes + expand these functions too far. Furthermore, because we have used the term + [smaller_wf_transparent] inside the definition of [cps], expanding [cps] + definition would often give rise to uncontrollably large terms. *) + +Global Opaque cps cpsv. diff --git a/coq/CPSIndifference.v b/coq/CPSIndifference.v new file mode 100644 index 0000000..28d1d69 --- /dev/null +++ b/coq/CPSIndifference.v @@ -0,0 +1,262 @@ +Require Import MyTactics. +Require Import Sequences. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusValues. +Require Import LambdaCalculusReduction. +Require Import CPSDefinition. + +(* In a CPS term (i.e., a term produced by the CPS translation), the + right-hand side of every application is a value, and the left-hand + side of every [let] construct is a value. *) + +Inductive is_cps : term -> Prop := +| IsCPSVar: + forall x, + is_cps (Var x) +| IsCPSLam: + forall t, + is_cps t -> + is_cps (Lam t) +| IsCPSApp: + forall t1 t2, + is_cps t1 -> + is_cps t2 -> + is_value t2 -> + is_cps (App t1 t2) +| IsCPSLet: + forall t1 t2, + is_cps t1 -> + is_cps t2 -> + is_value t1 -> + is_cps (Let t1 t2) +. + +(* To prove that the above invariant holds, we must also define what it means + for a continuation [c] to satisfy this invariant. *) + +Inductive is_cps_continuation : continuation -> Prop := +| IsCPSO: + forall k, + is_value k -> + is_cps k -> + is_cps_continuation (O k) +| IsCPSM: + forall K, + is_cps K -> + is_cps_continuation (M K). + +Local Hint Constructors is_cps is_cps_continuation. + +(* [is_cps] is preserved by renamings. *) + +Lemma is_cps_renaming: + forall t, + is_cps t -> + forall sigma, + is_ren sigma -> + is_cps t.[sigma]. +Proof. + induction 1; intros sigma Hsigma; asimpl; + try solve [ econstructor; obvious ]. + (* Var *) + { destruct Hsigma as [ xi ? ]. subst sigma. asimpl. econstructor. } +Qed. + +Local Hint Resolve is_cps_renaming. + +Lemma is_cps_continuation_renaming: + forall c i, + is_cps_continuation c -> + is_cps_continuation (liftc i c). +Proof. + induction 1; simpl; econstructor; obvious. +Qed. + +Local Hint Resolve is_cps_continuation_renaming. + +(* [is_cps] is preserved by substitution. *) + +Lemma is_cps_substitution_aux: + forall sigma, + (forall x, is_cps (sigma x)) -> + (forall x, is_cps (up sigma x)). +Proof. + intros sigma H [|x]; asimpl. + { econstructor. } + { eapply is_cps_renaming; obvious. } +Qed. + +Lemma is_cps_substitution: + forall K, + is_cps K -> + forall sigma, + (forall x, is_cps (sigma x)) -> + is_value_subst sigma -> + is_cps K.[sigma]. +Proof. + induction 1; intros; asimpl; eauto; + econstructor; eauto using is_cps_substitution_aux with obvious. +Qed. + +Lemma is_cps_substitution_0: + forall K v, + is_cps K -> + is_cps v -> + is_value v -> + is_cps K.[v/]. +Proof. + intros. eapply is_cps_substitution; obvious. + intros [|x]; asimpl; eauto. +Qed. + +(* Inversion lemmas for [is_cps]. *) + +Lemma is_cps_Lam_inversion: + forall t, + is_cps (Lam t) -> + is_cps t. +Proof. + inversion 1; eauto. +Qed. + +(* A CPS term reduces in the same manner in call-by-value and in call-by-name. + Thus, the CPS transformation produces terms that are "indifferent" to which + of these two reduction strategies is chosen. *) + +Lemma cps_indifference_1: + forall t1, is_cps t1 -> + forall t2, cbv t1 t2 -> cbn t1 t2. +Proof. + induction 1; intros; invert_cbv; obvious. +Qed. + +Lemma cps_indifference_2: + forall t1, is_cps t1 -> + forall t2, cbn t1 t2 -> cbv t1 t2. +Proof. + induction 1; intros; invert_cbn; obvious. +Qed. + +(* [is_cps] is preserved by call-by-value and call-by-name reduction. *) + +Lemma is_cps_cbv: + forall t, + is_cps t -> + forall t', + cbv t t' -> + is_cps t'. +Proof. + induction 1; intros; invert_cbv; + eauto using is_cps, is_cps_substitution_0, is_cps_Lam_inversion. +Qed. + +Lemma is_cps_cbn: + forall t, + is_cps t -> + forall t', + cbn t t' -> + is_cps t'. +Proof. + induction 1; intros; invert_cbn; + eauto using is_cps, is_cps_substitution_0, is_cps_Lam_inversion. +Qed. + +(* A CPS term reduces in the same manner in call-by-value and in call-by-name. + The statement is here generalized to a sequence of reduction steps. *) + +Lemma cps_star_indifference_1: + forall t1 t2, + star cbv t1 t2 -> + is_cps t1 -> + star cbn t1 t2. +Proof. + induction 1; intros; + eauto using cps_indifference_1, is_cps_cbv with sequences. +Qed. + +Lemma cps_star_indifference_2: + forall t1 t2, + star cbn t1 t2 -> + is_cps t1 -> + star cbv t1 t2. +Proof. + induction 1; intros; + eauto using cps_indifference_2, is_cps_cbn with sequences. +Qed. + +(* The main auxiliary lemmas. *) + +Lemma is_cps_apply: + forall c v, + is_cps_continuation c -> + is_cps v -> + is_value v -> + is_cps (apply c v). +Proof. + inversion 1; intros; simpl; eauto using is_cps_substitution_0. +Qed. + +Lemma is_cps_reify: + forall c, + is_cps_continuation c -> + is_cps (reify c). +Proof. + inversion 1; simpl; eauto. +Qed. + +Lemma is_value_reify: + forall c, + is_cps_continuation c -> + is_value (reify c). +Proof. + inversion 1; simpl; eauto. +Qed. + +Local Hint Resolve is_cps_apply is_cps_reify is_value_reify. + +(* The main lemma. *) + +Lemma cps_form: +( + forall v, + is_value v -> + is_cps (cpsv v) +) /\ ( + forall t c, + is_cps_continuation c -> + is_cps (cps t c) +). +Proof. + eapply mutual_induction. + (* [cpsv] *) + { intros n IHcps v Hvn ?. + destruct v; [ | | false; obvious | false; obvious ]. + { cpsv; eauto. } + { cpsv; eauto 6 with size. } + } + (* [cps] *) + { intros n IHcpsv IHcps t c Htn Hc. + value_or_app_or_let t; cps. + (* Case: [t] is a value. *) + { obvious. } + (* Case: [t] is an application. *) + { eapply IHcps; [ size | econstructor ]. + eapply IHcps; [ size | econstructor ]. + econstructor; obvious. } + (* Case: [t] is a [let] construct. *) + { eauto 8 with obvious. } + } +Qed. + +Lemma cps_form_main: + forall t, + is_cps (cpsinit t). +Proof. + simpl. intros. eapply cps_form. unfold init. obvious. +Qed. + +(* One property of CPS terms that we do not prove is that all applications are + in tail position, or, in other words, that there is no need for reduction + under a context. In fact, because a CPS-translated function expects two + arguments, there *is* a need for reduction under a context, but only under + a context of depth zero or one. *) diff --git a/coq/CPSKubstitution.v b/coq/CPSKubstitution.v new file mode 100644 index 0000000..5cda6d2 --- /dev/null +++ b/coq/CPSKubstitution.v @@ -0,0 +1,120 @@ +Require Import MyTactics. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusValues. +Require Import CPSDefinition. +Require Import CPSContextSubstitution. +Require Import CPSRenaming. + +(* The [substitution] lemma in CPSSubstitution pushes a substitution + into [cps t k]. The substitution is pushed into both [t] and [k]. + Because it is pushed into [t], this substitution must be of the + form [sigma >>> cpsv], so that, once pushed into [t], it becomes + just [sigma]. *) + +(* Here, we prove another substitution lemma, where the substitution + need not be of the form [sigma >>> cpsv]. It can be an arbitrary + substitution. We require [sigma] to not affect the term [t], so + [sigma] is not pushed into [t]: it is pushed into [k] only. For + this reason, we refer to this lemma as the [kubstitution] lemma. + + In order to express the idea that [sigma] does not affect a term, + more precisely, we write this term under the form [t.[theta]] + and we require that [theta] and [sigma] cancel out, that is, + + theta >> sigma = ids + + (This condition implies [is_ren theta], that is, [theta] must be + a renaming.) Then, we are able to prove the following result: + + (cps t.[theta] (O k)).[sigma] = cps t (O k.[sigma]) + + That is, the substitution [sigma], when pushed into [t], meets [theta] + and they cancel out. *) + +(* [apply] commutes with kubstitutions. *) + +Lemma apply_kubstitution: + forall c theta sigma c' v, + theta >> sigma = ids -> + substc sigma c = c' -> + (apply c v.[theta]).[sigma] = apply c' v. +Proof. + intros. subst. + destruct c; asimpl; pick @eq ltac:(fun h => rewrite h); autosubst. +Qed. + +Local Hint Resolve up_theta_sigma_ids : obvious. + +(* The main result: [cpsv] and [cps] commute with kubstitutions. *) + +Lemma kubstitution: + ( + forall v theta sigma, + theta >> sigma = ids -> + (cpsv v.[theta]).[sigma] = cpsv v + ) /\ ( + forall t c theta sigma c', + theta >> sigma = ids -> + substc sigma c = c' -> + (cps t.[theta] c).[sigma] = cps t c' + ). +Proof. + eapply mutual_induction. + (* [cpsv] *) + { intros n IHcps v Hvn theta sigma Hid. clear IHcps. + rewrite <- cpsv_renaming by obvious. + asimpl. rewrite Hid. + asimpl. reflexivity. } + (* [cps] *) + { intros n IHcpsv IHcps t c Htn theta sigma c' Hid Hkubstc. clear IHcpsv. + value_or_app_or_let t; asimpl; cps. + (* Case: [t] is a value. *) + { rewrite <- cpsv_renaming by obvious. + eauto using apply_kubstitution. } + (* Case: [t] is an application. *) + { eapply IHcps; obvious. + simpl. f_equal. + erewrite <- lift_up by tc. + eapply IHcps; obvious. + asimpl. do 2 f_equal. + rewrite lift_reify. + eapply reify_substitution. + subst. rewrite substc_substc. + reflexivity. } + (* Case: [t] is a [let] construct. *) + { eapply IHcps; obvious. + simpl. do 2 f_equal. + rewrite fold_up_up. + rewrite up_sigma_up_ren by tc. simpl. + eapply IHcps; obvious. } + } +Qed. + +(* The projections of the above result. *) + +Definition cpsv_kubstitution := proj1 kubstitution. +Definition cps_kubstitution := proj2 kubstitution. + +(* A corollary where the substitution [sigma] is [v .: ids], that is, a + substitution of the value [v] for the variable 0. *) + +Lemma cps_kubstitution_0: + forall t c v, + (cps (lift 1 t) c).[v/] = cps t (substc (v .: ids) c). +Proof. + intros. eapply cps_kubstitution. + { autosubst. } + { reflexivity. } +Qed. + +(* A corollary where the substitution [sigma] is [up (v .: ids)], that is, a + substitution of the value [v] for the variable 1. *) + +Lemma cps_kubstitution_1: + forall t c v, + (cps t.[up (ren (+1))] c).[up (v .: ids)] = cps t (substc (up (v .: ids)) c). +Proof. + intros. eapply cps_kubstitution. + { autosubst. } + { reflexivity. } +Qed. diff --git a/coq/CPSRenaming.v b/coq/CPSRenaming.v new file mode 100644 index 0000000..9af3ae3 --- /dev/null +++ b/coq/CPSRenaming.v @@ -0,0 +1,92 @@ +Require Import MyTactics. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusValues. +Require Import CPSDefinition. +Require Import CPSContextSubstitution. + +(* The CPS transformation commutes with renamings, where a renaming [sigma] is + a substitution that maps variables to variables. (Note that [sigma] is not + necessarily injective.) *) + +Lemma renaming: + ( + forall v sigma, + is_ren sigma -> + (cpsv v).[sigma] = cpsv v.[sigma] + ) /\ ( + forall t c sigma c', + is_ren sigma -> + substc sigma c = c' -> + (cps t c).[sigma] = cps t.[sigma] c' + ). +Proof. + eapply mutual_induction. + (* [cpsv] *) + { intros n IHcps v Hvn sigma Hsigma. + destruct v; asimpl; cpsv; asimpl; try reflexivity. + (* [Var] *) + (* The CPS transformation maps variables to variables. *) + { destruct Hsigma as [ xi ? ]. subst sigma. reflexivity. } + (* [Lam] *) + { erewrite IHcps by obvious. asimpl. reflexivity. } + } + (* [cps] *) + { intros n IHcpsv IHcps t c Htn sigma c' Hsigma Hsubstc. + (* Perform case analysis on [t]. The first two cases, [Var] and [Lam], + can be shared by treating the case where [t] is a value. *) + value_or_app_or_let t; asimpl; cps. + (* Case: [t] is a value. *) + { erewrite apply_substitution by eauto. + rewrite IHcpsv by obvious. + reflexivity. } + (* Case: [t] is an application. *) + { eapply IHcps; obvious. + erewrite <- lift_upn by tc. + simpl. f_equal. + eapply IHcps; obvious. + simpl. + rewrite fold_up_upn, lift_upn by tc. + do 3 f_equal. + eauto using reify_substitution. } + (* Case: [t] is a [let] construct. *) + { eapply IHcps; obvious. + simpl. do 2 f_equal. + rewrite fold_up_up. + erewrite IHcps by first [ eapply substc_liftc_liftc; eauto | obvious ]. + autosubst. } + } +Qed. + +(* The projections of the above result. *) + +Definition cpsv_renaming := proj1 renaming. +Definition cps_renaming := proj2 renaming. + +(* A point-free reformulation of the above result: [cpsv] commutes with + an arbitrary renaming [xi]. *) + +Goal + forall sigma, + is_ren sigma -> + cpsv >>> subst sigma = subst sigma >>> cpsv. +Proof. + intros. f_ext; intros t. asimpl. eauto using cpsv_renaming. +Qed. + +(* Corollaries. *) + +Lemma up_sigma_cpsv: + forall sigma, + up (sigma >>> cpsv) = up sigma >>> cpsv. +Proof. + eauto using up_sigma_f, cpsv_renaming with is_ren typeclass_instances. +Qed. + +Lemma upn_sigma_cpsv: + forall i sigma, + upn i (sigma >>> cpsv) = upn i sigma >>> cpsv. +Proof. + eauto using upn_sigma_f, cpsv_renaming with is_ren typeclass_instances. +Qed. + +Hint Resolve up_sigma_cpsv upn_sigma_cpsv : obvious. diff --git a/coq/CPSSimulation.v b/coq/CPSSimulation.v new file mode 100644 index 0000000..bb3a758 --- /dev/null +++ b/coq/CPSSimulation.v @@ -0,0 +1,177 @@ +Require Import MyTactics. +Require Import Sequences. +Require Import Relations. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusValues. +Require Import LambdaCalculusReduction. +Require Import CPSDefinition. +Require Import CPSContextSubstitution. +Require Import CPSRenaming. +Require Import CPSSubstitution. +Require Import CPSKubstitution. +Require Import CPSSpecialCases. + +(* We now prepare for the statement of the "magic step" lemma [pcbv_cps]. This + lemma states that if the continuations [c1] and [c2] are similar, then [cps + t c1] is able to reduce via [pcbv] to [cps t c2]. We use parallel reduction + [pcbv] because we must allow reduction to take place under [Lam] and in the + right-hand side of [Let]. We do not need the full power of [pcbv]: we only + reduce zero or one redexes, never more. *) + +(* A simplified copy of this file, where we pretend that the [Let] construct + does not exist, can be found in [CPSSimulationWithoutLet.v]. There, there + is no need for parallel reduction; a simpler simulation diagram holds. *) + +(* Similarity of continuations is defined as follows: *) + +Inductive similar : continuation -> continuation -> Prop := +| SimilarReify: + forall c, + similar (O (reify c)) c +| SimilarM: + forall K1 K2, + pcbv K1 K2 -> + similar (M K1) (M K2). + +(* Similarity is preserved by lifting. *) + +Lemma similar_liftc_liftc: + forall i c1 c2, + similar c1 c2 -> + similar (liftc i c1) (liftc i c2). +Proof. + induction 1; intros; simpl. + { rewrite lift_reify. econstructor. } + { econstructor. eapply red_subst; obvious. } +Qed. + +(* The lemmas [pcbv_apply] and [pcbv_reify] are preliminaries for the + "magic step" lemma. *) + +Lemma pcbv_apply: + forall c1 c2, + similar c1 c2 -> + forall v, + pcbv (apply c1 (cpsv v)) (apply c2 (cpsv v)). +Proof. + inversion 1; subst; intros; [ destruct c2 |]; simpl. + (* Case: both [c1] and [c2] are an object-level continuation [k]. + No computation step is taken. *) + { eapply red_refl; obvious. } + (* Case: [c1] is a two-level eta-expansion of [c2], which is a + meta-level continuation [K]. One beta-reduction step is taken. *) + { eapply pcbv_RedBetaV; obvious. } + (* Case: [c1] and [c2] are similar meta-level continuations. The + required reduction steps are provided directly by the similarity + hypothesis. *) + { eapply red_subst; obvious. } +Qed. + +Lemma pcbv_reify: + forall c1 c2, + similar c1 c2 -> + pcbv (reify c1) (reify c2). +Proof. + inversion 1; subst; intros; [ destruct c2 |]; simpl. + (* Case: both [c1] and [c2] are an object-level continuation [k]. + No computation step is taken. *) + { eapply red_refl; obvious. } + (* Case: [c1] is a two-level eta-expansion of [c2], which is a + meta-level continuation [K]. No computation step is taken. *) + { eapply red_refl; obvious. } + (* Case: [c1] and [c2] are similar meta-level continuations. The + required reduction steps are provided directly by the similarity + hypothesis, applied under a lambda-abstraction. *) + { eapply RedLam; obvious. } + (* We could arrange to just write [obvious] in each of the above + cases and finish the entire proof in one line, but we prefer to + explicitly show what happens in each case. *) +Qed. + +Local Hint Resolve red_refl : obvious. + +(* The "magic step" lemma. *) + +Lemma pcbv_cps: + forall t c1 c2, + similar c1 c2 -> + pcbv (cps t c1) (cps t c2). +Proof. + (* The proof is by induction on the size of [t]. *) + size_induction. intros c1 c2 Hsimilar. + value_or_app_or_let t; cps. + (* Case: [t] is a value. *) + { eauto using pcbv_apply. } + (* Case: [t] is an application. *) + { eapply IH; [ size | econstructor ]. + eapply IH; [ size | econstructor ]. + eapply RedAppLR; obvious. + eapply red_subst; obvious. + eauto using pcbv_reify. } + (* Case: [t] is a [let] construct. *) + { eapply IH; [ size | econstructor ]. + eapply RedLetLR; obvious. + eapply IH; [ size |]. + eauto using similar_liftc_liftc. } +Qed. + +(* The small-step simulation theorem: if [t1] reduces to [t2], then [cps t1 c] + reduces to [cps t2 c] via at least one step of [cbv], followed with one + step of [pcbv]. *) + +(* Although the reduction strategies [cbv] and [pcbv] allow reduction in the + left-hand side of applications, at an arbitrary depth, in reality the CPS + transformation exploits this only at depth 0 or 1. We do not formally + establish this result (but could, if desired). *) + +Notation plus_cbv_pcbv := + (composition (plus cbv) pcbv). + +Lemma cps_simulation: + forall t1 t2, + cbv t1 t2 -> + forall c, + is_value (reify c) -> + plus_cbv_pcbv + (cps t1 c) + (cps t2 c). +Proof. + induction 1; intros; subst; try solve [ tauto ]. + (* Beta-reduction. *) + { rewrite cps_app_value_value by eauto. cpsv. + (* We are looking at two beda redexes. Perform exactly two steps of [cbv]. *) + eexists. split; [ eapply plus_left; [ obvious | eapply star_step; [ obvious | eapply star_refl ]] |]. + (* There remains one step of [pcbv]. *) + rewrite cps_substitution_1_O_Var_0 by eauto. + rewrite lift_up by tc. + rewrite cps_kubstitution_0. asimpl. + eapply pcbv_cps. econstructor. + } + (* Let *) + { rewrite cps_let_value by eauto. + (* We are looking at a let-redex. Perform exactly one step of [cbv]. *) + eexists. split; [ eapply plus_left; [ obvious | eapply star_refl ] |]. + (* There remains a trivial (reflexive) step of [pcbv]. *) + rewrite cps_substitution_0 by eauto. + eapply red_refl; obvious. + } + (* Reduction in the left-hand side of an application. *) + { cps. eapply IHred. eauto. } + (* Reduction in the right-hand side of an application. *) + { rewrite !cps_app_value by eauto. eapply IHred. tauto. } + (* Reduction in the left-hand side of [Let]. *) + { cps. eapply IHred. tauto. } +Qed. + +(* We now specialize the above result to the identity continuation and + state it as a commutative diagram. *) + +Lemma cps_init_simulation: + let sim t t' := (cps t init = t') in + diamond22 + cbv sim + plus_cbv_pcbv sim. +Proof. + assert (is_value (reify init)). { simpl. eauto. } + unfold diamond22. intros. subst. eauto using cps_simulation. +Qed. diff --git a/coq/CPSSimulationWithoutLet.v b/coq/CPSSimulationWithoutLet.v new file mode 100644 index 0000000..ccbe944 --- /dev/null +++ b/coq/CPSSimulationWithoutLet.v @@ -0,0 +1,137 @@ +Require Import MyTactics. +Require Import Sequences. +Require Import Relations. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusValues. +Require Import LambdaCalculusReduction. +Require Import CPSDefinition. +Require Import CPSContextSubstitution. +Require Import CPSRenaming. +Require Import CPSSubstitution. +Require Import CPSKubstitution. +Require Import CPSSpecialCases. + +(* This file is a simplified copy of [CPSSimulation]. Here, we consider how + the proof of the simulation lemma is simplified in the absence of a [Let] + construct. We simply pretend that this construct does not exist, and skip + the proof cases where it appears. *) + +(* -------------------------------------------------------------------------- *) + +(* The definition of similarity of continuations boils down to just one rule: + [O (reify c)] is similar to [c]. *) + +Inductive similar : continuation -> continuation -> Prop := +| SimilarReify: + forall c, + similar (O (reify c)) c. + +(* -------------------------------------------------------------------------- *) + +(* The lemma [pcbv_apply] is simplified: its conclusion uses [star cbv] instead + of [pcbv]. In fact, zero or one step of reduction is needed. *) + +Lemma pcbv_apply: + forall c1 c2, + similar c1 c2 -> + forall v, + star cbv (apply c1 (cpsv v)) (apply c2 (cpsv v)). +Proof. + inversion 1; subst; intros; destruct c2; simpl. + (* Case: both [c1] and [c2] are an object-level continuation [k]. + No computation step is taken. *) + { eauto with sequences. } + (* Case: [c1] is a two-level eta-expansion of [c2], which is a + meta-level continuation [K]. One beta-reduction step is taken. *) + { eauto with sequences obvious. } +Qed. + +(* The lemma [pcbv_reify] is simplified: its conclusion becomes an equality. *) + +Lemma pcbv_reify: + forall c1 c2, + similar c1 c2 -> + reify c1 = reify c2. +Proof. + inversion 1; subst; intros; destruct c2; simpl; reflexivity. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* The "magic step" lemma is simplified: its conclusion uses [star cbv] instead + of [pcbv]. In fact, zero or one step of reduction is needed. The magic lies + in the case of applications, where [pcbv_reify] is used. *) + +Lemma pcbv_cps: + forall t c1 c2, + similar c1 c2 -> + star cbv (cps t c1) (cps t c2). +Proof. + (* The proof does NOT require an induction. *) + intros t c1 c2 Hsimilar. + value_or_app_or_let t; cps. + (* Case: [t] is a value. *) + { eauto using pcbv_apply. } + (* It turns out by magic that this proof case is trivial: it suffices to + take zero reduction steps. (That took me an evening to find out.) Thus, + no induction hypothesis is needed! *) + { erewrite pcbv_reify by eauto. + eauto with sequences. } + (* Case: [t] is a [let] construct. We pretend this case is not there. *) + { admit. } +Admitted. (* normal *) + +(* -------------------------------------------------------------------------- *) + +(* The small-step simulation theorem: if [t1] reduces to [t2], then [cps t1 c] + reduces to [cps t2 c] via at least one step of [cbv]. (In fact, two or three + steps are required.) *) + +Lemma cps_simulation: + forall t1 t2, + cbv t1 t2 -> + forall c, + is_value (reify c) -> + plus cbv + (cps t1 c) + (cps t2 c). +Proof. + induction 1; intros; subst; try solve [ tauto ]. + (* Beta-reduction. *) + { rewrite cps_app_value_value by eauto. cpsv. + (* We are looking at two beda redexes. Perform exactly two steps of [cbv]. *) + eapply plus_left. obvious. + eapply star_step. obvious. + (* Push the inner substitution (the actual argument) into [cps]. *) + rewrite cps_substitution_1_O_Var_0 by eauto. + rewrite lift_up by tc. + (* Push the outer substitution (the continuation) into [cps]. *) + rewrite cps_kubstitution_0. + asimpl. + (* Conclude. *) + eapply pcbv_cps. econstructor. + } + (* Let. We pretend this case is not there. *) + { admit. } + (* Reduction in the left-hand side of an application. *) + { cps. eapply IHred. eauto. } + (* Reduction in the right-hand side of an application. *) + { rewrite !cps_app_value by eauto. eapply IHred. tauto. } + (* Reduction in the left-hand side of [Let]. We pretend this case is not there. *) + { admit. } +Admitted. (* normal *) + +(* -------------------------------------------------------------------------- *) + +(* We now specialize the above result to the identity continuation and + state it as a commutative diagram. *) + +Lemma cps_init_simulation: + let sim t t' := (cps t init = t') in + diamond22 + cbv sim + (plus cbv) sim. +Proof. + assert (is_value (reify init)). { simpl. eauto. } + unfold diamond22. intros. subst. eauto using cps_simulation. +Qed. diff --git a/coq/CPSSpecialCases.v b/coq/CPSSpecialCases.v new file mode 100644 index 0000000..33fc4e3 --- /dev/null +++ b/coq/CPSSpecialCases.v @@ -0,0 +1,48 @@ +Require Import MyTactics. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusValues. +Require Import CPSDefinition. +Require Import CPSContextSubstitution. +Require Import CPSKubstitution. + +(* The translation of an application whose left-hand side is a value. *) + +Lemma cps_app_value: + forall v1 t2 c, + is_value v1 -> + cps (App v1 t2) c = + cps t2 (M (App (App (lift 1 (cpsv v1)) (Var 0)) (lift 1 (reify c)))). +Proof. + intros. cps. simpl. + rewrite cps_kubstitution_0. asimpl. + reflexivity. +Qed. + +(* The translation of a value-value application. *) + +Lemma cps_app_value_value: + forall v1 v2 c, + is_value v1 -> + is_value v2 -> + cps (App v1 v2) c = + App (App (cpsv v1) (cpsv v2)) (reify c). +Proof. + intros. + rewrite cps_app_value by obvious. + rewrite cps_value by eauto. asimpl. + reflexivity. +Qed. + +(* The translation of a [Let] construct whose left-hand side is a value. *) + +Lemma cps_let_value: + forall v1 t2 c, + is_value v1 -> + cps (Let v1 t2) c = + Let (cpsv v1) (cps t2 (liftc 1 c)). +Proof. + intros. cps. simpl. f_equal. + eapply cps_kubstitution. (* [cps_substitution] could be used too *) + { autosubst. } + { rewrite substc_substc. autosubst. } +Qed. diff --git a/coq/CPSSubstitution.v b/coq/CPSSubstitution.v new file mode 100644 index 0000000..a292aeb --- /dev/null +++ b/coq/CPSSubstitution.v @@ -0,0 +1,149 @@ +Require Import MyTactics. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusValues. +Require Import CPSDefinition. +Require Import CPSContextSubstitution. +Require Import CPSRenaming. + +(* The CPS transformation commutes with certain substitutions. More precisely, + it commutes with a substitution [sigma] of values for variables, up to a + transformation of the values in the codomain of [sigma]. + + In the case of [cpsv], we have the following diagram: applying [sigma] + first, followed with [cpsv], is the same as applying [cpsv] first, followed + with [sigma >>> cpsv]. + + cpsv v.[sigma] = (cpsv v).[sigma >>> cpsv] + + This can also be written in point-free style, that is, without mentioning + the value [v]: + + subst sigma >>> cpsv = cpsv >>> subst (sigma >>> cpsv) + + As in the case of the renaming lemma (see CPSRenaming.v), this statement is + proved by induction on the size of terms, together with an analogous + statement about the function [cps]. *) + +(* The proof depends on [CPSRenaming] via the lemmas [up_sigma_cpsv] and + [upn_sigma_cpsv], which are declared as hints for [obvious]. *) + +Lemma substitution: + ( + forall v sigma sigma', + sigma' = sigma >>> cpsv -> + is_value_subst sigma -> + (cpsv v).[sigma'] = cpsv v.[sigma] + ) /\ ( + forall t c sigma c' sigma', + sigma' = sigma >>> cpsv -> + is_value_subst sigma -> + substc sigma' c = c' -> + (cps t c).[sigma'] = cps t.[sigma] c' + ). +Proof. + eapply mutual_induction. + (* [cpsv] *) + { intros n IHcps v Hvn sigma sigma' Heq Hsigma. subst. + destruct v; asimpl; cpsv; asimpl; try reflexivity. + (* Lam *) + { erewrite IHcps by obvious. asimpl. reflexivity. } + } + (* [cps] *) + { intros n IHcpsv IHcps t c Htn sigma c' sigma' Heq Hsigma Hsubstc. subst. + value_or_app_or_let t; asimpl; cps. + (* Case: [t] is a value. *) + { erewrite apply_substitution by eauto. + erewrite IHcpsv by obvious. + reflexivity. } + (* Case: [t] is an application. *) + { eapply IHcps; obvious. + simpl. f_equal. + erewrite <- lift_up by tc. + eapply IHcps; obvious. + asimpl. do 2 f_equal. + rewrite lift_reify. + eapply reify_substitution. + rewrite substc_substc. + reflexivity. } + (* Case: [t] is a [let] construct. *) + { eapply IHcps; obvious. + simpl. + rewrite fold_up_up. + do 2 f_equal. + erewrite IHcps by first [ eapply substc_liftc_liftc; eauto | obvious ]. + autosubst. } + } +Qed. + +(* The projections of the above result. *) + +Definition cpsv_substitution := proj1 substitution. +Definition cps_substitution := proj2 substitution. + +(* A point-free reformulation of the above result: [cpsv] commutes with an + arbitrary substitution [sigma], up to a transformation of the values in the + codomain of [sigma]. *) + +Goal + forall sigma, + is_value_subst sigma -> + cpsv >>> subst (sigma >>> cpsv) = + subst sigma >>> cpsv. +Proof. + intros. f_ext; intros v. asimpl. eauto using cpsv_substitution. +Qed. + +(* This technical lemma is used below. *) + +Lemma cpsv_cons: + forall v, + cpsv v .: ids = (v .: ids) >>> cpsv. +Proof. + intros. f_ext; intros [|x]; autosubst. +Qed. + +(* A corollary where the substitution [sigma] is [v .: ids], that is, a + substitution of the value [v] for the variable 0. This one is about + [cpsv]. *) + +Lemma cpsv_substitution_0: + forall v w, + is_value v -> + (cpsv w).[cpsv v/] = + cpsv w.[v/]. +Proof. + intros. rewrite cpsv_cons. erewrite cpsv_substitution by obvious. reflexivity. +Qed. + +(* Another corollary where the substitution [sigma] is [v .: ids], that is, a + substitution of the value [v] for the variable 0. This one is about [cps] + and concerns the case where the continuation is of the form [liftc 1 c], so + it is unaffected. *) + +Lemma cps_substitution_0: + forall t c v, + is_value v -> + (cps t (liftc 1 c)).[cpsv v/] = + cps t.[v/] c. +Proof. + intros. eapply cps_substitution. + { autosubst. } + { obvious. } + { eauto using substc_liftc_single. } +Qed. + +(* A corollary where the substitution [sigma] is [up (v .: ids)], that is, a + substitution of the value [v] for the variable 1, and the continuation is + the variable 0, so it is unaffected. *) + +Lemma cps_substitution_1_O_Var_0: + forall t v, + is_value v -> + (cps t (O (Var 0))).[up (cpsv v .: ids)] = + cps t.[up (v .: ids)] (O (Var 0)). +Proof. + intros. eapply cps_substitution. + { rewrite cpsv_cons. obvious. } + { obvious. } + { reflexivity. } +Qed. diff --git a/coq/FixExtra.v b/coq/FixExtra.v new file mode 100644 index 0000000..43c57a3 --- /dev/null +++ b/coq/FixExtra.v @@ -0,0 +1,20 @@ +Require Import Coq.Logic.FunctionalExtensionality. + +(* This is a simplified version of the lemma [Fix_eq], which is defined in + [Coq.Init.Wf]. We use functional extensionality to remove one hypothesis. + Furthermore, we introduce the auxiliary equality [f = Fix Rwf P F] so as + to avoid duplicating the (usually large) term [F] in the right-hand side + of the conclusion. *) + +Lemma Fix_eq_simplified + (A : Type) (R : A -> A -> Prop) (Rwf : well_founded R) + (P : A -> Type) + (F : forall x : A, (forall y : A, R y x -> P y) -> P x) + (f : forall x, P x) : + f = Fix Rwf P F -> + forall x : A, + f x = F x (fun (y : A) (_ : R y x) => f y). +Proof. + intros. subst. eapply Fix_eq. intros. f_equal. + eauto using functional_extensionality_dep, functional_extensionality. +Qed. diff --git a/coq/LambdaCalculusParallelReduction.v b/coq/LambdaCalculusParallelReduction.v new file mode 100644 index 0000000..e66fd15 --- /dev/null +++ b/coq/LambdaCalculusParallelReduction.v @@ -0,0 +1,174 @@ +Require Import Relations. +Require Import Sequences. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusValues. +Require Import LambdaCalculusReduction. +Require Import MyTactics. (* TEMPORARY cannot be declared earlier; why? *) + +(* -------------------------------------------------------------------------- *) + +(* Parallel call-by-value reduction is stable by substitution. In fact, + if [t1] parallel-reduces to [t2] and [sigma1] parallel-reduces to + [sigma2], then [t1.[sigma1]] parallel-reduces to [t2.[sigma2]]. *) + +Notation pcbv_subst sigma1 sigma2 := + (forall x, pcbv (sigma1 x) (sigma2 x)). + +Lemma pcbv_subst_up: + forall sigma1 sigma2, + pcbv_subst sigma1 sigma2 -> + pcbv_subst (up sigma1) (up sigma2). +Proof. + intros ? ? ? [|x]; asimpl. + { eapply red_refl; obvious. } + { eapply red_subst; obvious. } +Qed. + +Lemma pcbv_subst_cons: + forall v1 v2 sigma1 sigma2, + pcbv v1 v2 -> + pcbv_subst sigma1 sigma2 -> + pcbv_subst (v1 .: sigma1) (v2 .: sigma2). +Proof. + intros ? ? ? ? ? ? [|x]; asimpl; eauto. +Qed. + +Hint Resolve pcbv_subst_up pcbv_subst_cons : red obvious. + +Lemma pcbv_parallel_subst: + forall t1 t2, + pcbv t1 t2 -> + forall sigma1 sigma2, + pcbv_subst sigma1 sigma2 -> + is_value_subst sigma1 -> + is_value_subst sigma2 -> + pcbv t1.[sigma1] t2.[sigma2]. +Proof. + induction 1; try solve [ tauto ]; intros; subst. + { rewrite subst_app, subst_lam. + eapply RedParBetaV. obvious. obvious. + { eapply IHred1; obvious. } + { eapply IHred2; obvious. } + autosubst. } + { rewrite subst_let. + eapply RedParLetV. obvious. obvious. + { eapply IHred1; obvious. } + { eapply IHred2; obvious. } + autosubst. } + { rewrite !subst_var. obvious. } + { rewrite !subst_lam. eauto 6 with obvious. } + { rewrite !subst_app. obvious. } + { rewrite !subst_let. eauto 7 with obvious. } +Qed. + +Hint Resolve pcbv_parallel_subst : red obvious. + +(* -------------------------------------------------------------------------- *) + +(* Parallel call-by-value reduction enjoys the diamond property. *) + +(* The proof is by Takahashi's method (1995). We first define the function + [fpbcv], for "full parallel call-by-value reduction". This function + performs as much reduction as is possible in one step of [pcbv]. We prove + that this is indeed the case: if [t1] reduces to [t2] by [pcbv], then [t2] + reduces to [fpcbv t1]. The diamond property follows immediately. *) + +Fixpoint fpcbv (t : term) : term := + match t with + | Var x => + Var x + | Lam t => + Lam (fpcbv t) + | App (Lam t1) t2 => + if_value t2 + (fpcbv t1).[fpcbv t2/] + (App (Lam (fpcbv t1)) (fpcbv t2)) + | App t1 t2 => + App (fpcbv t1) (fpcbv t2) + | Let t1 t2 => + if_value t1 + (fpcbv t2).[fpcbv t1/] + (Let (fpcbv t1) (fpcbv t2)) + end. + +Ltac fpcbv := + simpl; if_value. + +Lemma pcbv_takahashi: + forall t1 t2, + pcbv t1 t2 -> + pcbv t2 (fpcbv t1). +Proof. + induction 1; try solve [ tauto ]; subst; + try solve [ fpcbv; eauto 9 with obvious ]. + (* RedAppLR *) + { destruct t1; try solve [ fpcbv; obvious ]. + value_or_nonvalue u1; fpcbv; [ | obvious ]. + (* [t1] is a lambda-abstraction, and [u1] is a value. We have a redex. *) + (* [pcbv (Lam _) t2] implies that [t2] is a lambda-abstraction, too. *) + match goal with h: pcbv (Lam _) ?t2 |- _ => invert h end. + (* Thus, the reduction of [t1] to [t2] is a reduction under lambda. *) + simpl in IHred1. inversion IHred1; subst. + (* The result is then... *) + obvious. } + (* RedLetLR *) + { value_or_nonvalue t1; fpcbv; obvious. } +Qed. + +Lemma diamond_pcbv: + diamond pcbv. +Proof. + intros t u1 ? u2 ?. + exists (fpcbv t). + split; eauto using pcbv_takahashi. +Qed. + +Lemma diamond_star_pcbv: + diamond (star pcbv). +Proof. + eauto using diamond_pcbv, star_diamond. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* Parallel reduction preserves the property of being stuck and the property + of being irreducible. *) + +Lemma pcbv_preserves_stuck: + forall t1 t2, + pcbv t1 t2 -> + stuck t1 -> + stuck t2. +Proof. + induction 1; intros; subst; try solve [ tauto ]. + (* RedParBetaV *) + { false. eapply stuck_irred; eauto 2 with obvious. } + (* RedPatLetV *) + { false. eapply stuck_irred; eauto 2 with obvious. } + (* RedLam *) + { inv stuck. } + (* RedAppLR *) + { inv stuck. + { assert (forall t, t2 <> Lam t). + { do 2 intro. subst. + inv red; (* invert [pcbv _ (Lam _)] *) + try solve [ false; eauto 2 with obvious | false; congruence ]. } + eauto with stuck obvious. } + { eauto with stuck. } + { eauto with stuck obvious. } + } + (* RedLetLR *) + { inv stuck. + eauto with stuck. } +Qed. + +Lemma pcbv_preserves_irred: + forall t1 t2, + pcbv t1 t2 -> + irred cbv t1 -> + irred cbv t2. +Proof. + intros t1 t2 ?. + rewrite !irred_cbv_characterization. + intuition eauto 2 using pcbv_preserves_stuck with obvious. +Qed. diff --git a/coq/LambdaCalculusStandardization.v b/coq/LambdaCalculusStandardization.v new file mode 100644 index 0000000..5c066dc --- /dev/null +++ b/coq/LambdaCalculusStandardization.v @@ -0,0 +1,679 @@ +Require Import Sequences. +Require Import Relations. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusValues. +Require Import LambdaCalculusReduction. +Require Import LambdaCalculusParallelReduction. +Require Import MyTactics. + +(* This is an adaptation of the paper "A Simple Proof of Call-by-Value + Standardization", by Karl Crary (2009). We establish two main results: + + First, parallel call-by-value reduction is adequate, i.e., is contained in + contextual equivalence: if [t1] parallel-reduces to [t2], then [t1] halts + if and only if [t2] halts (where halting is considered with respect to + ordinary call-by-value reduction, [cbv]). + + Second, every call-by-value reduction sequence can be put in a standard + form, as defined by the predicate [stdred]. *) + +(* -------------------------------------------------------------------------- *) + +(* "Evaluation" in Crary's paper is [cbv] here. Parallel reduction in Crary's + paper is [pcbv] here. Internal parallel reduction, [ipcbv], is defined as + follows. It is a restricted version of parallel reduction: it is allowed to + act only under lambda, in the right-hand side of an application whose + left-hand side is not a value, and in the right-hand side of [Let]. *) + +Inductive ipcbv : term -> term -> Prop := +| IRedVar: + forall x, + ipcbv (Var x) (Var x) +| IRedLam: + forall t1 t2, + pcbv t1 t2 -> + ipcbv (Lam t1) (Lam t2) +| IRedAppLRNonValue: + forall t1 t2 u1 u2, + ~ is_value t1 -> + ipcbv t1 t2 -> + pcbv u1 u2 -> + ipcbv (App t1 u1) (App t2 u2) +| IRedAppLR: + forall t1 t2 u1 u2, + is_value t1 -> (* wlog; see [ipcbv_IRedAppLR] below *) + ipcbv t1 t2 -> + ipcbv u1 u2 -> + ipcbv (App t1 u1) (App t2 u2) +| IRedLetLR: + forall t1 t2 u1 u2, + ipcbv t1 t2 -> + pcbv u1 u2 -> + ipcbv (Let t1 u1) (Let t2 u2) +. + +Local Hint Constructors ipcbv : red obvious. + +(* [ipcbv] is a subset of [pcbv]. *) + +Lemma ipcbv_subset_pcbv: + forall t1 t2, + ipcbv t1 t2 -> + pcbv t1 t2. +Proof. + induction 1; obvious. +Qed. + +Local Hint Resolve ipcbv_subset_pcbv : red obvious. + +(* The side condition [is_value t1] in [IRedAppLR] does not cause any loss + of expressiveness, as the previous rule covers the case where [t1] is + not a value. *) + +Lemma ipcbv_IRedAppLR: + forall t1 t2 u1 u2, + ipcbv t1 t2 -> + ipcbv u1 u2 -> + ipcbv (App t1 u1) (App t2 u2). +Proof. + intros. value_or_nonvalue t1; obvious. +Qed. + +Local Hint Resolve ipcbv_IRedAppLR : red obvious. + +(* [ipcbv] is reflexive. *) + +Lemma ipcbv_refl: + forall t, + ipcbv t t. +Proof. + induction t; eauto using red_refl with obvious. +Qed. + +Local Hint Resolve ipcbv_refl. + +(* [ipcbv] preserves values, both ways. *) + +Lemma ipcbv_preserves_values: + forall v1 v2, ipcbv v1 v2 -> is_value v1 -> is_value v2. +Proof. + induction 1; is_value. +Qed. + +Lemma ipcbv_preserves_values_reversed: + forall v1 v2, ipcbv v1 v2 -> is_value v2 -> is_value v1. +Proof. + induction 1; is_value. +Qed. + +Lemma ipcbv_preserves_values_reversed_contrapositive: + forall v1 v2, ipcbv v1 v2 -> ~ is_value v1 -> ~ is_value v2. +Proof. + induction 1; is_value. +Qed. + +Local Hint Resolve ipcbv_preserves_values ipcbv_preserves_values_reversed + ipcbv_preserves_values_reversed_contrapositive. + +Lemma star_ipcbv_preserves_values_reversed: + forall v1 v2, star ipcbv v1 v2 -> is_value v2 -> is_value v1. +Proof. + induction 1; eauto. +Qed. + +Local Hint Resolve star_ipcbv_preserves_values_reversed. + +(* Reverse internal parallel reduction preserves the property of being stuck + and (therefore) the property of being irreducible. *) + +Lemma reverse_ipcbv_preserves_stuck: + forall t1 t2, + ipcbv t1 t2 -> + stuck t2 -> + stuck t1. +Proof. + induction 1; inversion 1; subst; eauto with stuck. + { false. obvious. } + { false. obvious. } + { eapply StuckApp; eauto. + do 2 intro; subst. inv ipcbv. congruence. } +Qed. + +Lemma reverse_star_ipcbv_preserves_stuck: + forall t1 t2, + star ipcbv t1 t2 -> + stuck t2 -> + stuck t1. +Proof. + induction 1; eauto using reverse_ipcbv_preserves_stuck. +Qed. + +Lemma reverse_ipcbv_preserves_irred: + forall t1 t2, + ipcbv t1 t2 -> + irred cbv t2 -> + irred cbv t1. +Proof. + do 3 intro. rewrite !irred_cbv_characterization. + intuition eauto 2 using reverse_ipcbv_preserves_stuck. +Qed. + +Local Hint Resolve + pcbv_preserves_irred + reverse_ipcbv_preserves_irred + (star_implication (irred cbv)) + (star_implication_reversed (irred cbv)) +: irred. + +(* -------------------------------------------------------------------------- *) + +(* Strong parallel reduction requires both (1) parallel reduction; and (2) a + decomposition as an ordinary call-by-value reduction sequence, followed + with an internal parallel reduction step. Our goal is to prove that strong + parallel reduction in fact coincides with parallel reduction, which means + that this decomposition always exists. *) + +Inductive spcbv : term -> term -> Prop := +| SPCbv: + forall t1 u t2, + pcbv t1 t2 -> + star cbv t1 u -> + ipcbv u t2 -> + spcbv t1 t2. + +Local Hint Constructors spcbv. + +(* By definition, [spcbv] is a subset of [pcbv]. *) + +Lemma spcbv_subset_pcbv: + forall t1 t2, + spcbv t1 t2 -> + pcbv t1 t2. +Proof. + inversion 1; eauto. +Qed. + +Local Hint Resolve spcbv_subset_pcbv. + +(* [spcbv] is reflexive. *) + +Lemma spcbv_refl: + forall t, + spcbv t t. +Proof. + econstructor; eauto using red_refl with sequences obvious. +Qed. + +Local Hint Resolve spcbv_refl. + +(* -------------------------------------------------------------------------- *) + +(* The main series of technical lemmas begins here. *) + +Lemma crarys_lemma2: + forall t1 t2 u1 u2, + spcbv t1 t2 -> + pcbv u1 u2 -> + ~ is_value t2 -> + spcbv (App t1 u1) (App t2 u2). +Proof. + inversion 1; intros; subst. econstructor; obvious. +Qed. + +Lemma crarys_lemma3_App: + forall t1 t2 u1 u2, + spcbv t1 t2 -> + spcbv u1 u2 -> + spcbv (App t1 u1) (App t2 u2). +Proof. + inversion 1; inversion 1; intros; subst. + value_or_nonvalue t2. + { eauto 6 with obvious. } + { eauto using crarys_lemma2. } +Qed. + +Lemma crarys_lemma3_Let: + forall t1 t2 u1 u2, + spcbv t1 t2 -> + pcbv u1 u2 -> + spcbv (Let t1 u1) (Let t2 u2). +Proof. + inversion 1; intros; subst; obvious. +Qed. + +Lemma crarys_lemma4: + forall {u1 u2}, + spcbv u1 u2 -> + is_value u1 -> + forall {t1 t2}, + ipcbv t1 t2 -> + spcbv t1.[u1/] t2.[u2/]. +Proof. + induction 3; intros. + (* Var. *) + { destruct x as [|x]; asimpl; eauto. } + (* Lam *) + { rewrite !subst_lam. inv spcbv. + econstructor; eauto 11 with sequences obvious. (* slow *) } + (* App (nonvalue) *) + { asimpl. eapply crarys_lemma2; obvious. eauto 9 with obvious. } + (* App *) + { asimpl. eapply crarys_lemma3_App; obvious. } + (* Let *) + { rewrite !subst_let. + eapply crarys_lemma3_Let; eauto 12 with obvious. } +Qed. + +Lemma crarys_lemma5: + forall {t1 t2 u1 u2}, + spcbv t1 t2 -> + spcbv u1 u2 -> + is_value u1 -> + spcbv t1.[u1/] t2.[u2/]. +Proof. + intros _ _ u1 u2 [ t1 t t2 Hpcbvt Hstarcbv Hipcbv ] Hpcbvu Hvalue. + generalize (crarys_lemma4 Hpcbvu Hvalue Hipcbv). + inversion 1; subst. + econstructor; [| | obvious ]. + { eauto 11 with obvious. } + { eauto using star_red_subst with sequences obvious. } +Qed. + +Lemma crarys_lemma6: + forall {t1 t2}, + pcbv t1 t2 -> + spcbv t1 t2. +Proof. + induction 1; try solve [ tauto ]; subst. + (* RedParBetaV *) + { match goal with hv: is_value _ |- _ => + generalize (crarys_lemma5 IHred1 IHred2 hv) + end. + inversion 1; subst. + econstructor; obvious. + eauto with sequences obvious. } + (* RedParLetV *) + { match goal with hv: is_value _ |- _ => + generalize (crarys_lemma5 IHred1 IHred2 hv) + end. + inversion 1; subst. + econstructor; obvious. + eauto with sequences obvious. } + (* RedVar *) + { obvious. } + (* RedLam *) + { clear IHred. eauto with sequences obvious. } + (* RedAppLR *) + { eauto using crarys_lemma3_App. } + (* RedLetLR *) + { eauto using crarys_lemma3_Let. } +Qed. + +(* A reformulation of Lemma 6. We can now forget about [spcbv]. *) + +Lemma crarys_main_lemma: + forall t1 t2, + pcbv t1 t2 -> + exists t, star cbv t1 t /\ ipcbv t t2. +Proof. + intros ? ? H. + generalize (crarys_lemma6 H); inversion 1; subst. + eauto. +Qed. + +Lemma crarys_main_lemma_plus: + commutation22 + cbv pcbv + (plus cbv) ipcbv. +Proof. + unfold commutation22. intros ? ? Hstarcbv ? Hpcbv. + forward1 crarys_main_lemma. + eauto with sequences. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* Postponement. *) + +Lemma crarys_lemma7: + commutation22 + ipcbv cbv + cbv pcbv. +Local Ltac ih7 := + match goal with IH: forall u, cbv _ u -> _, h: cbv _ _ |- _ => + generalize (IH _ h) + end; intros (?&?&?). +Proof. + unfold commutation22. + induction 1; intros; subst; + try solve [ false; eauto 2 with obvious ]. + (* IRedAppLRNonValue *) + { invert_cbv. ih7. obvious. } + (* IRedAppLR *) + { (* [t1] and [t2] are values. *) + clear IHipcbv1. + invert_cbv. + (* Case: [u1] and [u2] are values. (Case 5 in Crary's proof.) *) + { assert (is_value u1). { obvious. } + inv ipcbv. + eexists; split. + { eapply RedBetaV; obvious. } + { eauto 7 with obvious. } + } + (* Case: [u1] and [u2] are nonvalues. (Case 4 in Crary's proof.) *) + { ih7. eexists; split; obvious. } + } + (* IRedLetLR *) + { invert_cbv. + (* Case: [t1] and [t2] are values. *) + { eexists; split; eauto 8 with obvious. } + (* Case: [t1] and [t2] are nonvalues. *) + { ih7. eexists; split; obvious. } + } +Qed. + +(* Internal parallel reduction commutes with reduction, as follows. *) + +Lemma crarys_lemma8_plus: + commutation22 + ipcbv cbv + (plus cbv) ipcbv. +Proof. + eauto using crarys_lemma7, crarys_main_lemma_plus, + commutation22_transitive. +Qed. + +Lemma crarys_lemma8: + commutation22 + ipcbv cbv + (star cbv) ipcbv. +Proof. + eauto using crarys_lemma8_plus, commutation22_variance with sequences. +Qed. + +Lemma crarys_lemma8b_plus: + commutation22 + ipcbv (plus cbv) + (plus cbv) ipcbv. +Proof. + eauto using commute_R_Splus, crarys_lemma8_plus. +Qed. + +Lemma crarys_lemma8b: + commutation22 + ipcbv (star cbv) + (star cbv) ipcbv. +Proof. + eauto using commute_R_Sstar, crarys_lemma8. +Qed. + +Lemma crarys_lemma8b_plus_star: + commutation22 + (star ipcbv) (plus cbv) + (plus cbv) (star ipcbv). +Proof. + eapply commute_Rstar_Splus. + eauto using crarys_lemma8b_plus, commutation22_variance with sequences. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* Bifurcation. *) + +(* A sequence of parallel reduction steps can be reformulated as a sequence + of ordinary reduction steps, followed with a sequence of internal parallel + reduction steps. *) + +Lemma crarys_lemma9: + forall t1 t2, + star pcbv t1 t2 -> + exists t, + star cbv t1 t /\ star ipcbv t t2. +Proof. + induction 1. + { eauto with sequences. } + { unpack. + forward1 crarys_main_lemma. + forward2 crarys_lemma8b. + eauto with sequences. } +Qed. + +(* The following result does not seem to explicitly appear in Crary's paper. *) + +Lemma pcbv_cbv_commutation1: + commutation22 + (star pcbv) cbv + (plus cbv) (star pcbv). +Proof. + intros t1 t2 ? t3 ?. + forward1 crarys_lemma9. + assert (plus cbv t2 t3). { eauto with sequences. } + forward2 crarys_lemma8b_plus_star. + eauto 6 using ipcbv_subset_pcbv, star_covariant with sequences. +Qed. + +Lemma pcbv_cbv_commutation: + commutation22 + (star pcbv) (plus cbv) + (plus cbv) (star pcbv). +Proof. + eauto using pcbv_cbv_commutation1, commute_R_Splus. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* The notion of "reducing (in zero or more steps) to a value" is the same + under [pcbv] and under [cbv]. *) + +Lemma equiconvergence: + forall t v, + star pcbv t v -> + is_value v -> + exists v', + star cbv t v' /\ is_value v'. +Proof. + intros. forward1 crarys_lemma9. eauto. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* "Adequacy of reduction". In Crary's terminology, "reduction" is the + compatible closure of "evaluation", and "evaluation" is [cbv]. A + relation is adequate iff it is contained in contextual equivalence. *) + +(* The adequacy theorem. (Crary's lemma 10.) *) + +Theorem pcbv_adequacy: + forall t1 t2, + star pcbv t1 t2 -> + (halts cbv t1) <-> (halts cbv t2). +Proof. + split. + (* Case: [t1] reduces to an irreducible term [u1]. *) + { intros (u1&?&?). + (* [t1] reduces via [pcbv*] to both [u1] and [t2], so they must both + reduce via [pcbv*] to some common term [u]. *) + assert (star pcbv t1 u1). { eauto using star_covariant, cbv_subset_pcbv. } + forward2 diamond_star_pcbv. + (* The reduction of [t2] to [u] can be bifurcated. That is, [t2] first + reduces via [cbv*], then via [ipbcv], to [u]. *) + forward1 crarys_lemma9. + (* Because [pcbv] and [ipcbv] (reversed) both preserve irreducibility, + this establishes that [t2] halts. *) + eexists. split; eauto with irred. + } + (* Case: [t2] reduces to an irreducible term [u2]. *) + { intros (u2&?&?). + (* Therefore, [t1] reduces via [pcbv*] to [u2]. *) + assert (star pcbv t1 u2). + { eauto using cbv_subset_pcbv, star_covariant with sequences. } + (* This reduction can be bifurcated. That is, [t1] first reduces via + [cbv*], then via [ipcbv], to [u2]. *) + forward1 crarys_lemma9. + (* Because [ipcbv] (reversed) preserves irreducibility, this proves + that [t1] halts. *) + eexists. split; eauto with irred. + } +Qed. + +(* The previous result implies that [pcbv] and [star pcbv] are contained in + contextual equivalence. We do not establish this result, because we do + not need it, and we have not defined contextual equivalence. *) + +(* -------------------------------------------------------------------------- *) + +(* Preservation of divergence. *) + +(* If we have an infinite [cbv] reduction sequence with [pcbv] steps in it, + then we have an infinite [cbv] reduction sequence. *) + +Lemma pcbv_preserves_divergence: + forall t, + infseq (composition (plus cbv) pcbv) t -> + infseq cbv t. +Proof. + intros ? Hinfseq. + (* We generalize the statement slightly by allowing any number of initial + [pcbv] steps from [t] to [u] before finding an infinite reduction sequence + out of [u]. *) + eapply infseq_coinduction_principle with (P := fun t => + exists u, star pcbv t u /\ infseq (composition (plus cbv) pcbv) u + ); [| eauto with sequences ]. + (* We have to show that, under this hypothesis, we are able to take one step + of [cbv] out of [t] and reach a term that satisfies this hypothesis again. *) + clear dependent t. intros t (u&?&hInfSeq). + pick infseq invert. + pick @composition invert. unpack. + (* Out of [t], we have [pcbv* . cbv+ . pcbv ...]. *) + (* Thus, we have [cbv+ . pcbv* . pcbv ...]. *) + forward2 pcbv_cbv_commutation. + (* Thus, we have [cbv . pcbv* ...]. *) + pick plus invert. + (* We are happy. *) + eexists. split; [ eauto |]. + eexists. split; [| eauto ]. + eauto 6 using cbv_subset_pcbv, star_covariant with sequences. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* The final result in Crary's paper is a standardization theorem for + call-by-value reduction. The theorem states that any sequence of parallel + reduction steps can be put in a "standard" form, as defined by the relation + [stdred] below. *) + +Inductive stdred : term -> term -> Prop := +| StdNil: + forall t, + stdred t t +| StdCons: + forall t1 t2 t3, + cbv t1 t2 -> + stdred t2 t3 -> + stdred t1 t3 +| StdLam: + forall t1 t2, + stdred t1 t2 -> + stdred (Lam t1) (Lam t2) +| StdApp: + forall t1 t2 u1 u2, + stdred t1 u1 -> + stdred t2 u2 -> + stdred (App t1 t2) (App u1 u2) +| StdLet: + forall t1 t2 u1 u2, + stdred t1 u1 -> + stdred t2 u2 -> + stdred (Let t1 t2) (Let u1 u2) +. + +Hint Constructors stdred : stdred. + +(* A couple of more flexible constructors for [stdred]. *) + +Lemma star_cbv_subset_stdred: + forall t1 t2, + star cbv t1 t2 -> + stdred t1 t2. +Proof. + induction 1; eauto with stdred. +Qed. + +Lemma StdConsStar: + forall t1 t2 t3, + star cbv t1 t2 -> + stdred t2 t3 -> + stdred t1 t3. +Proof. + induction 1; eauto with stdred. +Qed. + +Hint Resolve star_cbv_subset_stdred StdConsStar : stdred. + +(* The following four lemmas analyze a reduction sequence of the form [star + ipcbv t1 t2], where the head constructor of the term [t2] is known. In + every case, it can be concluded that the term [t1] exhibits the same head + constructor. *) + +Lemma star_ipcbv_into_Var: + forall {t1 t2}, star ipcbv t1 t2 -> + forall {x}, t2 = Var x -> t1 = Var x. +Proof. + induction 1; intros; subst. + { eauto. } + { forward (IHstar _ eq_refl). inv ipcbv. eauto. } +Qed. + +Lemma star_ipcbv_into_Lam: + forall {t1 t2}, star ipcbv t1 t2 -> + forall {u2}, t2 = Lam u2 -> + exists u1, t1 = Lam u1 /\ star pcbv u1 u2. +Proof. + induction 1; intros; subst. + { eauto with sequences. } + { forward (IHstar _ eq_refl). inv ipcbv. eauto with sequences. } +Qed. + +Lemma star_ipcbv_into_App: + forall {t1 t2}, star ipcbv t1 t2 -> + forall {t21 t22}, t2 = App t21 t22 -> + exists t11 t12, + t1 = App t11 t12 /\ star pcbv t11 t21 /\ star pcbv t12 t22. +Proof. + induction 1; intros; subst. + { eauto with sequences. } + { forward (IHstar _ _ eq_refl). inv ipcbv; + eauto 9 using ipcbv_subset_pcbv with sequences. } +Qed. + +Lemma star_ipcbv_into_Let: + forall {t1 t2}, star ipcbv t1 t2 -> + forall {t21 t22}, t2 = Let t21 t22 -> + exists t11 t12, + t1 = Let t11 t12 /\ star ipcbv t11 t21 /\ star pcbv t12 t22. +Proof. + induction 1; intros; subst. + { eauto with sequences. } + { forward (IHstar _ _ eq_refl). inv ipcbv. eauto 9 with sequences. } +Qed. + +Ltac star_ipcbv_into := + pick (star ipcbv) ltac:(fun h => first [ + forward (star_ipcbv_into_Var h eq_refl) + | forward (star_ipcbv_into_Lam h eq_refl) + | forward (star_ipcbv_into_App h eq_refl) + | forward (star_ipcbv_into_Let h eq_refl) + ]). + +(* The standardization theorem. (Crary's lemma 12.) *) + +Theorem cbv_standardization: + forall t2 t1, + star pcbv t1 t2 -> + stdred t1 t2. +Proof. + induction t2; intros; + forward1 crarys_lemma9; + star_ipcbv_into; + eauto 8 using ipcbv_subset_pcbv, star_covariant with stdred. +Qed. diff --git a/coq/Relations.v b/coq/Relations.v new file mode 100644 index 0000000..2251b7e --- /dev/null +++ b/coq/Relations.v @@ -0,0 +1,498 @@ +Require Import Coq.Setoids.Setoid. +Require Import MyTactics. +Require Import Sequences. + +(* This file offers a few definitions and tactics that help deal with + relations and commutative diagrams. *) + +(* -------------------------------------------------------------------------- *) + +Section Relations. + +Context {A : Type}. + +Implicit Types R S : A -> A -> Prop. + +(* Composition of relations. *) + +Definition composition R S a c := + exists b, R a b /\ S b c. + +(* Transposition of relations. *) + +Definition transpose R a b := + R b a. + +(* Inclusion of relations. *) + +Definition inclusion R S := + forall a b, R a b -> S a b. + +(* A typical (square) commutative diagram, where the composition [R; S] can be + replaced with the composition [S; R]. This notion can be stated in several + equivalent ways; see [commutation22_eq] and [commutation22_reverse]. *) + +Definition commutation22 R S S' R' := + forall a1 b1, + R a1 b1 -> + forall b2, + S b1 b2 -> + exists a2, + S' a1 a2 /\ R' a2 b2. + +(* A typical diamond diagram, where a divergence [R | S] is resolved + via [S' | R']. *) + +Definition diamond22 R S R' S' := + forall a1 b1, + R a1 b1 -> + forall a2, + S a1 a2 -> + exists b2, + R' a2 b2 /\ S' b1 b2. + +Definition diamond R := + diamond22 R R R R. + +End Relations. + +(* -------------------------------------------------------------------------- *) + +(* The tactic [forward1 lemma] applies [lemma], forwards, to a hypothesis + found in the context. The lemma must have one hypothesis. *) + +Ltac forward1 lemma := + match type of lemma with + | (forall _ _, ?R _ _ -> _) => + match goal with hR: R ?a1 ?b1 |- _ => + generalize (lemma _ _ hR); intro + end + | (forall _, ?R _ _ -> _) => + match goal with hR: R ?a1 ?b1 |- _ => + generalize (lemma _ hR); intro + end + end; + unpack. + +(* The tactic [forward2 lemma] applies [lemma], forwards, to two hypotheses + found in the context. The lemma must be a commutation lemma or a diamond + lemma, as defined above. *) + +Ltac forward2 lemma := + match type of lemma with + | (forall a1 b1, ?R a1 b1 -> forall b2, ?S b1 b2 -> _) => + match goal with hR: R ?a1 ?b1, hS: S ?b1 ?b2 |- _ => + generalize (lemma _ _ hR _ hS); intro + end + | commutation22 ?R ?S _ _ => + match goal with hR: R ?a1 ?b1, hS: S ?b1 ?b2 |- _ => + generalize (lemma _ _ hR _ hS); intro + end + | (forall a1 b1, ?R a1 b1 -> forall a2, ?S a1 a2 -> _) => + match goal with hR: R ?a1 ?b1, hS: S ?a1 ?a2 |- _ => + generalize (lemma _ _ hR _ hS); intro + end + | diamond22 ?R ?S _ _ => + match goal with hR: R ?a1 ?b1, hS: S ?a1 ?a2 |- _ => + generalize (lemma _ _ hR _ hS); intro + end + | diamond ?R => + match goal with hR: R ?a1 ?b1, hS: R ?a1 ?a2 |- _ => + generalize (lemma _ _ hR _ hS); intro + end + end; + unpack. + +(* -------------------------------------------------------------------------- *) + +Section RelationLemmas. + +Context {A : Type}. + +Implicit Types R S : A -> A -> Prop. + +(* Inclusion of relations is transitive. *) + +Lemma inclusion_transitive: + forall R S T, + inclusion R S -> + inclusion S T -> + inclusion R T. +Proof. + unfold inclusion. eauto. +Qed. + +(* [star] is covariant with respect to inclusion. *) + +Lemma star_covariant_inclusion: + forall R S, + inclusion R S -> + inclusion (star R) (star S). +Proof. + unfold inclusion. eauto using star_covariant. +Qed. + +(* If [R] is reflexive and transitive, then [star R] is [R]. *) + +Lemma star_of_reflexive_transitive_relation: + forall {A} (R : A -> A -> Prop), + (forall a, R a a) -> + (forall a b c, R a b -> R b c -> R a c) -> + inclusion (star R) R. +Proof. + intros. induction 1; eauto. +Qed. + +(* Thus, [star (star R)] is [star R]. *) + +Lemma inclusion_star_star: + forall {A} (R : A -> A -> Prop), + inclusion (star (star R)) (star R). +Proof. + intros. + eapply star_of_reflexive_transitive_relation; eauto with sequences. +Qed. + +(* Composition is associative. *) + +Lemma composition_assoc_direct: + forall R S T, + inclusion + (composition R (composition S T)) + (composition (composition R S) T). +Proof. + unfold inclusion, composition. intros. unpack. eauto. +Qed. + +Lemma composition_assoc_reverse: + forall R S T, + inclusion + (composition (composition R S) T) + (composition R (composition S T)). +Proof. + unfold inclusion, composition. intros. unpack. eauto. +Qed. + +(* Composition is covariant. *) + +Lemma composition_covariant: + forall R1 R2 S1 S2, + inclusion R1 R2 -> + inclusion S1 S2 -> + inclusion (composition R1 S1) (composition R2 S2). +Proof. + unfold inclusion, composition. intros. unpack. eauto. +Qed. + +(* A commutative diagram can be stated in terms of inclusion of relations. *) + +Lemma commutation22_eq: + forall R S S' R', + commutation22 R S S' R' <-> + inclusion (composition R S) (composition S' R'). +Proof. + intros. unfold commutation22, inclusion, composition. + split; intros; unpack. + { forward2 H. eauto. } + { eauto. } +Qed. + +(* Thus, two commutative diagrams can be glued. *) + +Lemma commutation22_transitive: + forall R S S' R' S'' R'', + commutation22 R S S' R' -> + commutation22 S' R' S'' R'' -> + commutation22 R S S'' R''. +Proof. + intros. rewrite !commutation22_eq in *. + eauto using inclusion_transitive. +Qed. + +(* A commutation diagram can also be stated with its two hypotheses in reverse + order. This can be useful, e.g. when the diagram must be established by + induction on the second hypothesis. *) + +Lemma commutation22_reverse: + forall R S S' R', + commutation22 R S S' R' <-> + ( + forall b1 b2, + S b1 b2 -> + forall a1, + R a1 b1 -> + exists a2, + S' a1 a2 /\ R' a2 b2 + ). +Proof. + unfold commutation22. split; eauto. +Qed. + +(* [commutation22 R S S' R'] is contravariant in [R] and [S] and + covariant in [S'] and [R']. *) + +Lemma commutation22_variance: + forall R1 S1 S'1 R'1 R2 S2 S'2 R'2, + commutation22 R1 S1 S'1 R'1 -> + (forall a b, R2 a b -> R1 a b) -> + (forall a b, S2 a b -> S1 a b) -> + (forall a b, S'1 a b -> S'2 a b) -> + (forall a b, R'1 a b -> R'2 a b) -> + commutation22 R2 S2 S'2 R'2. +Proof. + do 8 intro. intros Hcomm. do 4 intro. intros a1 b1 ? b2 ?. + assert (R1 a1 b1). { eauto. } + assert (S1 b1 b2). { eauto. } + forward2 Hcomm. eauto. +Qed. + +(* If [S] can move left through [R], giving rise to (zero or more) [S'], + then [star S] can move left through [R] in the same manner. Think of + many [S] sheep jumping right-to-left above one [R] barrier. *) + +(* If [R S ] rewrites to [S'* R] + then [R S*] rewrites to [S'* R]. *) + +(* If desired, [star S'] could be replaced in this statement with any + reflexive and transitive relation. *) + +Lemma commute_R_Sstar: + forall {R S S'}, + commutation22 + R S + (star S') R + -> + commutation22 + R (star S) + (star S') R. +Proof. + intros ? ? ? Hdiagram. + rewrite commutation22_reverse. + induction 1; intros. + { eauto with sequences. } + { forward2 Hdiagram. + forward1 IHstar. + eauto with sequences. } +Qed. + +(* An analogous result, with [plus] instead of [star]. *) + +(* If [R S ] rewrites to [S'+ R] + then [R S+] rewrites to [S'+ R]. *) + +(* If desired, [plus S'] could be replaced in this statement with any + transitive relation. *) + +Lemma commute_R_Splus: + forall {R S S'}, + commutation22 + R S + (plus S') R + -> + commutation22 + R (plus S) + (plus S') R. +Proof. + intros ? ? ? Hcomm. + rewrite commutation22_reverse. + induction 1 using plus_ind_direct; intros. + (* Case: one step. *) + { forward2 Hcomm. eauto. } + (* Case: more than one step. *) + { forward2 Hcomm. + forward1 IHplus. + eauto with sequences. } +Qed. + +(* If [S] can move left through [R], giving rise to (zero or more) [S], + then [S] can move left through [star R]. Think of many [S] sheep jumping + right-to-left above many [R] barriers. *) + +(* If [R S ] rewrites to [S* R ] + then [R* S*] rewrites to [S* R*]. *) + +Lemma commute_Rstar_Sstar: + forall {R S}, + commutation22 + R S + (star S) R + -> + commutation22 + (star R) (star S) + (star S) (star R). +Proof. + intros ? ? Hdiagram. + induction 1; intros. + { eauto with sequences. } + { forward1 IHstar. + forward2 (commute_R_Sstar Hdiagram). + eauto with sequences. } +Qed. + +(* If [R S] rewrites to [S+ R ] + then [R* S] rewrites to [S+ R*]. *) + +Lemma commute_Rstar_S: + forall {R S}, + commutation22 + R S + (plus S) R + -> + commutation22 + (star R) S + (plus S) (star R). +Proof. + intros ? ? Hdiagram. + induction 1; intros. + { eauto with sequences. } + { forward1 IHstar. + forward2 (commute_R_Splus Hdiagram). + eauto with sequences. } +Qed. + +(* If [R S ] rewrites to [S+ R ] + then [R* S+] rewrites to [S+ R*]. *) + +Lemma commute_Rstar_Splus: + forall {R S}, + commutation22 + R S + (plus S) R + -> + commutation22 + (star R) (plus S) + (plus S) (star R). +Proof. + intros ? ? Hdiagram. + assert (Hdiagram2: + commutation22 + (star R) (star S) + (star S) (star R) + ). + { eapply commute_Rstar_Sstar. + eauto using commutation22_variance with sequences. } + (* We have [R* S+]. *) + induction 2; intros. + (* We have [R* S S*]. *) + forward2 (commute_Rstar_S Hdiagram). + (* We have [S+ R* S*]. *) + forward2 Hdiagram2. + (* We have [S+ S* R*]. *) + eauto with sequences. +Qed. + +(* [transpose] is involutive. *) + +Lemma transpose_transpose: + forall R, + transpose (transpose R) = R. +Proof. + reflexivity. (* it's just eta-expansion *) +Qed. + +(* [diamond22] can be viewed as an instance of [commutation22]. *) + +Lemma diamond22_as_commutation22: + forall R S R' S', + diamond22 R S R' S' <-> + commutation22 (transpose R) S S' (transpose R'). +Proof. + unfold diamond22, commutation22. split; intros H; intros. + { unfold transpose in *. forward2 H. eauto. } + { assert (transpose R b1 a1). { eauto. } + forward2 H. eauto. } +Qed. + +Lemma commutation22_as_diamond22: + forall R S R' S', + commutation22 R S S' R' <-> + diamond22 (transpose R) S (transpose R') S'. +Proof. + intros. + rewrite diamond22_as_commutation22. + rewrite !transpose_transpose. tauto. +Qed. + +(* [diamond22 is symmetric. *) + +Lemma diamond22_symmetric: + forall R S R' S', + diamond22 R S R' S' -> + diamond22 S R S' R'. +Proof. + intros ? ? ? ? Hcon. + unfold diamond22. intros. + forward2 Hcon. eauto. +Qed. + +(* If [R] is diamond, then [star R] is diamond. *) + +Lemma star_diamond_left: + forall R R' S, + diamond22 R S R' S -> + diamond22 (star R) S (star R') S. +Proof. + intros R R' S Hcon. induction 1; intros. + { eauto with sequences. } + { forward2 Hcon. forward1 IHstar. eauto with sequences. } +Qed. + +Lemma star_diamond_right: + forall R S S', + diamond22 R S R S' -> + diamond22 R (star S) R (star S'). +Proof. + eauto using star_diamond_left, diamond22_symmetric. +Qed. + +Lemma star_diamond_both: + forall R S, + diamond22 R S R S -> + diamond22 (star R) (star S) (star R) (star S). +Proof. + eauto using star_diamond_left, star_diamond_right. +Qed. + +Lemma star_diamond: + forall R, + diamond R -> + diamond (star R). +Proof. + unfold diamond. eauto using star_diamond_both. +Qed. + +(* If, through a simulation diagram, a step of [R] in the source is + translated to (at least) one step of [R'] in the target, then + divergence in the source implies divergence in the target. *) + +Lemma infseq_simulation: + forall R R' S, + diamond22 R S R' S -> + forall a, + infseq R a -> + forall b, + S a b -> + infseq R' b. +Proof. + intros. + eapply infseq_coinduction_principle + with (P := fun b => exists a, S a b /\ infseq R a); [| eauto ]. + clear dependent a. clear b. intros b (a&?&?). + pick infseq invert. + pick @diamond22 forward2. + eauto with sequences. +Qed. + +Lemma infseq_simulation_plus: + forall R R' S, + diamond22 R S (plus R') S -> + forall a, + infseq R a -> + forall b, + S a b -> + infseq R' b. +Proof. + eauto using infseq_simulation, infseq_plus. +Qed. + +End RelationLemmas. From b88b92d33cb01f7e5185ed2a69f561093686fcbb Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Franc=CC=A7ois=20Pottier?= Date: Wed, 11 Oct 2017 19:47:20 +0200 Subject: [PATCH 48/81] Add new OCaml files. --- ocaml/EvalCBNCPS.ml | 164 ++++++++++++++++++++++++++++++++++ ocaml/EvalCBVCPS.ml | 208 ++++++++++++++++++++++++++++++++++++++++++++ ocaml/Graph.ml | 203 ++++++++++++++++++++++++++++++++++++++++++ 3 files changed, 575 insertions(+) create mode 100644 ocaml/EvalCBNCPS.ml create mode 100644 ocaml/EvalCBVCPS.ml create mode 100644 ocaml/Graph.ml diff --git a/ocaml/EvalCBNCPS.ml b/ocaml/EvalCBNCPS.ml new file mode 100644 index 0000000..4945eeb --- /dev/null +++ b/ocaml/EvalCBNCPS.ml @@ -0,0 +1,164 @@ +(* -------------------------------------------------------------------------- *) + +(* The type of lambda-terms, in de Bruijn's representation. *) + +type var = int (* a de Bruijn index *) +type term = + | Var of var + | Lam of (* bind: *) term + | App of term * term + | Let of (* bind: *) term * term + +(* -------------------------------------------------------------------------- *) + +(* Under a call-by-name regime, in a function call, the actual argument is not + evaluated immediately; instead, a thunk is built (a pair of the argument + and the environment in which it must be evaluated). Thus, an environment is + a list of thunks. As in call-by-value, a closure is a pair of a term and an + environment. (Closures and thunks differ in that a closure binds a + variable, the formal argument, in the term. A thunk does not.) *) + +type cvalue = + | Clo of (* bind: *) term * cenv + +and cenv = + thunk list + +and thunk = + | Thunk of term * cenv + +(* -------------------------------------------------------------------------- *) + +(* Environments. *) + +let empty : cenv = + [] + +exception RuntimeError + +let lookup (e : cenv) (x : var) : thunk = + try + List.nth e x + with Failure _ -> + raise RuntimeError + +(* -------------------------------------------------------------------------- *) + +(* An environment-based big-step call-by-name interpreter. *) + +let rec eval (e : cenv) (t : term) : cvalue = + match t with + | Var x -> + let Thunk (t, e) = lookup e x in + eval e t + | Lam t -> + Clo (t, e) + | App (t1, t2) -> + let cv1 = eval e t1 in + let Clo (u1, e') = cv1 in + eval (Thunk(t2, e) :: e') u1 + | Let (t1, t2) -> + eval (Thunk (t1, e) :: e) t2 + +(* -------------------------------------------------------------------------- *) + +(* The CPS-transformed interpreter. *) + +let rec evalk (e : cenv) (t : term) (k : cvalue -> 'a) : 'a = + match t with + | Var x -> + let Thunk (t, e) = lookup e x in + evalk e t k + | Lam t -> + k (Clo (t, e)) + | App (t1, t2) -> + evalk e t1 (fun cv1 -> + let Clo (u1, e') = cv1 in + evalk (Thunk(t2, e) :: e') u1 k) + | Let (t1, t2) -> + evalk (Thunk (t1, e) :: e) t2 k + +let eval (e : cenv) (t : term) : cvalue = + evalk e t (fun cv -> cv) + +(* -------------------------------------------------------------------------- *) + +(* The CPS-transformed, defunctionalized interpreter. *) + +type kont = + | AppL of { e: cenv; t2: term; k: kont } + | Init + +let rec evalkd (e : cenv) (t : term) (k : kont) : cvalue = + match t with + | Var x -> + let Thunk (t, e) = lookup e x in + evalkd e t k + | Lam t -> + apply k (Clo (t, e)) + | App (t1, t2) -> + evalkd e t1 (AppL { e; t2; k }) + | Let (t1, t2) -> + evalkd (Thunk (t1, e) :: e) t2 k + +and apply (k : kont) (cv : cvalue) : cvalue = + match k with + | AppL { e; t2; k } -> + let cv1 = cv in + let Clo (u1, e') = cv1 in + evalkd (Thunk(t2, e) :: e') u1 k + | Init -> + cv + +let eval (e : cenv) (t : term) : cvalue = + evalkd e t Init + +(* -------------------------------------------------------------------------- *) + +(* Because [apply] has only one call site, it can be inlined, yielding a + slightly more compact and readable definition. *) + +let rec evalkd (e : cenv) (t : term) (k : kont) : cvalue = + match t, k with + | Var x, _ -> + let Thunk (t, e) = lookup e x in + evalkd e t k + | Lam t, AppL { e; t2; k } -> + let cv1 = Clo (t, e) in + let Clo (u1, e') = cv1 in + evalkd (Thunk(t2, e) :: e') u1 k + | Lam t, Init -> + Clo (t, e) + | App (t1, t2), _ -> + evalkd e t1 (AppL { e; t2; k }) + | Let (t1, t2), _ -> + evalkd (Thunk (t1, e) :: e) t2 k + +let eval (e : cenv) (t : term) : cvalue = + evalkd e t Init + +(* -------------------------------------------------------------------------- *) + +(* The type [kont] is isomorphic to [(cenv * term) list]. Using the latter + type makes the code slightly prettier, although slightly less efficient. *) + +(* At this point, one recognizes Krivine's machine. *) + +let rec evalkd (e : cenv) (t : term) (k : (cenv * term) list) : cvalue = + match t, k with + | Var x, _ -> + let Thunk (t, e) = lookup e x in + evalkd e t k + | Lam t, (e, t2) :: k -> + let cv1 = Clo (t, e) in + let Clo (u1, e') = cv1 in + evalkd (Thunk(t2, e) :: e') u1 k + | Lam t, [] -> + Clo (t, e) + | App (t1, t2), _ -> + evalkd e t1 ((e, t2) :: k) + | Let (t1, t2), _ -> + evalkd (Thunk (t1, e) :: e) t2 k + +let eval (e : cenv) (t : term) : cvalue = + evalkd e t [] diff --git a/ocaml/EvalCBVCPS.ml b/ocaml/EvalCBVCPS.ml new file mode 100644 index 0000000..1bba7eb --- /dev/null +++ b/ocaml/EvalCBVCPS.ml @@ -0,0 +1,208 @@ +(* -------------------------------------------------------------------------- *) + +(* The type of lambda-terms, in de Bruijn's representation. *) + +type var = int (* a de Bruijn index *) +type term = + | Var of var + | Lam of (* bind: *) term + | App of term * term + | Let of (* bind: *) term * term + +(* -------------------------------------------------------------------------- *) + +(* An environment-based big-step interpreter. This is the same interpreter + that we programmed in Coq, except here, in OCaml, fuel is not needed. *) + +type cvalue = + | Clo of (* bind: *) term * cenv + +and cenv = + cvalue list + +let empty : cenv = + [] + +exception RuntimeError + +let lookup (e : cenv) (x : var) : cvalue = + try + List.nth e x + with Failure _ -> + raise RuntimeError + +let rec eval (e : cenv) (t : term) : cvalue = + match t with + | Var x -> + lookup e x + | Lam t -> + Clo (t, e) + | App (t1, t2) -> + let cv1 = eval e t1 in + let cv2 = eval e t2 in + let Clo (u1, e') = cv1 in + eval (cv2 :: e') u1 + | Let (t1, t2) -> + eval (eval e t1 :: e) t2 + +(* -------------------------------------------------------------------------- *) + +(* Term/value/environment printers. *) + +open Printf + +let rec print_term f = function + | Var x -> + fprintf f "(Var %d)" x + | Lam t -> + fprintf f "(Lam %a)" print_term t + | App (t1, t2) -> + fprintf f "(App %a %a)" print_term t1 print_term t2 + | Let (t1, t2) -> + fprintf f "(Let %a %a)" print_term t1 print_term t2 + +let rec print_cvalue f = function + | Clo (t, e) -> + fprintf f "< %a | %a >" print_term t print_cenv e + +and print_cenv f = function + | [] -> + fprintf f "[]" + | cv :: e -> + fprintf f "%a :: %a" print_cvalue cv print_cenv e + +let print_cvalue cv = + fprintf stdout "%a\n" print_cvalue cv + +(* -------------------------------------------------------------------------- *) + +(* A tiny test suite. *) + +let id = + Lam (Var 0) + +let idid = + App (id, id) + +let apply = + Lam (Lam (App (Var 1, Var 0))) + +let test1 eval t = + print_cvalue (eval empty t) + +let test name eval = + printf "Testing %s...\n%!" name; + test1 eval idid; + test1 eval (App (apply, id)); + test1 eval (App (App (apply, id), id)); + () + +(* -------------------------------------------------------------------------- *) + +(* Test. *) + +let () = + test "the direct-style evaluator" eval + +(* -------------------------------------------------------------------------- *) + +(* A CPS-transformed, environment-based big-step interpreter. *) + +(* In this code, every recursive call to [evalk] is a tail call. Thus, + it is compiled by the OCaml compiler to a loop, and requires only O(1) + space on the OCaml stack. *) + +let rec evalk (e : cenv) (t : term) (k : cvalue -> 'a) : 'a = + match t with + | Var x -> + k (lookup e x) + | Lam t -> + k (Clo (t, e)) + | App (t1, t2) -> + evalk e t1 (fun cv1 -> + evalk e t2 (fun cv2 -> + let Clo (u1, e') = cv1 in + evalk (cv2 :: e') u1 k)) + | Let (t1, t2) -> + evalk e t1 (fun cv1 -> + evalk (cv1 :: e) t2 k) + +(* It is possible to explicitly assert that these calls are tail calls. + The compiler would tell us if we were wrong. *) + +let rec evalk (e : cenv) (t : term) (k : cvalue -> 'a) : 'a = + match t with + | Var x -> + (k[@tailcall]) (lookup e x) + | Lam t -> + (k[@tailcall]) (Clo (t, e)) + | App (t1, t2) -> + (evalk[@tailcall]) e t1 (fun cv1 -> + (evalk[@tailcall]) e t2 (fun cv2 -> + let Clo (u1, e') = cv1 in + (evalk[@tailcall]) (cv2 :: e') u1 k)) + | Let (t1, t2) -> + (evalk[@tailcall]) e t1 (fun cv1 -> + (evalk[@tailcall]) (cv1 :: e) t2 k) + +let eval (e : cenv) (t : term) : cvalue = + evalk e t (fun cv -> cv) + +(* -------------------------------------------------------------------------- *) + +(* Test. *) + +let () = + test "the CPS evaluator" eval + +(* -------------------------------------------------------------------------- *) + +(* The above code uses heap-allocated closures, which form a linked list in the + heap. In fact, the interpreter's "stack" is now heap-allocated. To see this + more clearly, let us defunctionalize the CPS-transformed interpreter. *) + +(* There are four places in the above code where an anonymous continuation is + built, so defunctionalization yields a data type of four possible kinds of + continuations. This data type describes a linked list of stack frames! *) + +type kont = + | AppL of { e: cenv; t2: term; k: kont } + | AppR of { cv1: cvalue; k: kont } + | LetL of { e: cenv; t2: term; k: kont } + | Init + +let rec evalkd (e : cenv) (t : term) (k : kont) : cvalue = + match t with + | Var x -> + apply k (lookup e x) + | Lam t -> + apply k (Clo (t, e)) + | App (t1, t2) -> + evalkd e t1 (AppL { e; t2; k }) + | Let (t1, t2) -> + evalkd e t1 (LetL { e; t2; k }) + +and apply (k : kont) (cv : cvalue) : cvalue = + match k with + | AppL { e; t2; k } -> + let cv1 = cv in + evalkd e t2 (AppR { cv1; k }) + | AppR { cv1; k } -> + let cv2 = cv in + let Clo (u1, e') = cv1 in + evalkd (cv2 :: e') u1 k + | LetL { e; t2; k } -> + let cv1 = cv in + evalkd (cv1 :: e) t2 k + | Init -> + cv + +let eval e t = + evalkd e t Init + +(* -------------------------------------------------------------------------- *) + +(* Test. *) + +let () = + test "the defunctionalized CPS evaluator" eval diff --git a/ocaml/Graph.ml b/ocaml/Graph.ml new file mode 100644 index 0000000..3b4f08c --- /dev/null +++ b/ocaml/Graph.ml @@ -0,0 +1,203 @@ +open Printf + +(* -------------------------------------------------------------------------- *) + +(* A simple type of binary trees. *) + +type tree = + | Leaf + | Node of { data: int; left: tree; right: tree } + +(* -------------------------------------------------------------------------- *) + +(* Constructors. *) + +let node data left right = + Node { data; left; right } + +let singleton data = + node data Leaf Leaf + +(* -------------------------------------------------------------------------- *) + +(* A sample tree. *) + +let christmas = + node 6 + (node 2 (singleton 0) (singleton 1)) + (node 5 (singleton 3) (singleton 4)) + +(* -------------------------------------------------------------------------- *) + +(* A test procedure. *) + +let test name walk = + printf "Testing %s...\n%!" name; + walk christmas; + walk christmas; + flush stdout + +(* -------------------------------------------------------------------------- *) + +(* A recursive depth-first traversal, with postfix printing. *) + +let rec walk (t : tree) : unit = + match t with + | Leaf -> + () + | Node { data; left; right } -> + walk left; + walk right; + printf "%d\n" data + +let () = + test "walk" walk + +(* -------------------------------------------------------------------------- *) + +(* A CPS traversal. *) + +let rec walkk (t : tree) (k : unit -> 'a) : 'a = + match t with + | Leaf -> + k() + | Node { data; left; right } -> + walkk left (fun () -> + walkk right (fun () -> + printf "%d\n" data; + k())) + +let walk t = + walkk t (fun t -> t) + +let () = + test "walkk" walk + +(* -------------------------------------------------------------------------- *) + +(* A CPS-defunctionalized traversal. *) + +type kont = + | Init + | GoneL of { data: int; tail: kont; right: tree } + | GoneR of { data: int; tail: kont } + +let rec walkkd (t : tree) (k : kont) : unit = + match t with + | Leaf -> + apply k () + | Node { data; left; right } -> + walkkd left (GoneL { data; tail = k; right }) + +and apply k () = + match k with + | Init -> + () + | GoneL { data; tail; right } -> + walkkd right (GoneR { data; tail }) + | GoneR { data; tail } -> + printf "%d\n" data; + apply tail () + +let walk t = + walkkd t Init + +let () = + test "walkkd" walk + +(* CPS, defunctionalized, with in-place allocation of continuations. *) + +(* [Init] is encoded by [Leaf]. + + [GoneL { data; tail; right }] is encoded by: + - setting [status] to [GoneL]; and + - storing [tail] in the [left] field. + - the [data] and [right] fields retain their original value. + + [GoneR { data; tail }] is encoded by: + - setting [status] to [GoneR]; and + - storing [tail] in the [right] field. + - the [data] and [left] fields retain their original value. + + The [status] field is meaningful only when the memory block is + being viewed as a continuation. If it is being viewed as a tree, + then (by convention) [status] must be [GoneL]. (We could also + let the type [status] have three values, but I prefer to use just + two, for the sake of economy.) + + Does that sound crazy? Well, it is, in a way. *) + +type status = GoneL | GoneR +type mtree = Leaf | Node of { + data: int; mutable status: status; + mutable left: mtree; mutable right: mtree + } +type mkont = mtree + +(* Constructors. *) + +let node data left right = + Node { data; status = GoneL; left; right } + +let singleton data = + node data Leaf Leaf + +(* A sample tree. *) + +let christmas = + node 6 + (node 2 (singleton 0) (singleton 1)) + (node 5 (singleton 3) (singleton 4)) + +(* A test. *) + +let test name walk = + printf "Testing %s...\n%!" name; + walk christmas; + walk christmas; + flush stdout + +(* The code. *) + +let rec walkkdi (t : mtree) (k : mkont) : unit = + match t with + | Leaf -> + (* We decide to let [apply] takes a tree as a second argument, + instead of just a unit value. Indeed, in order to restore + the [left] or [right] fields of [k], we need the address + of the child [t] out of which we are coming. *) + apply k t + | Node ({ left; _ } as n) -> + (* At this point, [t] is a tree. + [n] is a name for its root record. *) + (* Change this tree to a [GoneL] continuation. *) + assert (n.status = GoneL); + n.left (* n.tail *) <- k; + (* [t] now represents a continuation. Go down into the left + child, with this continuation. *) + walkkdi left (t : mkont) + +and apply (k : mkont) (child : mtree) : unit = + match k with + | Leaf -> () + | Node ({ status = GoneL; left = tail; right; _ } as n) -> + (* We are popping a [GoneL] frame, that is, coming out of + a left child. *) + n.status <- GoneR; (* update continuation! *) + n.left <- child; (* restore orig. left child! *) + n.right (* n.tail *) <- tail; + (* [k] now represents a [GoneR] continuation. Go down into + the right child, with [k] as a continuation. *) + walkkdi right k + | Node ({ data; status = GoneR; right = tail; _ } as n) -> + printf "%d\n" data; + n.status <- GoneL; (* change back to a tree! *) + n.right <- child; (* restore orig. right child! *) + (* [k] now represents a valid tree again. *) + apply tail (k : mtree) + +let walk t = + walkkdi t Leaf + +let () = + test "walkkdi" walk From 8c50463ec4f468369c698b6023b64599e907cfd9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Thu, 12 Oct 2017 15:40:11 +0200 Subject: [PATCH 49/81] New OCaml exercise. --- ocaml/EvalCBVExercise.ml | 71 ++++++++++++++++++++++++++++++++++++++++ 1 file changed, 71 insertions(+) create mode 100644 ocaml/EvalCBVExercise.ml diff --git a/ocaml/EvalCBVExercise.ml b/ocaml/EvalCBVExercise.ml new file mode 100644 index 0000000..36b11ca --- /dev/null +++ b/ocaml/EvalCBVExercise.ml @@ -0,0 +1,71 @@ +(* -------------------------------------------------------------------------- *) + +(* The type of lambda-terms, in de Bruijn's representation. *) + +type var = int (* a de Bruijn index *) +type term = + | Var of var + | Lam of (* bind: *) term + | App of term * term + | Let of (* bind: *) term * term + +(* -------------------------------------------------------------------------- *) + +(* An environment-based big-step interpreter. This is the same interpreter + that we programmed in Coq, except here, in OCaml, fuel is not needed. *) + +type cvalue = + | Clo of (* bind: *) term * cenv + +and cenv = + cvalue list + +let empty : cenv = + [] + +exception RuntimeError + +let lookup (e : cenv) (x : var) : cvalue = + try + List.nth e x + with Failure _ -> + raise RuntimeError + +let rec eval (e : cenv) (t : term) : cvalue = + match t with + | Var x -> + lookup e x + | Lam t -> + Clo (t, e) + | App (t1, t2) -> + let cv1 = eval e t1 in + let cv2 = eval e t2 in + let Clo (u1, e') = cv1 in + eval (cv2 :: e') u1 + | Let (t1, t2) -> + eval (eval e t1 :: e) t2 + +(* -------------------------------------------------------------------------- *) + +(* The CPS-transformed interpreter. *) + +let rec evalk (e : cenv) (t : term) (k : cvalue -> 'a) : 'a = + assert false + +let eval (e : cenv) (t : term) : cvalue = + evalk e t (fun cv -> cv) + +(* -------------------------------------------------------------------------- *) + +(* The CPS-transformed, defunctionalized interpreter. *) + +type kont + +let rec evalkd (e : cenv) (t : term) (k : kont) : cvalue = + assert false + +and apply (k : kont) (cv : cvalue) : cvalue = + assert false + +let eval e t = + evalkd e t (assert false) From 521f29c442ecc2e15ab9412a4ecf37f7811d3a85 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Franc=CC=A7ois=20Pottier?= Date: Thu, 12 Oct 2017 15:47:05 +0200 Subject: [PATCH 50/81] README. --- README.md | 10 +++++++++- 1 file changed, 9 insertions(+), 1 deletion(-) diff --git a/README.md b/README.md index aafe38e..d3940d3 100644 --- a/README.md +++ b/README.md @@ -82,7 +82,15 @@ We also show the limits of dependently-typed functional programming. closure conversion, defunctionalization ([slides 03](slides/fpottier-03.pdf)) ([Coq repo](coq/)). -* (13/10/2017) Compiling away the call stack: the CPS transformation. +* (13/10/2017) Making the stack explicit: the CPS transformation + ([slides 04](slides/fpottier-04.pdf)) + ([Coq repo](coq/)). + Transforming a call-by-value interpreter + ([exercise](ocaml/EvalCBVExercise.ml), [solution](ocaml/EvalCBVCPS.ml)). + Transforming a call-by-name interpreter + ([solution](ocaml/EvalCBNCPS.ml)). + Transforming a graph traversal + ([solution](ocaml/Graph.ml)). * (20/10/2017) Equational reasoning and program optimizations. ### Metatheory of Typed Programming Languages From de8d6fedd9c0d9456aae082e42e5d8d4a3d84173 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Franc=CC=A7ois=20Pottier?= Date: Fri, 13 Oct 2017 09:26:06 +0200 Subject: [PATCH 51/81] Add today's slides. --- slides/fpottier-04.pdf | Bin 0 -> 717559 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 slides/fpottier-04.pdf diff --git a/slides/fpottier-04.pdf b/slides/fpottier-04.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f9ee4453209534d108967af076a478441c726133 GIT binary patch literal 717559 zcmeFa2|QJ87dE~%C>j){NgU0S(1bFCkV0vmC8J&l1PJ6 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=?UTF-8?q?Franc=CC=A7ois=20Pottier?= Date: Fri, 13 Oct 2017 09:28:58 +0200 Subject: [PATCH 52/81] Nested lists. --- README.md | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/README.md b/README.md index d3940d3..00bcbd5 100644 --- a/README.md +++ b/README.md @@ -68,9 +68,9 @@ We also show the limits of dependently-typed functional programming. Introduction ([slides 00](slides/fpottier-00.pdf)). Syntax and operational semantics, on paper and on a machine ([slides 01a](slides/fpottier-01a.pdf)) - ([slides 01b](slides/fpottier-01b.pdf)) - ([OCaml solution to Newton-Raphson exercise](ocaml/NewtonRaphson.ml)) - ([Even.v](coq/Even.v)). + ([slides 01b](slides/fpottier-01b.pdf)). + * Newton-Raphson in OCaml ([solution](ocaml/NewtonRaphson.ml)). + * 1 is not even in Coq ([Even.v](coq/Even.v)). * (29/09/2017) From a small-step semantics down to an efficient verified interpreter, in several stages @@ -85,12 +85,12 @@ We also show the limits of dependently-typed functional programming. * (13/10/2017) Making the stack explicit: the CPS transformation ([slides 04](slides/fpottier-04.pdf)) ([Coq repo](coq/)). - Transforming a call-by-value interpreter - ([exercise](ocaml/EvalCBVExercise.ml), [solution](ocaml/EvalCBVCPS.ml)). - Transforming a call-by-name interpreter - ([solution](ocaml/EvalCBNCPS.ml)). - Transforming a graph traversal - ([solution](ocaml/Graph.ml)). + * Transforming a call-by-value interpreter + ([exercise](ocaml/EvalCBVExercise.ml), [solution](ocaml/EvalCBVCPS.ml)). + * Transforming a call-by-name interpreter + ([solution](ocaml/EvalCBNCPS.ml)). + * Transforming a graph traversal + ([solution](ocaml/Graph.ml)). * (20/10/2017) Equational reasoning and program optimizations. ### Metatheory of Typed Programming Languages From c0c83a490df25222016abb6eac267e34b8dbce0b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Fri, 20 Oct 2017 10:36:47 +0200 Subject: [PATCH 53/81] Add a demo of equational reasoning in Coq. --- coq/DemoEqReasoning.v | 73 +++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 73 insertions(+) create mode 100644 coq/DemoEqReasoning.v diff --git a/coq/DemoEqReasoning.v b/coq/DemoEqReasoning.v new file mode 100644 index 0000000..eade91a --- /dev/null +++ b/coq/DemoEqReasoning.v @@ -0,0 +1,73 @@ +Require Import List. + +Section Demo. + +(* -------------------------------------------------------------------------- *) + +Variables A B : Type. +Variable p : B -> bool. +Variable f : A -> B. + +(* The composition of [filter] and [map] can be computed by the specialized + function [filter_map]. *) + +Fixpoint filter_map xs := + match xs with + | nil => + nil + | cons x xs => + let y := f x in + if p y then y :: filter_map xs else filter_map xs + end. + +Lemma filter_map_spec: + forall xs, + filter p (map f xs) = filter_map xs. +Proof. + induction xs as [| x xs ]; simpl. + { reflexivity. } + { rewrite IHxs. reflexivity. } +Qed. + +(* -------------------------------------------------------------------------- *) + +(* [filter] and [map] commute in a certain sense. *) + +Variable q : A -> bool. + +Lemma filter_map_commute: + (forall x, p (f x) = q x) -> + forall xs, + filter p (map f xs) = map f (filter q xs). +Proof. + intros h. + induction xs as [| x xs ]; simpl; intros. + (* Case: [nil]. *) + { reflexivity. } + (* Case: [x :: xs]. *) + { rewrite h. + rewrite IHxs. + (* Case analysis: [q x] is either true or false. + In either case, the result is immediate. *) + destruct (q x); reflexivity. } +Qed. + +(* In a slightly stronger version of the lemma, the equality [p (f x) = q x] + needs to be proved only under the hypothesis that [x] is an element of the + list [xs]. *) + +Lemma filter_map_commute_stronger: + forall xs, + (forall x, In x xs -> p (f x) = q x) -> + filter p (map f xs) = map f (filter q xs). +Proof. + induction xs as [| x xs ]; simpl; intro h. + { reflexivity. } + { (* The proof is the same as above, except the two rewriting steps have + side conditions, which are immediately proved by [eauto]. *) + rewrite h by eauto. + rewrite IHxs by eauto. + destruct (q x); reflexivity. } +Qed. + +End Demo. 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z=CH|+iuJPV^=ESGPsq*PGI!;++7tKV8ZKVYt+RP2y`Ft$f9Qv8Rn>;m5;UhBWZ(H# zv|ys=?XRkDw?;_p+W*xnCd>3%f_2#e`?YW97A*9wHvg8Of9T3UuD|thmso!FR{qv^@BbFZyplQhSV-?N z*SbeiG5nYIHO?;hb#Y&J_rl-qsWDIcF8+F1T_|U_GuQrzvQ3-0`)#3&rFT_wC~P>$>EtrM&dxK0i6|P3A^i&Ek(rVBN(X)qI8L72h58 z&V!hiUa?(}e@*k<;*a6ykH+t1?~w1aK4ROxf7?`_ zVE^7U^~uux+o%7#7x$~W`sMq#Fb;F|&Bo-fMJ7G9dA{}c@BjCB{hzUTx-c{5Xi;%V ZVo^y&5pZ(u_-0)z#mP3jnZWN*@3K literal 0 HcmV?d00001 From 9db653f6440b1fa597fd12270fc1bccb6a07a682 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Fri, 20 Oct 2017 11:28:57 +0200 Subject: [PATCH 55/81] Expose today's slides. --- README.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/README.md b/README.md index 00bcbd5..7e35712 100644 --- a/README.md +++ b/README.md @@ -91,7 +91,8 @@ We also show the limits of dependently-typed functional programming. ([solution](ocaml/EvalCBNCPS.ml)). * Transforming a graph traversal ([solution](ocaml/Graph.ml)). -* (20/10/2017) Equational reasoning and program optimizations. +* (20/10/2017) Equational reasoning and program optimizations + ([slides 05](slides/fpottier-05.pdf)). ### Metatheory of Typed Programming Languages From 639d7951fc0e9a4ba4cc4c9f15480ef58ee0adad Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Franc=CC=A7ois=20Pottier?= Date: Fri, 20 Oct 2017 12:38:59 +0200 Subject: [PATCH 56/81] Link to Coq mini-demo. --- README.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/README.md b/README.md index 7e35712..f04249f 100644 --- a/README.md +++ b/README.md @@ -92,7 +92,8 @@ We also show the limits of dependently-typed functional programming. * Transforming a graph traversal ([solution](ocaml/Graph.ml)). * (20/10/2017) Equational reasoning and program optimizations - ([slides 05](slides/fpottier-05.pdf)). + ([slides 05](slides/fpottier-05.pdf)) + ([Coq mini-demo](coq/DemoEqReasoning.v)). ### Metatheory of Typed Programming Languages From 054e451133572239b5a69ca34878dab8e304d042 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Franc=CC=A7ois=20Pottier?= Date: Fri, 20 Oct 2017 21:00:49 +0200 Subject: [PATCH 57/81] Fix the statement of rule SR1. --- slides/fpottier-05.pdf | Bin 769473 -> 769497 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/slides/fpottier-05.pdf b/slides/fpottier-05.pdf index 44ca37665dfef18b887f81c97b8f3a67708e1caf..f92e5906f7a4d2487ab74c70092328dfd1c128bb 100644 GIT binary patch delta 15588 zcmb7q1yCH__azqG-GU^zyF+l7V8I=N%V5DLxVr}l8r^XsY|&_in$c=|16K9v~uodN{rT z@Me_PswIsDt$xJ7hviJ^W*f{;zBp8W>b~n4#gvx(0=wj(r3IYVTwgr?c7}-iOg6QU zyqUatO{7PMI3A9L>@kJ}B95RRga4GJ+)>ppiwF!4+QQOhzQu!*qnqs*q9uJ~y_sl= zz2%WSYynmSR!HpbU4%St;~C?q*cH1y;J;Xr!7fIzl^Q=ny`aAXcD!&!c-VzxexC+{ zfE3u0HetAOrXc&y@)5iEC)rWEbiKs4vMbG4pMN1qqFM&XonC-sOHJ5KUDI~fgQHTH zSI{QHF~70jEym~dS?GM6?Tz)_u_J8^l{E{&4FD$fr$JrxT}DRoX9?QPfg+wpO}|v8 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z?nXyMliL_9+X%H=1V*T{U4dCH_cL8lQ3VgOmPWg=GxvwSt|TIDj-vgPLSekJYZx{| zm>Nq6O|PWp0|Jc?o_Ez1uMJF@Of~>h?<#EPFgBknMpt{X=&t9$R&G3*&1bDIcHs@3 zQ(KE_c!G_5PRQXvgQmHwJyrCpOzai`)8qZU!5S8#tz5}3QA_*%aXsTT)=r&X-ieeO zf+0U+HT=NedicgVP0gI}w+qX*;Ih`fC-o&zU^Twn>A4|VHuftzr}*95J;E2ob&C9S zK+);XXx3gW=&04f>+a{eQv)WQuo0xOg8jnbt#bQlw_6WW=XFdZTV9c=MQfL3H;uq+ z(KBw#os#aBwC4@udtcwfvy*?mK>Dr(nqRwdMf*D`1Dl&Ka^GLT-P153)9O1BNfCJY Nc@U_nrB!7R{uk~3okRct From 12d27e9bfd7d6b07929a87567525784a7265518d Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Fri, 27 Oct 2017 10:38:45 +0200 Subject: [PATCH 58/81] Slides du cours 2. --- README.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/README.md b/README.md index ea12d46..f58fdbe 100644 --- a/README.md +++ b/README.md @@ -75,7 +75,9 @@ We also show the limits of dependently-typed functional programming. and chap [1](http://gallium.inria.fr/~remy/mpri/cours1.pdf) and [2](http://gallium.inria.fr/~remy/mpri/cours2.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)) -* (27/10/2017) ADTs, existential types, GADTs. +* (27/10/2017) + [ADTs, existential types, GADTs] + (http://gallium.inria.fr/~remy/mpri/slides2.pdf). * (03/11/2017) Logical relations. * (10/11/2017) Subtyping. Rows. * (17/11/2017) References, Value restriction, Side effects. From 1abcc4dc850eb8b14134360a71a440baff6a99f3 Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Sun, 29 Oct 2017 16:39:41 +0100 Subject: [PATCH 59/81] - Adding anchors for Remy's lecture 2 - Adding Makefile to produce README.html from README.md so that we can see the output in a browser before commiting to gitlab. requires pandoc installed. --- Makefile | 4 ++++ 1 file changed, 4 insertions(+) create mode 100644 Makefile diff --git a/Makefile b/Makefile new file mode 100644 index 0000000..0f984d0 --- /dev/null +++ b/Makefile @@ -0,0 +1,4 @@ + +README.html: README.md + pandoc -s -f markdown -t html -o $@ $< + From 1f4d3ff456b4bfe28ee561a07badf72be1bf2ff6 Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Sun, 29 Oct 2017 16:42:23 +0100 Subject: [PATCH 60/81] Adding anchors lecture 2 of Remy's course --- README.md | 16 ++++++++++------ 1 file changed, 10 insertions(+), 6 deletions(-) diff --git a/README.md b/README.md index 276f84c..c5365b8 100644 --- a/README.md +++ b/README.md @@ -99,13 +99,17 @@ We also show the limits of dependently-typed functional programming. * (15/09/2017) [Metatheory of System F](http://gallium.inria.fr/~remy/mpri/slides1.pdf) - (See also [intro](http://gallium.inria.fr/~remy/mpri/slides8.pdf), - and chap [1](http://gallium.inria.fr/~remy/mpri/cours1.pdf) - and [2](http://gallium.inria.fr/~remy/mpri/cours2.pdf) - of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)) + (see also [intro](http://gallium.inria.fr/~remy/mpri/slides8.pdf), + and chap [1,2,3](http://gallium.inria.fr/~remy/mpri/cours1.pdf) + and [4](http://gallium.inria.fr/~remy/mpri/cours2.pdf) + of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). * (27/10/2017) - [ADTs, existential types, GADTs] - (http://gallium.inria.fr/~remy/mpri/slides2.pdf). + [ADTs, existential types, GADTs](http://gallium.inria.fr/~remy/mpri/slides2.pdf) + ([without](http://gallium.inria.fr/~remy/mpri/slides2a.pdf) or + [only](http://gallium.inria.fr/~remy/mpri/slides2b.pdf) + the extra material; + and [chap 6](http://gallium.inria.fr/~remy/mpri/cours4.pdf)). + of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)) * (03/11/2017) Logical relations. * (10/11/2017) Subtyping. Rows. * (17/11/2017) References, Value restriction, Side effects. From bacf138196dcba02b504579d88567d27b362004b Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Sun, 29 Oct 2017 16:43:39 +0100 Subject: [PATCH 61/81] fixing ). --- README.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index c5365b8..a77e8aa 100644 --- a/README.md +++ b/README.md @@ -108,9 +108,10 @@ We also show the limits of dependently-typed functional programming. ([without](http://gallium.inria.fr/~remy/mpri/slides2a.pdf) or [only](http://gallium.inria.fr/~remy/mpri/slides2b.pdf) the extra material; - and [chap 6](http://gallium.inria.fr/~remy/mpri/cours4.pdf)). + and [chap 6](http://gallium.inria.fr/~remy/mpri/cours4.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)) -* (03/11/2017) Logical relations. +* (03/11/2017) Logical relations + (See also [chap ](http://gallium.inria.fr/~remy/mpri/cours6.pdf)). * (10/11/2017) Subtyping. Rows. * (17/11/2017) References, Value restriction, Side effects. From 50b685a8d666460804131b1f1f8f77aeceb8c714 Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Sun, 29 Oct 2017 16:44:50 +0100 Subject: [PATCH 62/81] Anchor for lecture 3. --- README.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index a77e8aa..da7cc08 100644 --- a/README.md +++ b/README.md @@ -109,9 +109,10 @@ We also show the limits of dependently-typed functional programming. [only](http://gallium.inria.fr/~remy/mpri/slides2b.pdf) the extra material; and [chap 6](http://gallium.inria.fr/~remy/mpri/cours4.pdf) - of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)) + of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). * (03/11/2017) Logical relations - (See also [chap ](http://gallium.inria.fr/~remy/mpri/cours6.pdf)). + (See also [chap ](http://gallium.inria.fr/~remy/mpri/cours6.pdf) + of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). * (10/11/2017) Subtyping. Rows. * (17/11/2017) References, Value restriction, Side effects. From aee8182a5c56195e03caded0390fd784250951a9 Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Sun, 29 Oct 2017 16:48:50 +0100 Subject: [PATCH 63/81] Anchor for Lecture 3. --- Makefile | 1 - README.md | 7 ++++--- 2 files changed, 4 insertions(+), 4 deletions(-) diff --git a/Makefile b/Makefile index 0f984d0..e22c835 100644 --- a/Makefile +++ b/Makefile @@ -1,4 +1,3 @@ README.html: README.md pandoc -s -f markdown -t html -o $@ $< - diff --git a/README.md b/README.md index da7cc08..22f7bf5 100644 --- a/README.md +++ b/README.md @@ -105,13 +105,14 @@ We also show the limits of dependently-typed functional programming. of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). * (27/10/2017) [ADTs, existential types, GADTs](http://gallium.inria.fr/~remy/mpri/slides2.pdf) - ([without](http://gallium.inria.fr/~remy/mpri/slides2a.pdf) or + (slides + [without](http://gallium.inria.fr/~remy/mpri/slides2a.pdf) or [only](http://gallium.inria.fr/~remy/mpri/slides2b.pdf) the extra material; and [chap 6](http://gallium.inria.fr/~remy/mpri/cours4.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). -* (03/11/2017) Logical relations - (See also [chap ](http://gallium.inria.fr/~remy/mpri/cours6.pdf) +* (03/11/2017) [Logical relations](http://gallium.inria.fr/~remy/mpri/slides2.pdf) + ([chap 8](http://gallium.inria.fr/~remy/mpri/cours6.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). * (10/11/2017) Subtyping. Rows. * (17/11/2017) References, Value restriction, Side effects. From 0f340ed1fdfd8927a79d3cf5d49b9d85fc53dc8d Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Sun, 29 Oct 2017 16:50:40 +0100 Subject: [PATCH 64/81] lecture 3 --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 22f7bf5..4d768d3 100644 --- a/README.md +++ b/README.md @@ -111,7 +111,7 @@ We also show the limits of dependently-typed functional programming. the extra material; and [chap 6](http://gallium.inria.fr/~remy/mpri/cours4.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). -* (03/11/2017) [Logical relations](http://gallium.inria.fr/~remy/mpri/slides2.pdf) +* (03/11/2017) [Logical relations](http://gallium.inria.fr/~remy/mpri/slides3.pdf) ([chap 8](http://gallium.inria.fr/~remy/mpri/cours6.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). * (10/11/2017) Subtyping. Rows. From c8961ac16261087d11e44c65f5b1efda66648b66 Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Tue, 31 Oct 2017 10:03:09 +0100 Subject: [PATCH 65/81] Exams, dates and documents allowed. --- README.md | 10 ++++++---- 1 file changed, 6 insertions(+), 4 deletions(-) diff --git a/README.md b/README.md index 4d768d3..2e71ec0 100644 --- a/README.md +++ b/README.md @@ -135,10 +135,12 @@ We also show the limits of dependently-typed functional programming. ## Evaluation of the course -Two written exams (a partial and a final exam) and one programming project -or several programming exercises are used to evaluate the students that -follow the full course. Only the partial exam will count to grade students -who split the course. +Two written exams (a partial exam on Friday Dec 12 and a final exam) and one +programming project or several programming exercises are used to evaluate +the students that follow the full course. Only the partial exam will count +to grade students who split the course. + +Only course notes and hand written notes are allowed for the exams. Although the course has changed, you may still have a look at previous exams available with solutions: From da4900bdf5dcc72c74b28bd73827ea994a9a3683 Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Wed, 1 Nov 2017 17:34:39 +0100 Subject: [PATCH 66/81] handout --- README.md | 11 +++++++---- 1 file changed, 7 insertions(+), 4 deletions(-) diff --git a/README.md b/README.md index 2e71ec0..52e6dab 100644 --- a/README.md +++ b/README.md @@ -99,19 +99,22 @@ We also show the limits of dependently-typed functional programming. * (15/09/2017) [Metatheory of System F](http://gallium.inria.fr/~remy/mpri/slides1.pdf) + [(handout)](http://gallium.inria.fr/~remy/mpri/handout1.pdf) (see also [intro](http://gallium.inria.fr/~remy/mpri/slides8.pdf), and chap [1,2,3](http://gallium.inria.fr/~remy/mpri/cours1.pdf) and [4](http://gallium.inria.fr/~remy/mpri/cours2.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). * (27/10/2017) [ADTs, existential types, GADTs](http://gallium.inria.fr/~remy/mpri/slides2.pdf) - (slides - [without](http://gallium.inria.fr/~remy/mpri/slides2a.pdf) or - [only](http://gallium.inria.fr/~remy/mpri/slides2b.pdf) + ([handout](http://gallium.inria.fr/~remy/mpri/handout2.pdf) + [without](http://gallium.inria.fr/~remy/mpri/handout2a.pdf) or + [only](http://gallium.inria.fr/~remy/mpri/handout2b.pdf) the extra material; and [chap 6](http://gallium.inria.fr/~remy/mpri/cours4.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). -* (03/11/2017) [Logical relations](http://gallium.inria.fr/~remy/mpri/slides3.pdf) +* (03/11/2017) + [Logical relations](http://gallium.inria.fr/~remy/mpri/slides3.pdf) + [(handout)](http://gallium.inria.fr/~remy/mpri/handout3.pdf) ([chap 8](http://gallium.inria.fr/~remy/mpri/cours6.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). * (10/11/2017) Subtyping. Rows. 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[(handout)](http://gallium.inria.fr/~remy/mpri/handout3.pdf) ([chap 8](http://gallium.inria.fr/~remy/mpri/cours6.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). -* (10/11/2017) Subtyping. Rows. +# * (10/11/2017) Subtyping. Rows. +* (10/11/2017) + [Overloading](http://gallium.inria.fr/~remy/mpri/slides4.pdf) + [(handout)](http://gallium.inria.fr/~remy/mpri/handout4.pdf) + (see also [chap 7](http://gallium.inria.fr/~remy/mpri/slides5.pdf) + of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). + * (17/11/2017) References, Value restriction, Side effects. ### Advanced Aspects of Type Systems From 828f05db8b108daa3d41597f3e736b66c9bc5690 Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Thu, 9 Nov 2017 22:45:38 +0100 Subject: [PATCH 69/81] Change of topic for lesson 4 --- README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index c50ea04..8a0ee9b 100644 --- a/README.md +++ b/README.md @@ -118,10 +118,10 @@ We also show the limits of dependently-typed functional programming. ([chap 8](http://gallium.inria.fr/~remy/mpri/cours6.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). # * (10/11/2017) Subtyping. Rows. -* (10/11/2017) +* (10/11/2017) ~~Subtyping. Rows.~~ [Overloading](http://gallium.inria.fr/~remy/mpri/slides4.pdf) [(handout)](http://gallium.inria.fr/~remy/mpri/handout4.pdf) - (see also [chap 7](http://gallium.inria.fr/~remy/mpri/slides5.pdf) + (see also [chap 7](http://gallium.inria.fr/~remy/mpri/cours5.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). * (17/11/2017) References, Value restriction, Side effects. From 8d9111b95e487a8c37a4d9435c42f5c91a7e99ac Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Fri, 10 Nov 2017 08:04:53 +0100 Subject: [PATCH 70/81] Removing blank line --- README.md | 1 - 1 file changed, 1 deletion(-) diff --git a/README.md b/README.md index 8a0ee9b..faf223e 100644 --- a/README.md +++ b/README.md @@ -123,7 +123,6 @@ We also show the limits of dependently-typed functional programming. [(handout)](http://gallium.inria.fr/~remy/mpri/handout4.pdf) (see also [chap 7](http://gallium.inria.fr/~remy/mpri/cours5.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). - * (17/11/2017) References, Value restriction, Side effects. ### Advanced Aspects of Type Systems From ade3a69b24c14786fa325216b36d1a845913b1bd Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Fri, 10 Nov 2017 08:10:59 +0100 Subject: [PATCH 71/81] Removing comment --- README.md | 11 +++++------ 1 file changed, 5 insertions(+), 6 deletions(-) diff --git a/README.md b/README.md index faf223e..175cc51 100644 --- a/README.md +++ b/README.md @@ -103,7 +103,7 @@ We also show the limits of dependently-typed functional programming. (see also [intro](http://gallium.inria.fr/~remy/mpri/slides8.pdf), and chap [1,2,3](http://gallium.inria.fr/~remy/mpri/cours1.pdf) and [4](http://gallium.inria.fr/~remy/mpri/cours2.pdf) - of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). + of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)) * (27/10/2017) [ADTs, existential types, GADTs](http://gallium.inria.fr/~remy/mpri/slides2.pdf) ([handout](http://gallium.inria.fr/~remy/mpri/handout2.pdf) @@ -111,19 +111,18 @@ We also show the limits of dependently-typed functional programming. [only](http://gallium.inria.fr/~remy/mpri/handout2b.pdf) the extra material; and [chap 6](http://gallium.inria.fr/~remy/mpri/cours4.pdf) - of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). + of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)) * (03/11/2017) [Logical relations](http://gallium.inria.fr/~remy/mpri/slides3.pdf) [(handout)](http://gallium.inria.fr/~remy/mpri/handout3.pdf) ([chap 8](http://gallium.inria.fr/~remy/mpri/cours6.pdf) - of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). -# * (10/11/2017) Subtyping. Rows. + of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)) * (10/11/2017) ~~Subtyping. Rows.~~ [Overloading](http://gallium.inria.fr/~remy/mpri/slides4.pdf) [(handout)](http://gallium.inria.fr/~remy/mpri/handout4.pdf) (see also [chap 7](http://gallium.inria.fr/~remy/mpri/cours5.pdf) - of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)). -* (17/11/2017) References, Value restriction, Side effects. + of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)) +* (17/11/2017) References, Value restriction, Side effects ### Advanced Aspects of Type Systems From f36225d1f5be17661fb5b13d250ddd65bbbfdb10 Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Tue, 14 Nov 2017 16:46:09 +0100 Subject: [PATCH 72/81] Correction de l'ancre pour l'examen de 2010 --- README.md | 15 +++++++++++---- 1 file changed, 11 insertions(+), 4 deletions(-) diff --git a/README.md b/README.md index 175cc51..73b8d85 100644 --- a/README.md +++ b/README.md @@ -109,8 +109,8 @@ We also show the limits of dependently-typed functional programming. ([handout](http://gallium.inria.fr/~remy/mpri/handout2.pdf) [without](http://gallium.inria.fr/~remy/mpri/handout2a.pdf) or [only](http://gallium.inria.fr/~remy/mpri/handout2b.pdf) - the extra material; - and [chap 6](http://gallium.inria.fr/~remy/mpri/cours4.pdf) + the extra material) + (se also [chap 6](http://gallium.inria.fr/~remy/mpri/cours4.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)) * (03/11/2017) [Logical relations](http://gallium.inria.fr/~remy/mpri/slides3.pdf) @@ -122,7 +122,14 @@ We also show the limits of dependently-typed functional programming. [(handout)](http://gallium.inria.fr/~remy/mpri/handout4.pdf) (see also [chap 7](http://gallium.inria.fr/~remy/mpri/cours5.pdf) of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)) -* (17/11/2017) References, Value restriction, Side effects +* (17/11/2017) + [References, Value restriction, Side effects](http://gallium.inria.fr/~remy/mpri/slides5.pdf) + ([handout](http://gallium.inria.fr/~remy/mpri/handout5.pdf)) + (see also sections [3.6, 3.7](http://gallium.inria.fr/~remy/mpri/cours1.pdf) + and [4.5](http://gallium.inria.fr/~remy/mpri/cours2.pdf) + of [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf)) +* See exercises in [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf) + ### Advanced Aspects of Type Systems @@ -175,7 +182,7 @@ available with solutions: [Compiling a language with subtyping](http://gallium.inria.fr/~xleroy/mpri/2-4/exam-2010-2011.pdf) - mid-term exam 2010-2011: [Compilation - of polymorphic records](http://gallium.inria.fr/~remy/mpri/exams/2010/partiel-2010-2011.pdf) + of polymorphic records](http://gallium.inria.fr/~remy/mpri/exams/partiel-2010-2011.pdf) ## Recommended software From 20664156ac8d40650ecb3317021499a4e3e032c1 Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Thu, 23 Nov 2017 18:06:30 +0100 Subject: [PATCH 73/81] Remove pointer to my own page. --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 73b8d85..4e49d76 100644 --- a/README.md +++ b/README.md @@ -14,7 +14,7 @@ There is a 15-minute break in the middle of each lecture. ## Teachers * Functional Programming: Under the Hood (12h30, [François Pottier](http://gallium.inria.fr/~fpottier)) - * [Metatheory of Typed Programming Languages](http://gallium.inria.fr/~remy/mpri/) (12h30, [Didier Rémy](http://gallium.inria.fr/~remy/), *head*) + * Metatheory of Typed Programming Languages (12h30, [Didier Rémy](http://gallium.inria.fr/~remy/), *head*) * Advanced Aspects of Type Systems (12h30, [Yann Régis Gianas](http://www.pps.jussieu.fr/~yrg/)) * Dependently-typed Functional Programming (12h30, [Pierre-Evariste Dagand](https://pages.lip6.fr/Pierre-Evariste.Dagand/)) From 61142fa289b80b2f3aba5815131cea79ca06f3b0 Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Wed, 29 Nov 2017 16:28:14 +0100 Subject: [PATCH 74/81] 12<-1 --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 4e49d76..eebc21b 100644 --- a/README.md +++ b/README.md @@ -149,7 +149,7 @@ We also show the limits of dependently-typed functional programming. ## Evaluation of the course -Two written exams (a partial exam on Friday Dec 12 and a final exam) and one +Two written exams (a partial exam on Friday Dec 1 and a final exam) and one programming project or several programming exercises are used to evaluate the students that follow the full course. Only the partial exam will count to grade students who split the course. From ea28092e576e0f8ba2bcf019c24c62aa8abc55a4 Mon Sep 17 00:00:00 2001 From: Yann Regis-Gianas Date: Fri, 1 Dec 2017 09:50:23 +0100 Subject: [PATCH 75/81] Update part 3 syllabus. --- README.md | 26 ++++++++++++++++---------- 1 file changed, 16 insertions(+), 10 deletions(-) diff --git a/README.md b/README.md index eebc21b..8900d4a 100644 --- a/README.md +++ b/README.md @@ -50,10 +50,15 @@ reasoning by induction on the structure of types. Finally, we introduce subtyping, row polymorphism, and illustrate the problems induced by side effects (references) and the need for the value restriction. -The third part of the course describes more advanced features of type -systems: exceptions and effect handlers, including their typechecking and -static analyses: type inference, data flow and control flow analyses. -Finally, it introduces dependent types and refinement types. +The third part of the course introduces "rich" type systems designed +to guarantee extra properties in addition to safety: principality, +information hiding, modularity, extensionality, purity, control of +effects, algorithmic invariants, complexity, resource usage, or full +functional correctness. The expressivity of these systems sometimes +endangers the tractability, or even the feasibility, of type checking +and type inference: a common thread between these lectures discusses +the tradeoffs made on the design of these systems to balance +expressivity and tractability. The last part focuses on the use of dependent types for programming: effectful programming with monads and algebraic effects; tagless @@ -131,13 +136,14 @@ We also show the limits of dependently-typed functional programming. * See exercises in [course notes](http://gallium.inria.fr/~remy/mpri/cours.pdf) -### Advanced Aspects of Type Systems +### Rich types, tractable typing -* Exceptions and effect handlers. (Compiled away via CPS.) -* Typechecking exceptions and handlers. -* Type inference. (ML. Bidirectional. Elaboration.) -* Data/control flow analysis. -* Functional correctness. Intro to dependent/refinement types. +* Type inference +* Subtyping +* Effects and resources +* Modules +* Dependent types +* Functional correctness ### Dependently-typed Functional Programming From 0ee19236b0d172d98b4ab23f69326a3c25944f4d Mon Sep 17 00:00:00 2001 From: Yann Regis-Gianas Date: Wed, 13 Dec 2017 10:26:29 +0100 Subject: [PATCH 76/81] Slides cours YRG-1. --- README.md | 4 +++- slides/yrg-00-introduction.pdf | Bin 0 -> 97679 bytes slides/yrg-01-type-inference.pdf | Bin 0 -> 116583 bytes 3 files changed, 3 insertions(+), 1 deletion(-) create mode 100644 slides/yrg-00-introduction.pdf create mode 100644 slides/yrg-01-type-inference.pdf diff --git a/README.md b/README.md index 8900d4a..a131adb 100644 --- a/README.md +++ b/README.md @@ -138,7 +138,9 @@ We also show the limits of dependently-typed functional programming. ### Rich types, tractable typing -* Type inference +* (08/12/2017) + [Introduction](slides/yrg-00-introduction.pdf), + [ML and Type inference](slides/yrg-01-type-inference.pdf) * Subtyping * Effects and resources * Modules diff --git a/slides/yrg-00-introduction.pdf b/slides/yrg-00-introduction.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d232d8e0742ad48c2b7a881a5ebe122a9e6cbe4e GIT binary patch literal 97679 zcmcG#bx>s8lP!u{Tmg%;4bySuwJ+IZt`g%|Gb);Nv3dn1iIG;R&eg=kFeQNKOYh~ss8WjmC77kVb63zbA#S;>MoSod+%nk_%WYh5eV#%hX zY-VR^?#`y}VdnnNdnspE$A6vv871dn6BBdx(r5qp5`c%5ho6g_lb4f~gOkUQP0`fd z)z(X&<70@O{Ns$oZpfx)>E`U=YHsO9&hv3mjZMnd!QIl8P0GR4-BQBR+}XmCO~KO1 z+TDhnlS^0_$<5u>($o>j>!&)q$`@^R6&dYWv}qOkX$S~>DCiU_@2_9O4`^dt5iV!{ytkq z;5%luAch#$cm`sbP@ccdDxkwedP2mZK?#!NA|Y+qG6l|JDXwM=$6FrQMKvFH1oRa7 zcJw`ygiE7ww~S&MXTuzhuq6c5e)9p^lZEwMOsCDDxwnvO(7Q0vEVwFXLLEUVA7o&% zjQq|Mt0xu%nqkKd0A=m3N~0m2qpx^$&gC+`F&o>;Wt*2WK=Yy|3h>^Wd*cSh*+RQ# zc*Ik)2l{XY>2u>h+C!PwTq7IL2YA8(lPMlr@lyZ1uC7px?~$C`x)$Cg#g-E%B{EnG z-MeJlU*7vl%1U>KPXpgC`p(V)>AJ|mAoS)x!(&Tn>Np04Pn$Zo*0Sp!-JbFC3rzx8 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From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Wed, 13 Dec 2017 14:04:28 +0100 Subject: [PATCH 77/81] Publish the project. --- projet/src/.merlin | 9 + projet/src/CPS.ml | 7 + projet/src/CPS.mli | 4 + projet/src/Cook.ml | 54 +++++ projet/src/Cook.mli | 15 ++ projet/src/Defun.ml | 7 + projet/src/Defun.mli | 4 + projet/src/Error.ml | 40 ++++ projet/src/Error.mli | 30 +++ projet/src/Finish.ml | 323 ++++++++++++++++++++++++++ projet/src/Finish.mli | 5 + projet/src/Lambda.ml | 48 ++++ projet/src/Lexer.mll | 97 ++++++++ projet/src/Main.ml | 101 ++++++++ projet/src/Makefile | 28 +++ projet/src/Parser.mly | 155 +++++++++++++ projet/src/RawLambda.ml | 55 +++++ projet/src/Tail.ml | 132 +++++++++++ projet/src/Top.ml | 94 ++++++++ projet/src/_tags | 8 + projet/src/alphalib/Atom.ml | 215 ++++++++++++++++++ projet/src/alphalib/Atom.mli | 71 ++++++ projet/src/kremlin/C.ml | 126 ++++++++++ projet/src/kremlin/Common.ml | 19 ++ projet/src/kremlin/Constant.ml | 66 ++++++ projet/src/kremlin/Options.ml | 7 + projet/src/kremlin/PrintC.ml | 354 +++++++++++++++++++++++++++++ projet/src/kremlin/PrintCommon.ml | 96 ++++++++ projet/src/kremlin/Utils.ml | 73 ++++++ projet/src/prologue.h | 83 +++++++ projet/src/test/.gitignore | 1 + projet/src/test/.merlin | 1 + projet/src/test/Makefile | 21 ++ projet/src/test/_tags | 3 + projet/src/test/auxiliary.ml | 138 +++++++++++ projet/src/test/test.ml | 288 +++++++++++++++++++++++ projet/src/tests/.gitignore | 4 + projet/src/tests/bool.exp | 20 ++ projet/src/tests/bool.lambda | 88 +++++++ projet/src/tests/church.exp | 1 + projet/src/tests/church.lambda | 16 ++ projet/src/tests/hello.exp | 1 + projet/src/tests/hello.lambda | 1 + projet/src/tests/loop/loop.lambda | 10 + projet/src/tests/printprint.exp | 2 + projet/src/tests/printprint.lambda | 2 + projet/sujet.pdf | Bin 0 -> 112447 bytes 47 files changed, 2923 insertions(+) create mode 100644 projet/src/.merlin create mode 100644 projet/src/CPS.ml create mode 100644 projet/src/CPS.mli create mode 100644 projet/src/Cook.ml create mode 100644 projet/src/Cook.mli create mode 100644 projet/src/Defun.ml create mode 100644 projet/src/Defun.mli create mode 100644 projet/src/Error.ml create mode 100644 projet/src/Error.mli create mode 100644 projet/src/Finish.ml create mode 100644 projet/src/Finish.mli create mode 100644 projet/src/Lambda.ml create mode 100644 projet/src/Lexer.mll create mode 100644 projet/src/Main.ml create mode 100644 projet/src/Makefile create mode 100644 projet/src/Parser.mly create mode 100644 projet/src/RawLambda.ml create mode 100644 projet/src/Tail.ml create mode 100644 projet/src/Top.ml create mode 100644 projet/src/_tags create mode 100644 projet/src/alphalib/Atom.ml create mode 100644 projet/src/alphalib/Atom.mli create mode 100644 projet/src/kremlin/C.ml create mode 100644 projet/src/kremlin/Common.ml create mode 100644 projet/src/kremlin/Constant.ml create mode 100644 projet/src/kremlin/Options.ml create mode 100644 projet/src/kremlin/PrintC.ml create mode 100644 projet/src/kremlin/PrintCommon.ml create mode 100644 projet/src/kremlin/Utils.ml create mode 100644 projet/src/prologue.h create mode 100644 projet/src/test/.gitignore create mode 100644 projet/src/test/.merlin create mode 100644 projet/src/test/Makefile create mode 100644 projet/src/test/_tags create mode 100644 projet/src/test/auxiliary.ml create mode 100644 projet/src/test/test.ml create mode 100644 projet/src/tests/.gitignore create mode 100644 projet/src/tests/bool.exp create mode 100644 projet/src/tests/bool.lambda create mode 100755 projet/src/tests/church.exp create mode 100644 projet/src/tests/church.lambda create mode 100755 projet/src/tests/hello.exp create mode 100644 projet/src/tests/hello.lambda create mode 100644 projet/src/tests/loop/loop.lambda create mode 100755 projet/src/tests/printprint.exp create mode 100644 projet/src/tests/printprint.lambda create mode 100644 projet/sujet.pdf diff --git a/projet/src/.merlin b/projet/src/.merlin new file mode 100644 index 0000000..7de32a3 --- /dev/null +++ b/projet/src/.merlin @@ -0,0 +1,9 @@ +S kremlin +S alphalib +B _build +B _build/kremlin +B _build/alphalib +PKG unix +PKG process +PKG pprint +PKG ppx_deriving.std diff --git a/projet/src/CPS.ml b/projet/src/CPS.ml new file mode 100644 index 0000000..9b3d87d --- /dev/null +++ b/projet/src/CPS.ml @@ -0,0 +1,7 @@ +(* The source calculus. *) +module S = Lambda +(* The target calculus. *) +module T = Tail + +let cps_term (t : S.term) : T.term = + assert false diff --git a/projet/src/CPS.mli b/projet/src/CPS.mli new file mode 100644 index 0000000..2497ddd --- /dev/null +++ b/projet/src/CPS.mli @@ -0,0 +1,4 @@ +(* Through a CPS transformation, the surface language [Lambda] is translated + down to the intermediate language [Tail]. *) + +val cps_term: Lambda.term -> Tail.term diff --git a/projet/src/Cook.ml b/projet/src/Cook.ml new file mode 100644 index 0000000..89657ed --- /dev/null +++ b/projet/src/Cook.ml @@ -0,0 +1,54 @@ +open Error + +(* The source calculus. *) +module S = RawLambda +(* The target calculus. *) +module T = Lambda + +(* Environments map strings to atoms. *) +module Env = + Map.Make(String) + +(* [bind env x] creates a fresh atom [a] and extends the environment [env] + with a mapping of [x] to [a]. *) +let bind env x = + let a = Atom.fresh x in + Env.add x a env, a + +let rec cook_term env { S.place; S.value } = + match value with + | S.Var x -> + begin try + T.Var (Env.find x env) + with Not_found -> + error place "Unbound variable: %s" x + end + | S.Lam (x, t) -> + let env, x = bind env x in + T.Lam (T.NoSelf, x, cook_term env t) + | S.App (t1, t2) -> + T.App (cook_term env t1, cook_term env t2) + | S.Lit i -> + T.Lit i + | S.BinOp (t1, op, t2) -> + T.BinOp (cook_term env t1, op, cook_term env t2) + | S.Print t -> + T.Print (cook_term env t) + | S.Let (S.NonRecursive, x, t1, t2) -> + let t1 = cook_term env t1 in + let env, x = bind env x in + let t2 = cook_term env t2 in + T.Let (x, t1, t2) + | S.Let (S.Recursive, f, { S.value = S.Lam (x, t1); _ }, t2) -> + let env, f = bind env f in + let x, t1 = + let env, x = bind env x in + x, cook_term env t1 + in + let t2 = cook_term env t2 in + T.Let (f, T.Lam (T.Self f, x, t1), t2) + | S.Let (S.Recursive, _, { S.place; _ }, _) -> + error place "the right-hand side of 'let rec' must be a lambda-abstraction" + +let cook_term t = + cook_term Env.empty t diff --git a/projet/src/Cook.mli b/projet/src/Cook.mli new file mode 100644 index 0000000..617e408 --- /dev/null +++ b/projet/src/Cook.mli @@ -0,0 +1,15 @@ +(* This module translates [RawLambda] into [Lambda]. *) + +(* This involves ensuring that every name is properly bound (otherwise, an + error is reported) and switching from a representation of names as strings + to a representation of names as atoms. *) + +(* This also involves checking that the right-hand side of every [let] + construct is a function (otherwise, an error is reported) and switching + from a representation where [let] constructs can carry a [rec] annotation + to a representation where functions can carry such an annotation. *) + +(* This also involves dropping places (that is, source code locations), since + they are no longer used after this phase. *) + +val cook_term: RawLambda.term -> Lambda.term diff --git a/projet/src/Defun.ml b/projet/src/Defun.ml new file mode 100644 index 0000000..7264dac --- /dev/null +++ b/projet/src/Defun.ml @@ -0,0 +1,7 @@ +(* The source calculus. *) +module S = Tail +(* The target calculus. *) +module T = Top + +let defun_term (t : S.term) : T.program = + assert false diff --git a/projet/src/Defun.mli b/projet/src/Defun.mli new file mode 100644 index 0000000..0c02815 --- /dev/null +++ b/projet/src/Defun.mli @@ -0,0 +1,4 @@ +(* Through defunctionalization, the intermediate language [Tail] is translated + down to the next intermediate language, [Top]. *) + +val defun_term: Tail.term -> Top.program diff --git a/projet/src/Error.ml b/projet/src/Error.ml new file mode 100644 index 0000000..e4f1d8b --- /dev/null +++ b/projet/src/Error.ml @@ -0,0 +1,40 @@ +open Lexing + +type place = + position * position + +let place lexbuf : place = + lexbuf.lex_start_p, lexbuf.lex_curr_p + +let line p : int = + p.pos_lnum + +let column p : int = + p.pos_cnum - p.pos_bol + +let show place : string = + let startp, endp = place in + Printf.sprintf "File \"%s\", line %d, characters %d-%d" + startp.pos_fname + (line startp) + (column startp) + (endp.pos_cnum - startp.pos_bol) (* intentionally [startp.pos_bol] *) + +let display continuation header place format = + Printf.fprintf stderr "%s:\n" (show place); + Printf.kfprintf + continuation + stderr + (header ^^ format ^^ "\n%!") + +let error place format = + display + (fun _ -> exit 1) + "Error: " + place format + +let set_filename lexbuf filename = + lexbuf.lex_curr_p <- { lexbuf.lex_curr_p with pos_fname = filename } + +let pp_place formatter _place = + Format.fprintf formatter "<>" diff --git a/projet/src/Error.mli b/projet/src/Error.mli new file mode 100644 index 0000000..b8ba5f9 --- /dev/null +++ b/projet/src/Error.mli @@ -0,0 +1,30 @@ +open Lexing + +(* A place is a pair of a start position and an end position. *) + +type place = + position * position + +(* [set_filename lexbuf filename] updates [lexbuf] to record the + fact that the current file name is [filename]. This file name + is later used in error messages. *) + +val set_filename: lexbuf -> string -> unit + +(* [place lexbuf] produces a pair of the current token's start and + end positions. This function is useful when reporting an error + during lexing. *) + +val place: lexbuf -> place + +(* [error place format ...] displays an error message and exits. + The error message is located at [place]. The error message + is composed based on [format] and the extra arguments [...]. *) + +val error: place -> ('a, out_channel, unit, 'b) format4 -> 'a + +(* [pp_place formatter place] prints a place. It is used by + [@@deriving show] for data structures that contain places. + As of now, it prints nothing. *) + +val pp_place: Format.formatter -> place -> unit diff --git a/projet/src/Finish.ml b/projet/src/Finish.ml new file mode 100644 index 0000000..08779b3 --- /dev/null +++ b/projet/src/Finish.ml @@ -0,0 +1,323 @@ +(* The source calculus. *) +module S = Top +(* The target calculus. *) +module T = C + +(* -------------------------------------------------------------------------- *) + +(* [interval i j f] constructs the list [[f i; f (i + 1); ...; f (j - 1)]]. *) + +let rec interval i j (f : int -> 'a) : 'a list = + if i < j then + f i :: interval (i + 1) j f + else + [] + +(* -------------------------------------------------------------------------- *) + +(* [index xs] constructs a list of pairs, where each element of [xs] is paired + with its index. Indices are 0-based. *) + +let index (xs : 'a list) : (int * 'a) list = + let n = List.length xs in + let indices = interval 0 n (fun i -> i) in + List.combine indices xs + +(* -------------------------------------------------------------------------- *) + +(* The number of fields of a block, not counting its tag. *) + +let block_num_fields b = + match b with + | S.Con (_, vs) -> + List.length vs + +(* -------------------------------------------------------------------------- *) + +(* A simple-minded way of ensuring that every atom is printed as a + distinct string is to concatenate the atom's hint and identity, + with an underscore in between. This is guaranteed to rule out + collisions. *) + +let var (x : S.variable) : T.ident = + Printf.sprintf "%s_%d" (Atom.hint x) (Atom.identity x) + +let evar (x : S.variable) : T.expr = + T.Name (var x) + +(* -------------------------------------------------------------------------- *) + +(* Predefined C types and functions. *) + +(* A universal type: every value is translated to a C value of type [univ]. + This is a union type (i.e., an untagged sum) of integers and pointers to + memory blocks. *) + +let univ : T.type_spec = + T.Named "univ" + +(* The type of integers. *) + +let int : T.type_spec = + T.Named "int" + +(* The type [char] appears in the type of [main]. *) + +let char : T.type_spec = + T.Named "char" + +let answer : T.type_spec = + int + (* Our functions never actually return, since they are tail recursive. + We use [int] as their return type, since this is the return type of + [main]. *) + +let exit : T.expr = + T.Name "exit" + +let printf : T.expr = + T.Name "printf" + +(* -------------------------------------------------------------------------- *) + +(* [declare x init] constructs a local variable declaration for a variable [x] + of type [univ]. [x] is optionally initialized according to [init]. *) + +let declare (x : S.variable) (init : T.init option) : T.declaration = + univ, None, [ T.Ident (var x), init ] + +(* -------------------------------------------------------------------------- *) + +(* Macro invocations. *) + +let macro m es : T.expr = + (* We disguise a macro invocation as a procedure call. *) + T.Call (T.Name m, es) + +(* -------------------------------------------------------------------------- *) + +(* Integer literals; conversions between [univ] and [int]. *) + +let iconst i : T.expr = + T.Constant (Constant.Int64, string_of_int i) + +let to_int v : T.expr = + macro "TO_INT" [ v ] + (* This is an unsafe conversion, of course. *) + +let from_int v : T.expr = + macro "FROM_INT" [ v ] + +(* -------------------------------------------------------------------------- *) + +(* The translation of values. *) + +let finish_op = function + | S.OpAdd -> + T.K.Add + | S.OpSub -> + T.K.Sub + | S.OpMul -> + T.K.Mult + | S.OpDiv -> + T.K.Div + +let rec finish_value (v : S.value) : T.expr = + match v with + | S.VVar x -> + evar x + | S.VLit i -> + from_int (iconst i) + | S.VBinOp (v1, op, v2) -> + from_int ( + T.Op2 ( + finish_op op, + to_int (finish_value v1), + to_int (finish_value v2) + ) + ) + +let finish_values vs = + List.map finish_value vs + +(* -------------------------------------------------------------------------- *) + +(* A macro for allocating a memory block. *) + +let alloc b : T.expr = + T.Call (T.Name "ALLOC", [ iconst (block_num_fields b) ]) + +(* -------------------------------------------------------------------------- *) + +(* Macros for reading and initializing the tag of a memory block. *) + +let read_tag (v : S.value) : T.expr = + macro "GET_TAG" [ finish_value v ] + +let set_tag (x : S.variable) (tag : S.tag) : T.stmt = + T.Expr (macro "SET_TAG" [ evar x; iconst tag ]) + +(* -------------------------------------------------------------------------- *) + +(* Macros for reading and setting a field in a memory block. *) + +let read_field (v : S.value) (i : int) : T.expr = + (* [i] is a 0-based field index. *) + macro "GET_FIELD" [ finish_value v; iconst i ] + +let read_field (v : S.value) (i, x) (t : T.stmt list) : T.stmt list = + (* [x] is a variable, which is declared and initialized with + the content of the [i]th field of the block [v]. *) + T.DeclStmt (declare x (Some (T.InitExpr (read_field v i)))) :: + t + +let read_fields (v : S.value) xs (t : T.stmt list) : T.stmt list = + (* [xs] are variables, which are declared and initialized with + the contents of the fields of the block [v]. *) + List.fold_right (read_field v) (index xs) t + +let set_field x i (v : S.value) : T.stmt = + T.Expr (macro "SET_FIELD" [ evar x; iconst i; finish_value v ]) + +(* -------------------------------------------------------------------------- *) + +(* A sequence of instructions for initializing a memory block. *) + +let init_block (x : S.variable) (b : S.block) : T.stmt list = + match b with + | S.Con (tag, vs) -> + T.Comment "Initializing a memory block:" :: + set_tag x tag :: + List.mapi (set_field x) vs + +(* -------------------------------------------------------------------------- *) + +(* Function calls, as expressions and as statements. *) + +let ecall f args : T.expr = + T.Call (f, args) + +let scall f args : T.stmt = + T.Expr (ecall f args) + +(* -------------------------------------------------------------------------- *) + +(* The translation of terms. *) + +let rec finish_term (t : S.term) : C.stmt = + match t with + | S.Exit -> + T.Compound [ + scall exit [ iconst 0 ] + ] + | S.TailCall (f, vs) -> + T.Return (Some (ecall (evar f) (finish_values vs))) + | S.Print (v, t) -> + T.Compound [ + scall printf [ T.Literal "%d\\n"; to_int (finish_value v) ]; + finish_term t + ] + | S.LetVal (x, v1, t2) -> + T.Compound [ + T.DeclStmt (declare x (Some (T.InitExpr (finish_value v1)))); + finish_term t2 + ] + | S.LetBlo (x, b1, t2) -> + T.Compound ( + T.DeclStmt (declare x (Some (T.InitExpr (alloc b1)))) :: + init_block x b1 @ + [ finish_term t2 ] + ) + | S.Swi (v, bs) -> + T.Switch ( + read_tag v, + finish_branches v bs, + default + ) + +and default : T.stmt = + (* This default [switch] branch should never be taken. *) + T.Compound [ + scall printf [ T.Literal "Oops! A nonexistent case has been taken.\\n" ]; + scall exit [ iconst 42 ]; + ] + +and finish_branches v bs = + List.map (finish_branch v) bs + +and finish_branch v (S.Branch (tag, xs, t)) : T.expr * T.stmt = + iconst tag, + T.Compound (read_fields v xs [finish_term t]) + +(* -------------------------------------------------------------------------- *) + +(* Function declarations. *) + +(* We distinguish the function [main], whose type is imposed by the C standard, + and ordinary functions, whose parameters have type [univ]. *) + +(* A parameter of an ordinary function has type [univ]. *) + +let param (x : S.variable) : T.param = + univ, T.Ident (var x) + +(* A declaration of an ordinary function. *) + +let declare_ordinary_function f xs : T.declaration = + answer, None, [ T.Function (None, T.Ident (var f), List.map param xs), None ] + +(* The declaration of the main function. *) + +let declare_main_function : T.declaration = + let params = [ + int, T.Ident "argc"; + char, T.Pointer (T.Pointer (T.Ident "argv")) + ] in + int, None, [ T.Function (None, T.Ident "main", params), None ] + +(* -------------------------------------------------------------------------- *) + +(* A function definition. *) + +type decl_or_fun = + T.declaration_or_function + +let define (decl : T.declaration) (t : S.term) : decl_or_fun = + T.Function ( + [], (* no comments *) + false, (* not inlined *) + decl, + T.Compound [finish_term t] + ) + +let define_ordinary_function (S.Fun (f, xs, t)) : decl_or_fun = + define (declare_ordinary_function f xs) t + +let define_main_function (t : S.term) : decl_or_fun = + define declare_main_function t + +(* -------------------------------------------------------------------------- *) + +(* Because all functions are mutually recursive, their definitions must be + preceded with their prototypes. *) + +let prototype (f : decl_or_fun) : decl_or_fun = + match f with + | T.Function (_, _, declaration, _) -> + T.Decl ([], declaration) + | T.Decl _ -> + assert false + +let prototypes (fs : decl_or_fun list) : decl_or_fun list = + List.map prototype fs @ + fs + +(* -------------------------------------------------------------------------- *) + +(* The translation of a complete program. *) + +let finish_program (S.Prog (decls, main) : S.program) : T.program = + prototypes ( + define_main_function main :: + List.map define_ordinary_function decls + ) diff --git a/projet/src/Finish.mli b/projet/src/Finish.mli new file mode 100644 index 0000000..6785984 --- /dev/null +++ b/projet/src/Finish.mli @@ -0,0 +1,5 @@ +(* This function implements a translation of the intermediate language [Top] + down to [C]. This transformation is mostly a matter of choosing appropriate + C constructs to reflect the concepts of the language [Top]. *) + +val finish_program: Top.program -> C.program diff --git a/projet/src/Lambda.ml b/projet/src/Lambda.ml new file mode 100644 index 0000000..fdb9ecb --- /dev/null +++ b/projet/src/Lambda.ml @@ -0,0 +1,48 @@ +(* This language is the untyped lambda-calculus, extended with recursive + lambda-abstractions, nonrecursive [let] bindings, integer literals and + integer arithmetic operations, and the primitive operation [print]. *) + +(* This language is really the same language as [RawLambda], with the + following internal differences: + + 1. Instead of recursive [let] bindings, the language has recursive + lambda-abstractions. A [let rec] definition whose right-hand side is not + a lambda-abstraction is rejected during the translation of [RawLambda] + to [Lambda]. + + 2. Variables are represented by atoms (instead of strings). A term with an + unbound variable is rejected during the translation of [RawLambda] to + [Lambda]. + + 3. Terms are no longer annotated with places. *) + +(* Variables are atoms. *) + +type variable = + Atom.atom + + (* Every lambda-abstraction is marked recursive or nonrecursive. Whereas a + nonrecursive lambda-abstraction [fun x -> t] binds one variable [x], a + recursive lambda-abstraction [fix f. fun x -> t] binds two variables [f] + and [x]. The variable [f] is a self-reference. *) + +and self = + | Self of variable + | NoSelf + +and binop = RawLambda.binop = + | OpAdd + | OpSub + | OpMul + | OpDiv + +and term = + | Var of variable + | Lam of self * variable * term + | App of term * term + | Lit of int + | BinOp of term * binop * term + | Print of term + | Let of variable * term * term + +[@@deriving show { with_path = false }] diff --git a/projet/src/Lexer.mll b/projet/src/Lexer.mll new file mode 100644 index 0000000..ba016f0 --- /dev/null +++ b/projet/src/Lexer.mll @@ -0,0 +1,97 @@ +{ + +open Lexing +open Error +open Parser +open RawLambda + +} + +(* -------------------------------------------------------------------------- *) + +(* Regular expressions. *) + +let newline = + ('\010' | '\013' | "\013\010") + +let whitespace = + [ ' ' '\t' ] + +let lowercase = + ['a'-'z' '\223'-'\246' '\248'-'\255' '_'] + +let uppercase = + ['A'-'Z' '\192'-'\214' '\216'-'\222'] + +let identchar = + ['A'-'Z' 'a'-'z' '_' '\192'-'\214' '\216'-'\246' '\248'-'\255' '0'-'9'] + +let digit = + ['0'-'9'] + +(* -------------------------------------------------------------------------- *) + +(* The lexer. *) + +rule entry = parse +| "fun" + { FUN } +| "in" + { IN } +| "let" + { LET } +| "print" + { PRINT } +| "rec" + { REC } +| "->" + { ARROW } +| "=" + { EQ } +| "(" + { LPAREN } +| ")" + { RPAREN } +| "+" + { ADDOP OpAdd } +| "-" + { ADDOP OpSub } +| "*" + { MULOP OpMul } +| "/" + { MULOP OpDiv } +| (lowercase identchar *) as x + { IDENT x } +| digit+ as i + { try + INTLITERAL (int_of_string i) + with Failure _ -> + error (place lexbuf) "invalid integer literal." } +| "(*" + { ocamlcomment (place lexbuf) lexbuf; entry lexbuf } +| newline + { new_line lexbuf; entry lexbuf } +| whitespace+ + { entry lexbuf } +| eof + { EOF } +| _ as c + { error (place lexbuf) "unexpected character: '%c'." c } + +(* ------------------------------------------------------------------------ *) + + (* Skip OCaml-style comments. Comments can be nested. This sub-lexer is + parameterized with the place of the opening comment, so if an unterminated + comment is detected, we can show where it was opened. *) + +and ocamlcomment p = parse +| "*)" + { () } +| "(*" + { ocamlcomment (place lexbuf) lexbuf; ocamlcomment p lexbuf } +| newline + { new_line lexbuf; ocamlcomment p lexbuf } +| eof + { error p "unterminated comment." } +| _ + { ocamlcomment p lexbuf } diff --git a/projet/src/Main.ml b/projet/src/Main.ml new file mode 100644 index 0000000..9c8eed4 --- /dev/null +++ b/projet/src/Main.ml @@ -0,0 +1,101 @@ +(* -------------------------------------------------------------------------- *) + +(* Parse the command line. *) + +let debug = + ref false + +let filenames = + ref [] + +let record filename = + filenames := filename :: !filenames + +let options = + Arg.align [ + "--debug", Arg.Set debug, " Enable debugging output"; + ] + +let usage = + Printf.sprintf "Usage: %s " Sys.argv.(0) + +let () = + Arg.parse options record usage + +let debug = + !debug + +let filenames = + List.rev !filenames + +(* -------------------------------------------------------------------------- *) + +(* Printing a syntax tree in an intermediate language (for debugging). *) + +let print_delimiter () = + Printf.eprintf "----------------------------------------"; + Printf.eprintf "----------------------------------------\n" + +let dump (phase : string) (show : 'term -> string) (t : 'term) = + if debug then begin + print_delimiter(); + Printf.eprintf "%s:\n\n%s\n\n%!" phase (show t) + end; + t + +(* -------------------------------------------------------------------------- *) + +(* Reading and parsing a file. *) + +let read filename : RawLambda.term = + try + let contents = Utils.file_get_contents filename in + let lexbuf = Lexing.from_string contents in + Error.set_filename lexbuf filename; + try + Parser.entry Lexer.entry lexbuf + with + | Parser.Error -> + Error.error (Error.place lexbuf) "Syntax error." + with + | Sys_error msg -> + prerr_endline msg; + exit 1 + +(* -------------------------------------------------------------------------- *) + +(* Printing the final C program on the standard output channel. *) + +let output (p : C.program) : unit = + Printf.printf "#include\n"; + Printf.printf "#include\n"; + Printf.printf "#include \"prologue.h\"\n\n"; + let print_program = PrintCommon.print_program PrintC.p_decl_or_function in + let buf = Buffer.create 1024 in + PrintCommon.printf_of_pprint_pretty print_program buf p; + print_endline (Buffer.contents buf) + +(* -------------------------------------------------------------------------- *) + +(* The complete processing pipeline. Beautiful, isn't it? *) + +let process filename = + filename + |> read + |> dump "RawLambda" RawLambda.show_term + |> Cook.cook_term + |> dump "Lambda" Lambda.show_term + |> CPS.cps_term + |> dump "Tail" Tail.show_term + |> Defun.defun_term + |> dump "Top" Top.show_program + |> Finish.finish_program + |> dump "C" C.show_program + |> output + +(* -------------------------------------------------------------------------- *) + +(* The main program. *) + +let () = + List.iter process filenames diff --git a/projet/src/Makefile b/projet/src/Makefile new file mode 100644 index 0000000..73f39e1 --- /dev/null +++ b/projet/src/Makefile @@ -0,0 +1,28 @@ +SHELL := bash +TARGET := Main.native +JOUJOU := joujou +DIRS := kremlin,alphalib +OCAMLBUILD :=\ + ocamlbuild \ + -classic-display \ + -j 4 \ + -use-ocamlfind \ + -use-menhir \ + -menhir "menhir -lg 1 -la 1 --explain" \ + -Is $(DIRS) \ + +.PHONY: all test clean + +all: + @ $(OCAMLBUILD) -quiet $(TARGET) + @ ln -sf $(TARGET) $(JOUJOU) + +test: all + @ make -C test test + +clean: + rm -f *~ + rm -f tests/*.c tests/*.out + $(OCAMLBUILD) -clean + rm -f $(TARGET) $(JOUJOU) + $(MAKE) -C test clean diff --git a/projet/src/Parser.mly b/projet/src/Parser.mly new file mode 100644 index 0000000..4f032b2 --- /dev/null +++ b/projet/src/Parser.mly @@ -0,0 +1,155 @@ +%token IDENT +%token INTLITERAL +%token FUN IN LET PRINT REC +%token ARROW EQ LPAREN RPAREN +%token MULOP ADDOP +%token EOF + +%start entry + +%{ + +open RawLambda + +%} + +%% + +(* -------------------------------------------------------------------------- *) + +(* A toplevel phrase is just a term. *) + +entry: + t = any_term EOF + { t } + +(* -------------------------------------------------------------------------- *) + +(* The syntax of terms is stratified as follows: + + atomic_term -- unambiguously delimited terms + application_term -- n-ary applications of atomic terms + multiplicative_term -- built using multiplication & division + additive_term -- built using addition & subtraction + any_term -- everything + + A [match/with/end] construct is terminated with an [end] keyword, as in Coq, + so it is an atomic term. *) + +atomic_term_: +| LPAREN t = any_term RPAREN + { t.value } +| x = IDENT + { Var x } +| i = INTLITERAL + { Lit i } + +application_term_: +| t = atomic_term_ + { t } +| t1 = placed(application_term_) t2 = placed(atomic_term_) + { App (t1, t2) } +| PRINT t2 = placed(atomic_term_) + { Print t2 } + +%inline multiplicative_term_: + t = left_associative_level(application_term_, MULOP, mkbinop) + { t } + +%inline additive_term_: + t = left_associative_level(multiplicative_term_, ADDOP, mkbinop) + { t } + +any_term_: +| t = additive_term_ + { t } +| FUN x = IDENT ARROW t = any_term + { Lam (x, t) } +| LET mode = recursive x = IDENT EQ t1 = any_term IN t2 = any_term + { Let (mode, x, t1, t2) } + +%inline any_term: + t = placed(any_term_) + { t } + +(* -------------------------------------------------------------------------- *) + +(* An infix-left-associative-operator level in a hierarchy of arithmetic + expressions. *) + +(* [basis] is the next lower level in the hierarchy. + [op] is the category of binary operators. + [action] is a ternary sequencing construct. *) + +left_associative_level(basis, op, action): +| t = basis +| t = action( + left_associative_level(basis, op, action), + op, + basis + ) + { t } + +(* -------------------------------------------------------------------------- *) + +(* A ternary sequence whose semantic action builds a [BinOp] node. *) + +%inline mkbinop(term1, op, term2): + t1 = placed(term1) op = op t2 = placed(term2) + { BinOp (t1, op, t2) } + +(* -------------------------------------------------------------------------- *) + +(* A [let] construct carries an optional [rec] annotation. *) + +recursive: +| REC { Recursive } +| { NonRecursive } + +(* -------------------------------------------------------------------------- *) + +(* A term is annotated with its start and end positions, for use in error + messages. *) + +%inline placed(X): + x = X + { { place = ($startpos, $endpos); value = x } } + +(* -------------------------------------------------------------------------- *) + +(* In a right-flexible list, the last delimiter is optional, i.e., [delim] can + be viewed as a terminator or a separator, as desired. *) + +(* There are several ways of expressing this. One could say it is either a + separated list or a terminated list; this works if one uses right recursive + lists. Or, one could say that it is a separated list followed with an + optional delimiter; this works if one uses a left-recursive list. The + following formulation is direct and seems most natural. It should lead to + the smallest possible automaton. *) + +right_flexible_list(delim, X): +| (* nothing *) + { [] } +| x = X + { [x] } +| x = X delim xs = right_flexible_list(delim, X) + { x :: xs } + +(* In a left-flexible list, the first delimiter is optional, i.e., [delim] can + be viewed as an opening or as a separator, as desired. *) + +(* Again, there are several ways of expressing this, and again, I suppose the + following formulation is simplest. It is the mirror image of the above + definition, so it is naturally left-recursive, this time. *) + +reverse_left_flexible_list(delim, X): +| (* nothing *) + { [] } +| x = X + { [x] } +| xs = reverse_left_flexible_list(delim, X) delim x = X + { x :: xs } + +%inline left_flexible_list(delim, X): + xs = reverse_left_flexible_list(delim, X) + { List.rev xs } diff --git a/projet/src/RawLambda.ml b/projet/src/RawLambda.ml new file mode 100644 index 0000000..51f33e1 --- /dev/null +++ b/projet/src/RawLambda.ml @@ -0,0 +1,55 @@ +(* Variables are strings. *) + +type variable = + string + +(* Every [let] binding is marked recursive or nonrecursive. *) + +and recursive = + | Recursive + | NonRecursive + +(* The four standard integer arithmetic operations are supported. *) + +and binop = + | OpAdd + | OpSub + | OpMul + | OpDiv + +(* This language is the untyped lambda-calculus, extended with possibly + recursive [let] bindings, integer literals (that is, constants), integer + arithmetic operations, and the primitive operation [print], which prints an + integer value and returns it. *) + +and term_ = + | Var of variable + | Lam of variable * term + | App of term * term + | Lit of int + | BinOp of term * binop * term + | Print of term + | Let of recursive * variable * term * term + +(* Every abstract syntax tree node of type [term] is annotated with a place, + that is, a position in the source code. This allows us to produce a good + error message when a problem is detected. *) + +and term = + term_ placed + +(* A value of type ['a placed] can be thought of as a value of type ['a] + decorated with a place. *) + +and 'a placed = { + place: Error.place; + value: 'a + } + +(* The following annotation requests the automatic generation of a [show_] + function for each of the types defined above. For instance, the function + [show_term], of type [term -> string], converts a term to a string. These + functions can be useful during debugging. Running with [--debug] causes + every intermediate abstract syntax tree to be displayed in this way. *) + +[@@deriving show { with_path = false }] diff --git a/projet/src/Tail.ml b/projet/src/Tail.ml new file mode 100644 index 0000000..44b8b6d --- /dev/null +++ b/projet/src/Tail.ml @@ -0,0 +1,132 @@ +(* This intermediate language describes the result of the CPS transformation. + It is a lambda-calculus where the ordering of computations is explicit and + where every function call is a tail call. + + The following syntactic categories are distinguished: + + 1. "Values" include variables, integer literals, and applications of the + primitive integer operations to values. Instead of "values", they could + also be referred to as "pure expressions". They are expressions whose + evaluation terminates and has no side effect, not even memory + allocation. + + 2. "Blocks" include lambda-abstractions. Even though lambda-abstractions + are traditionally considered values, here, they are viewed as + expressions whose evaluation has a side effect, namely, the allocation + of a memory block. + + 3. "Terms" are expressions with side effects. Terms always appear in tail + position: an examination of the syntax of terms shows that a term can be + viewed as a sequence of [LetVal], [LetBlo] and [Print] instructions, + terminated with either [Exit] or [TailCall]. This implies, in + particular, that every call is a tail call. + + In contrast with the surface language, where every lambda-abstraction has + arity 1, in this calculus, lambda-abstractions of arbitrary arity are + supported. A lambda-abstraction [Lam] carries a list of formal arguments + and a function call [TailCall] carries a list of actual arguments. Partial + applications or over-applications are not supported: it is the programmer's + responsibility to ensure that every function call provides exactly as many + arguments as expected by the called function. *) + +type variable = + Atom.atom + +and self = Lambda.self = + | Self of variable + | NoSelf + +and binop = Lambda.binop = + | OpAdd + | OpSub + | OpMul + | OpDiv + +and value = + | VVar of variable + | VLit of int + | VBinOp of value * binop * value + +and block = + | Lam of self * variable list * term + +(* Terms include the following constructs: + + - The primitive operation [Exit] stops the program. + + - The tail call [TailCall (v, vs)] transfers control to the function [v] + with actual arguments [vs]. (The value [v] should be a function and its + arity should be the length of [vs].) + + - The term [Print (v, t)] prints the value [v], then executes the term [t]. + (The value [v] should be a primitive integer value.) + + - The term [LetVal (x, v, t)] binds the variable [x] to the value [v], then + executes the term [t]. + + - The term [LetBlo (x, b, t)] allocates the memory block [b] and binds the + variable [x] to its address, then executes the term [t]. *) + +and term = + | Exit + | TailCall of value * value list + | Print of value * term + | LetVal of variable * value * term + | LetBlo of variable * block * term + +[@@deriving show { with_path = false }] + +(* -------------------------------------------------------------------------- *) + +(* Constructor functions. *) + +let vvar x = + VVar x + +let vvars xs = + List.map vvar xs + +(* -------------------------------------------------------------------------- *) + +(* Computing the free variables of a value, block, or term. *) + +open Atom.Set + +let rec fv_value (v : value) = + match v with + | VVar x -> + singleton x + | VLit _ -> + empty + | VBinOp (v1, _, v2) -> + union (fv_value v1) (fv_value v2) + +and fv_values (vs : value list) = + union_many fv_value vs + +and fv_lambda (xs : variable list) (t : term) = + diff (fv_term t) (of_list xs) + +and fv_block (b : block) = + match b with + | Lam (NoSelf, xs, t) -> + fv_lambda xs t + | Lam (Self f, xs, t) -> + remove f (fv_lambda xs t) + +and fv_term (t : term) = + match t with + | Exit -> + empty + | TailCall (v, vs) -> + fv_values (v :: vs) + | Print (v1, t2) -> + union (fv_value v1) (fv_term t2) + | LetVal (x, v1, t2) -> + union + (fv_value v1) + (remove x (fv_term t2)) + | LetBlo (x, b1, t2) -> + union + (fv_block b1) + (remove x (fv_term t2)) diff --git a/projet/src/Top.ml b/projet/src/Top.ml new file mode 100644 index 0000000..edbb0fb --- /dev/null +++ b/projet/src/Top.ml @@ -0,0 +1,94 @@ +(* This intermediate language describes the result of defunctionalization. + It retains the key features of the previous calculus, [Tail], in that + the ordering of computations is explicit and every function call is a + tail call. Furthermore, lambda-abstractions disappear. A memory block + [Con] now contains an integer tag followed with a number of fields, + which hold values. A [switch] construct appears, which allows testing + the tag of a memory block. A number of (closed, mutually recursive) + functions can be defined at the top level. *) + +type tag = + int + +and variable = + Atom.atom + +and binop = Tail.binop = + | OpAdd + | OpSub + | OpMul + | OpDiv + +and value = Tail.value = + | VVar of variable + | VLit of int + | VBinOp of value * binop * value + +(* A block contains an integer tag, followed with a number of fields. *) + +and block = + | Con of tag * value list + +(* The construct [Swi (v, branches)] reads the integer tag stored in the + memory block at address [v] and performs a case analysis on this tag, + transferring control to the appropriate branch. (The value [v] should be a + pointer to a memory block.) *) + +and term = + | Exit + | TailCall of variable * value list + | Print of value * term + | LetVal of variable * value * term + | LetBlo of variable * block * term + | Swi of value * branch list + +(* A branch [tag xs -> t] is labeled with an integer tag [tag], and is + executed if the memory block carries this tag. The variables [xs] are + then bounds to the fields of the memory block. (The length of the list + [xs] should be the number of fields of the memory block.) *) + +and branch = + | Branch of tag * variable list * term + +(* A toplevel function declaration mentions the function's name, formal + parameters, and body. *) + +and function_declaration = + | Fun of variable * variable list * term + +(* A complete program consits of a set of toplevel function declarations + and a term (the "main program"). The functions are considered mutually + recursive: every function may refer to every function. *) + +and program = + | Prog of function_declaration list * term + +[@@deriving show { with_path = false }] + +(* -------------------------------------------------------------------------- *) + +(* Constructor functions. *) + +let vvar = + Tail.vvar + +let vvars = + Tail.vvars + +(* [let x_1 = v_1 in ... let x_n = v_n in t] *) + +let rec sequential_let (xs : variable list) (vs : value list) (t : term) = + match xs, vs with + | [], [] -> + t + | x :: xs, v :: vs -> + LetVal (x, v, sequential_let xs vs t) + | _ -> + assert false + +(* [let x_1 = v_1 and ... x_n = v_n in t] *) + +let parallel_let (xs : variable list) (vs : value list) (t : term) = + assert (List.length xs = List.length vs); + assert (Atom.Set.disjoint (Atom.Set.of_list xs) (Tail.fv_values vs)); + sequential_let xs vs t diff --git a/projet/src/_tags b/projet/src/_tags new file mode 100644 index 0000000..bf1bddf --- /dev/null +++ b/projet/src/_tags @@ -0,0 +1,8 @@ +true: \ + debug, \ + strict_sequence, \ + warn(A-3-4-30-44-42-45-50), \ + package(unix), \ + package(process), \ + package(pprint), \ + package(ppx_deriving.std) diff --git a/projet/src/alphalib/Atom.ml b/projet/src/alphalib/Atom.ml new file mode 100644 index 0000000..6502410 --- /dev/null +++ b/projet/src/alphalib/Atom.ml @@ -0,0 +1,215 @@ +(* -------------------------------------------------------------------------- *) + +(* We impose maximal sharing on strings so as to reduce the total amount of + space that they occupy. This is done using a weak hash set. *) + +module StringStorage = + Weak.Make(struct + type t = string + let equal (s1 : string) (s2 : string) = (s1 = s2) + let hash = Hashtbl.hash + end) + +let share : string -> string = + StringStorage.merge (StringStorage.create 128) + +(* -------------------------------------------------------------------------- *) + +(* Removing any trailing digits in a string. *) + +let is_digit c = + Char.code '0' <= Char.code c && Char.code c <= Char.code '9' + +let remove_trailing_digits (s : string) : string = + let n = ref (String.length s) in + while !n > 0 && is_digit s.[!n-1] do n := !n-1 done; + (* We assume that there is at least one non-digit character in the string. *) + assert (!n > 0); + String.sub s 0 !n + +(* -------------------------------------------------------------------------- *) + +(* An atom is implemented as a pair of an integer identity and a string that + serves as a printing hint. *) + +(* We maintain the invariant that a hint is nonempty and does not end in a + digit. This allows us to later produce unique identifiers, without risk of + collisions, by concatenating a hint and a unique number. *) + +(* To preserve space, hints are maximally shared. This is not essential for + correctness, though. *) + +type atom = { identity: int; hint: string } + +and t = atom + [@@deriving show { with_path = false }] + +let identity a = + a.identity + +let hint a = + a.hint + +(* -------------------------------------------------------------------------- *) + +(* A global integer counter holds the next available identity. *) + +let counter = + ref 0 + +let allocate () = + let number = !counter in + counter := number + 1; + assert (number >= 0); + number + +(* [fresh hint] produces a fresh atom. *) + +(* The argument [hint] must not be a string of digits. *) + +let fresh hint = + let identity = allocate() + and hint = share (remove_trailing_digits hint) in + { identity; hint } + +(* [copy a] returns a fresh atom modeled after the atom [a]. *) + +let copy a = + fresh a.hint + +(* -------------------------------------------------------------------------- *) + +(* Comparison of atoms. *) + +let equal a b = + a.identity = b.identity + +let compare a b = + (* Identities are always positive numbers (see [allocate] above) + so I believe overflow is impossible here. *) + a.identity - b.identity + +let hash a = + Hashtbl.hash a.identity + +(* -------------------------------------------------------------------------- *) + +(* A scratch buffer for printing. *) + +let scratch = + Buffer.create 1024 + +(* [print_separated_sequence] prints a sequence of elements into the [scratch] + buffer. The sequence is given by the higher-order iterator [iter], applied + to the collection [xs]. The separator is the string [sep]. Each element is + transformed to a string by the function [show]. *) + +let print_separated_sequence show sep iter xs : unit = + let first = ref true in + iter (fun x -> + if !first then begin + Buffer.add_string scratch (show x); + first := false + end + else begin + Buffer.add_string scratch sep; + Buffer.add_string scratch (show x) + end + ) xs + +(* -------------------------------------------------------------------------- *) + +(* Sets and maps. *) + +module Order = struct + type t = atom + let compare = compare +end + +module Set = struct + + include Set.Make(Order) + + (* A disjointness test. *) + + let disjoint xs ys = + is_empty (inter xs ys) + + (* Iterated union. *) + + let union_many (f : 'a -> t) (xs : 'a list) : t = + List.fold_left (fun accu x -> + union accu (f x) + ) empty xs + + (* Disjoint union. *) + + exception NonDisjointUnion of atom + + let disjoint_union xs ys = + match choose (inter xs ys) with + | exception Not_found -> + (* The intersection of [xs] and [ys] is empty. Return their union. *) + union xs ys + | x -> + (* The intersection contains [x]. Raise an exception. *) + raise (NonDisjointUnion x) + + let handle_NonDisjointUnion f x = + try + f x; true + with NonDisjointUnion a -> + Printf.eprintf "NonDisjointUnion: %s\n%!" (show a); + false + + (* Sets of atoms form a monoid under union. *) + + class ['z] union_monoid = object + method zero: 'z = empty + method plus: 'z -> 'z -> 'z = union + end + + (* Sets of atoms form a monoid under disjoint union. *) + + class ['z] disjoint_union_monoid = object + method zero: 'z = empty + method plus: 'z -> 'z -> 'z = disjoint_union + end + + (* These printing functions should be used for debugging purposes only. *) + + let print_to_scratch xs = + Buffer.clear scratch; + Buffer.add_string scratch "{"; + print_separated_sequence show ", " iter xs; + Buffer.add_string scratch "}" + + let show xs = + print_to_scratch xs; + let result = Buffer.contents scratch in + Buffer.reset scratch; + result + + let print oc xs = + print_to_scratch xs; + Buffer.output_buffer oc scratch; + Buffer.reset scratch + +end + +module Map = struct + + include Map.Make(Order) + + (* This is O(nlog n), whereas in principle O(n) is possible. + The abstraction barrier in OCaml's [Set] module hinders us. *) + let domain m = + fold (fun a _ accu -> Set.add a accu) m Set.empty + + let codomain f m = + fold (fun _ v accu -> Set.union (f v) accu) m Set.empty + +end + +type renaming = + atom Map.t diff --git a/projet/src/alphalib/Atom.mli b/projet/src/alphalib/Atom.mli new file mode 100644 index 0000000..6ec6206 --- /dev/null +++ b/projet/src/alphalib/Atom.mli @@ -0,0 +1,71 @@ +(* TEMPORARY document *) + +(* An atom is a pair of a unique integer identity and a (not necessarily + unique) string. *) + +type atom + +and t = atom + [@@deriving show] + +val identity: atom -> int +val hint: atom -> string + +(* Producing fresh atoms. *) + +val fresh: string -> atom +val copy: atom -> atom + +(* Comparison of atoms. *) + +val equal: atom -> atom -> bool +val compare: atom -> atom -> int +val hash: atom -> int + +(* Sets. *) + +module Set : sig + include Set.S with type elt = atom + + val disjoint: t -> t -> bool + + val union_many: ('a -> t) -> 'a list -> t + + (* Disjoint union. *) + + exception NonDisjointUnion of atom + val disjoint_union: t -> t -> t + + val handle_NonDisjointUnion: ('a -> unit) -> 'a -> bool + + (* Sets of atoms form monoids under union and disjoint union. *) + + class ['z] union_monoid : object + constraint 'z = t + method zero: 'z + method plus: 'z -> 'z -> 'z + end + + class ['z] disjoint_union_monoid : object + constraint 'z = t + method zero: 'z + method plus: 'z -> 'z -> 'z + end + + val show: t -> string + val print: out_channel -> t -> unit +end + +(* Maps. *) + +module Map : sig + + include Map.S with type key = atom + + val domain: 'a t -> Set.t + val codomain: ('a -> Set.t) -> 'a t -> Set.t + +end + +type renaming = + atom Map.t diff --git a/projet/src/kremlin/C.ml b/projet/src/kremlin/C.ml new file mode 100644 index 0000000..db29569 --- /dev/null +++ b/projet/src/kremlin/C.ml @@ -0,0 +1,126 @@ + + +module K = Constant +open Common + +(* This pretty-printer based on: http:/ /en.cppreference.com/w/c/language/declarations + * Many cases are omitted from this bare-bones C grammar; hopefully, to be extended. *) +type type_spec = + | Int of Constant.width + | Void + | Named of ident + | Struct of ident option * declaration list option + (** Note: the option allows for zero-sized structs (GCC's C, C++) but as + * of 2017/05/14 we never generate these. *) + | Union of ident option * declaration list + | Enum of ident option * ident list + (** not encoding all the invariants here *) + [@@deriving show { with_path = false }] + +and storage_spec = + | Typedef + | Extern + | Static + +and declarator_and_init = + declarator * init option + +and declarator_and_inits = + declarator_and_init list + +and declarator = + | Ident of ident + | Pointer of declarator + | Array of declarator * expr + | Function of calling_convention option * declarator * params + +and expr = + | Op1 of K.op * expr + | Op2 of K.op * expr * expr + | Index of expr * expr + | Deref of expr + | Address of expr + | Member of expr * expr + | MemberP of expr * expr + | Assign of expr * expr + (** not covering *=, +=, etc. *) + | Call of expr * expr list + | Name of ident + | Cast of type_name * expr + | Literal of string + | Constant of K.t + | Bool of bool + | Sizeof of expr + | SizeofType of type_spec + | CompoundLiteral of type_name * init list + | MemberAccess of expr * ident + | MemberAccessPointer of expr * ident + | InlineComment of string * expr * string + | Type of type_name + (** note: these two not in the C grammar *) + +(** this is a WILD approximation of the notion of "type name" in C _and_ a hack + * because there's the invariant that the ident found at the bottom of the + * [declarator] is irrelevant... *) +and type_name = + type_spec * declarator + +and params = + (* No support for old syntax, e.g. K&R, or [void f(void)]. *) + param list + +and param = + (** Also approximate. *) + type_spec * declarator + +and declaration = + type_spec * storage_spec option * declarator_and_inits + +and ident = + string + +and init = + | InitExpr of expr + | Designated of designator * init + | Initializer of init list + +and designator = + | Dot of ident + | Bracket of int + +(** Note: according to http:/ /en.cppreference.com/w/c/language/statements, + * declarations can only be part of a compound statement... we do not enforce + * this invariant via the type [stmt], but rather check it at runtime (see + * [mk_compound_if]), as the risk of messing things up, naturally. *) +type stmt = + | Compound of stmt list + | DeclStmt of declaration + | Expr of expr + | If of expr * stmt + | IfElse of expr * stmt * stmt + | While of expr * stmt + | For of declaration_or_expr * expr * expr * stmt + | Return of expr option + | Switch of expr * (expr * stmt) list * stmt + (** the last component is the default statement *) + | Break + | Comment of string + (** note: this is not in the C grammar *) + +and program = + declaration_or_function list + +and comment = + string + +and declaration_or_function = + | Decl of comment list * declaration + | Function of comment list * bool * declaration * stmt + (** [stmt] _must_ be a compound statement; boolean is inline *) + +and declaration_or_expr = [ + | `Decl of declaration + | `Expr of expr + | `Skip +] +[@@deriving show { with_path = false }] diff --git a/projet/src/kremlin/Common.ml b/projet/src/kremlin/Common.ml new file mode 100644 index 0000000..77638d7 --- /dev/null +++ b/projet/src/kremlin/Common.ml @@ -0,0 +1,19 @@ +type calling_convention = + | StdCall + | CDecl + | FastCall + [@@deriving show { with_path = false }] + +type lifetime = + | Eternal + | Stack + [@@deriving show { with_path = false }] + +type flag = + | Private + | NoExtract + | CInline + | Substitute + | GcType + | Comment of string + [@@deriving show { with_path = false }] diff --git a/projet/src/kremlin/Constant.ml b/projet/src/kremlin/Constant.ml new file mode 100644 index 0000000..5533d38 --- /dev/null +++ b/projet/src/kremlin/Constant.ml @@ -0,0 +1,66 @@ +(** Machine integers. Don't repeat the same thing everywhere. *) + +type t = width * string + [@@deriving show] + +and width = + | UInt8 | UInt16 | UInt32 | UInt64 | Int8 | Int16 | Int32 | Int64 + | Bool + | CInt (** Checked signed integers. *) + +let bytes_of_width (w: width) = + match w with + | UInt8 -> 1 + | UInt16 -> 2 + | UInt32 -> 4 + | UInt64 -> 8 + | Int8 -> 1 + | Int16 -> 2 + | Int32 -> 4 + | Int64 -> 8 + | CInt -> 4 + | _ -> invalid_arg "bytes_of_width" + +type op = + (* Arithmetic operations *) + | Add | AddW | Sub | SubW | Div | DivW | Mult | MultW | Mod + (* Bitwise operations *) + | BOr | BAnd | BXor | BShiftL | BShiftR | BNot + (* Arithmetic comparisons / boolean comparisons *) + | Eq | Neq | Lt | Lte | Gt | Gte + (* Boolean operations *) + | And | Or | Xor | Not + (* Effectful operations *) + | Assign | PreIncr | PreDecr | PostIncr | PostDecr + (* Misc *) + | Comma + [@@deriving show { with_path = false }] + +let unsigned_of_signed = function + | Int8 -> UInt8 + | Int16 -> UInt16 + | Int32 -> UInt32 + | Int64 -> UInt64 + | CInt | UInt8 | UInt16 | UInt32 | UInt64 | Bool -> raise (Invalid_argument "unsigned_of_signed") + +let is_signed = function + | Int8 | Int16 | Int32 | Int64 | CInt -> true + | UInt8 | UInt16 | UInt32 | UInt64 -> false + | Bool -> raise (Invalid_argument "is_signed") + +let is_unsigned w = not (is_signed w) + +let without_wrap = function + | AddW -> Add + | SubW -> Sub + | MultW -> Mult + | DivW -> Div + | _ -> raise (Invalid_argument "without_wrap") + +let uint8_of_int i = + assert (i < (1 lsl 8) && i >= 0); + UInt8, string_of_int i + +let uint32_of_int i = + assert (i < (1 lsl 32) && i >= 0); + UInt32, string_of_int i diff --git a/projet/src/kremlin/Options.ml b/projet/src/kremlin/Options.ml new file mode 100644 index 0000000..fd6bfbf --- /dev/null +++ b/projet/src/kremlin/Options.ml @@ -0,0 +1,7 @@ +(* By default, the C printer inserts a minimal amount of parentheses. + However, this can trigger GCC and Clang's -Wparentheses warning, + so there is an option to produce more parentheses than strictly + necessary. *) + +let parentheses = + ref false diff --git a/projet/src/kremlin/PrintC.ml b/projet/src/kremlin/PrintC.ml new file mode 100644 index 0000000..34583ee --- /dev/null +++ b/projet/src/kremlin/PrintC.ml @@ -0,0 +1,354 @@ +(** Pretty-printer that conforms with C syntax. Also defines the grammar of + * concrete C syntax, as opposed to our idealistic, well-formed C*. *) + +open PPrint +open PrintCommon +open C + +(* Pretty-printing of actual C syntax *****************************************) + +let p_storage_spec = function + | Typedef -> string "typedef" + | Extern -> string "extern" + | Static -> string "static" + +let rec p_type_spec = function + | Int w -> print_width w ^^ string "_t" + | Void -> string "void" + | Named s -> string s + | Union (name, decls) -> + group (string "union" ^/^ + (match name with Some name -> string name ^^ break1 | None -> empty)) ^^ + braces_with_nesting (separate_map hardline (fun p -> group (p_declaration p ^^ semi)) decls) + | Struct (name, decls) -> + group (string "struct" ^/^ + (match name with Some name -> string name | None -> empty)) ^^ + (match decls with + | Some decls -> + break1 ^^ braces_with_nesting (separate_map hardline (fun p -> group (p_declaration p ^^ semi)) decls) + | None -> + empty) + | Enum (name, tags) -> + group (string "enum" ^/^ + (match name with Some name -> string name ^^ break1 | None -> empty)) ^^ + braces_with_nesting (separate_map (comma ^^ break1) string tags) + + +and p_type_declarator d = + let rec p_noptr = function + | Ident n -> + string n + | Array (d, s) -> + p_noptr d ^^ lbracket ^^ p_expr s ^^ rbracket + | Function (cc, d, params) -> + let cc = match cc with Some cc -> print_cc cc ^^ break1 | None -> empty in + group (cc ^^ p_noptr d ^^ parens_with_nesting (separate_map (comma ^^ break 1) (fun (spec, decl) -> + group (p_type_spec spec ^/^ p_any decl) + ) params)) + | d -> + lparen ^^ p_any d ^^ rparen + and p_any = function + | Pointer d -> + star ^^ p_any d + | d -> + p_noptr d + in + p_any d + +and p_type_name (spec, decl) = + match decl with + | Ident "" -> + p_type_spec spec + | _ -> + p_type_spec spec ^^ space ^^ p_type_declarator decl + +(* http:/ /en.cppreference.com/w/c/language/operator_precedence *) +and prec_of_op2 op = + let open Constant in + match op with + | AddW | SubW | MultW | DivW -> failwith "[prec_of_op]: should've been desugared" + | Add -> 4, 4, 4 + | Sub -> 4, 4, 3 + | Div -> 3, 3, 2 + | Mult -> 3, 3, 3 + | Mod -> 3, 3, 2 + | BOr -> 10, 10, 10 + | BAnd -> + 8, 8, 8 + | BXor | Xor -> 9, 9, 9 + | BShiftL | BShiftR -> + 5, 5, 4 + | Eq | Neq -> 7, 7, 7 + | Lt | Lte | Gt | Gte -> 6, 6, 5 + | And -> 11, 11, 11 + | Or -> 12, 12, 12 + | Assign -> 14, 13, 14 + | Comma -> 15, 15, 14 + | PreIncr | PostIncr | PreDecr | PostDecr | Not | BNot -> raise (Invalid_argument "prec_of_op2") + +and prec_of_op1 op = + let open Constant in + match op with + | PreDecr | PreIncr | Not | BNot -> 2 + | PostDecr | PostIncr -> 1 + | _ -> raise (Invalid_argument "prec_of_op1") + +and is_prefix op = + let open Constant in + match op with + | PreDecr | PreIncr | Not | BNot -> true + | PostDecr | PostIncr -> false + | _ -> raise (Invalid_argument "is_prefix") + +(* The precedence level [curr] is the maximum precedence the current node should + * have. If it doesn't, then it should be parenthesized. Lower numbers bind + * tighter. *) +and paren_if curr mine doc = + if curr < mine then + group (lparen ^^ doc ^^ rparen) + else + doc + +(* [e] is an operand of [op]; is this likely to trigger GCC's -Wparentheses? If + * so, downgrade the current precedence to 0 to force parenthses. *) +and defeat_Wparentheses op e prec = + let open Constant in + if not !Options.parentheses then + prec + else match op, e with + | (BShiftL | BShiftR | BXor | BOr | BAnd), Op2 ((Add | Sub | BOr | BXor), _, _) -> + 0 + | _ -> + prec + +and p_expr' curr = function + | Op1 (op, e1) -> + let mine = prec_of_op1 op in + let e1 = p_expr' mine e1 in + paren_if curr mine (if is_prefix op then print_op op ^^ e1 else e1 ^^ print_op op) + | Op2 (op, e1, e2) -> + let mine, left, right = prec_of_op2 op in + let left = defeat_Wparentheses op e1 left in + let right = defeat_Wparentheses op e2 right in + let e1 = p_expr' left e1 in + let e2 = p_expr' right e2 in + paren_if curr mine (e1 ^/^ print_op op ^^ jump e2) + | Index (e1, e2) -> + let mine, left, right = 1, 1, 15 in + let e1 = p_expr' left e1 in + let e2 = p_expr' right e2 in + paren_if curr mine (e1 ^^ lbracket ^^ e2 ^^ rbracket) + | Assign (e1, e2) -> + let mine, left, right = 14, 13, 14 in + let e1 = p_expr' left e1 in + let e2 = p_expr' right e2 in + paren_if curr mine (group (e1 ^/^ equals) ^^ jump e2) + | Call (e, es) -> + let mine, left, arg = 1, 1, 14 in + let e = p_expr' left e in + let es = nest 2 (separate_map (comma ^^ break 1) (fun e -> group (p_expr' arg e)) es) in + paren_if curr mine (e ^^ lparen ^^ es ^^ rparen) + | Literal s -> + dquote ^^ string s ^^ dquote + | Constant (w, s) -> + string s ^^ if K.is_unsigned w then string "U" else empty + | Name s -> + string s + | Cast (t, e) -> + let mine, right = 2, 2 in + let e = group (p_expr' right e) in + let t = p_type_name t in + paren_if curr mine (lparen ^^ t ^^ rparen ^^ e) + | Type t -> + p_type_name t + | Sizeof e -> + let mine, right = 2, 2 in + let e = p_expr' right e in + paren_if curr mine (string "sizeof" ^^ space ^^ e) + | SizeofType t -> + string "sizeof" ^^ group (lparen ^^ p_type_spec t ^^ rparen) + | Address e -> + let mine, right = 2, 2 in + let e = p_expr' right e in + paren_if curr mine (ampersand ^^ e) + | Deref e -> + let mine, right = 2, 2 in + let e = p_expr' right e in + paren_if curr mine (star ^^ e) + | Member _ | MemberP _ -> + failwith "[p_expr']: not implemented" + | Bool b -> + string (string_of_bool b) + | CompoundLiteral (t, init) -> + (* NOTE: always parenthesize compound literal no matter what, because GCC + * parses an application of a function to a compound literal as an n-ary + * application. *) + parens_with_nesting ( + lparen ^^ p_type_name t ^^ rparen ^^ + braces_with_nesting (separate_map (comma ^^ break1) p_init init) + ) + | MemberAccess (expr, member) -> + p_expr' 1 expr ^^ dot ^^ string member + | MemberAccessPointer (expr, member) -> + p_expr' 1 expr ^^ string "->" ^^ string member + | InlineComment (s, e, s') -> + surround 2 1 (p_comment s) (p_expr' curr e) (p_comment s') + +and p_comment s = + (* TODO: escape *) + string "/* " ^^ nest 2 (flow space (words s)) ^^ string " */" + + +and p_expr e = p_expr' 15 e + +and p_init (i: init) = + match i with + | Designated (designator, i) -> + group (p_designator designator ^^ space ^^ equals ^^ space ^^ p_init i) + | InitExpr e -> + p_expr' 14 e + | Initializer inits -> + let inits = + if List.length inits > 4 then + flow (comma ^^ break1) (List.map p_init inits) + else + separate_map (comma ^^ break1) p_init inits + in + braces_with_nesting inits + +and p_designator = function + | Dot ident -> + dot ^^ string ident + | Bracket i -> + lbracket ^^ int i ^^ rbracket + +and p_decl_and_init (decl, init) = + group (p_type_declarator decl ^^ match init with + | Some init -> + space ^^ equals ^^ jump (p_init init) + | None -> + empty) + +and p_declaration (spec, stor, decl_and_inits) = + let stor = match stor with Some stor -> p_storage_spec stor ^^ space | None -> empty in + stor ^^ group (p_type_spec spec) ^/^ + separate_map (comma ^^ break 1) p_decl_and_init decl_and_inits + +(* This is abusing the definition of a compound statement to ensure it is printed with braces. *) +let nest_if f stmt = + match stmt with + | Compound _ -> + hardline ^^ f stmt + | _ -> + nest 2 (hardline ^^ f stmt) + +(* A note on the classic dangling else problem. Remember that this is how things + * are parsed (note the indentation): + * + * if (foo) + * if (bar) + * ... + * else + * ... + * + * And remember that this needs braces: + * + * if (foo) { + * if (bar) + * ... + * } else + * ... + * + * [protect_solo_if] adds braces to the latter case. However, GCC, unless + * -Wnoparentheses is given, will produce a warning for the former case. + * [protect_ite_if_needed] adds braces to the former case, when the user has + * requested the extra, unnecessary parentheses needed to silence -Wparentheses. + * *) +let protect_solo_if s = + match s with + | If _ -> Compound [ s ] + | _ -> s + +let protect_ite_if_needed s = + match s with + | IfElse _ when !Options.parentheses -> Compound [ s ] + | _ -> s + +let p_or p x = + match x with + | Some x -> p x + | None -> empty + +let rec p_stmt (s: stmt) = + (* [p_stmt] is responsible for appending [semi] and calling [group]! *) + match s with + | Compound stmts -> + lbrace ^^ nest 2 (hardline ^^ separate_map hardline p_stmt stmts) ^^ + hardline ^^ rbrace + | Expr expr -> + group (p_expr expr ^^ semi) + | Comment s -> + group (string "/*" ^/^ separate break1 (words s) ^/^ string "*/") + | For (decl, e2, e3, stmt) -> + let init = match decl with + | `Decl decl -> p_declaration decl + | `Expr expr -> p_expr expr + | `Skip -> empty + in + group (string "for" ^/^ lparen ^^ nest 2 ( + init ^^ semi ^^ break1 ^^ + p_expr e2 ^^ semi ^^ break1 ^^ + p_expr e3 + ) ^^ rparen) ^^ nest_if p_stmt stmt + | If (e, stmt) -> + group (string "if" ^/^ lparen ^^ p_expr e ^^ rparen) ^^ + nest_if p_stmt (protect_ite_if_needed stmt) + | IfElse (e, s1, s2) -> + group (string "if" ^/^ lparen ^^ p_expr e ^^ rparen) ^^ + nest_if p_stmt (protect_solo_if s1) ^^ hardline ^^ + string "else" ^^ + (match s2 with + | If _ | IfElse _ -> + space ^^ p_stmt s2 + | _ -> + nest_if p_stmt s2) + | While (e, s) -> + group (string "while" ^/^ lparen ^^ p_expr e ^^ rparen) ^^ + nest_if p_stmt s + | Return None -> + group (string "return" ^^ semi) + | Return (Some e) -> + group (string "return" ^^ jump (p_expr e ^^ semi)) + | DeclStmt d -> + group (p_declaration d ^^ semi) + | Switch (e, branches, default) -> + group (string "switch" ^/^ lparen ^^ p_expr e ^^ rparen) ^/^ + braces_with_nesting ( + separate_map hardline (fun (e, s) -> + group (string "case" ^/^ p_expr e ^^ colon) ^^ nest 2 ( + hardline ^^ p_stmt s + ) + ) branches ^^ hardline ^^ + string "default:" ^^ nest 2 ( + hardline ^^ p_stmt default + ) + ) + | Break -> + string "break" ^^ semi + +let p_comments cs = + separate_map hardline (fun c -> string ("/*\n" ^ c ^ "\n*/")) cs ^^ + if List.length cs > 0 then hardline else empty + +let p_decl_or_function (df: declaration_or_function) = + match df with + | Decl (comments, d) -> + p_comments comments ^^ + group (p_declaration d ^^ semi) + | Function (comments, inline, d, stmt) -> + p_comments comments ^^ + let inline = if inline then string "inline" ^^ space else empty in + inline ^^ group (p_declaration d) ^/^ p_stmt stmt + +let print_files = + PrintCommon.print_files p_decl_or_function diff --git a/projet/src/kremlin/PrintCommon.ml b/projet/src/kremlin/PrintCommon.ml new file mode 100644 index 0000000..84e2b40 --- /dev/null +++ b/projet/src/kremlin/PrintCommon.ml @@ -0,0 +1,96 @@ +open PPrint +open Constant +open Common + +let break1 = break 1 + +let jump ?(indent=2) body = + jump indent 1 body + +let parens_with_nesting contents = + surround 2 0 lparen contents rparen + +let braces_with_nesting contents = + surround 2 1 lbrace contents rbrace + +let int i = string (string_of_int i) + +let print_width = function + | UInt8 -> string "uint8" + | UInt16 -> string "uint16" + | UInt32 -> string "uint32" + | UInt64 -> string "uint64" + | CInt -> string "krml_checked_int" + | Int8 -> string "int8" + | Int16 -> string "int16" + | Int32 -> string "int32" + | Int64 -> string "int64" + | Bool -> string "bool" + +let print_constant = function + | w, s -> string s ^^ print_width w + +let print_op = function + | Add -> string "+" + | AddW -> string "+w" + | Sub -> string "-" + | SubW -> string "-" + | Div -> string "/" + | DivW -> string "/w" + | Mult -> string "*" + | MultW -> string "*w" + | Mod -> string "%" + | BOr -> string "|" + | BAnd -> string "&" + | BXor -> string "^" + | BNot -> string "~" + | BShiftL -> string "<<" + | BShiftR -> string ">>" + | Eq -> string "==" + | Neq -> string "!=" + | Lt -> string "<" + | Lte -> string "<=" + | Gt -> string ">" + | Gte -> string ">=" + | And -> string "&&" + | Or -> string "||" + | Xor -> string "^" + | Not -> string "!" + | PostIncr | PreIncr -> string "++" + | PostDecr | PreDecr -> string "--" + | Assign -> string "=" + | Comma -> string "," + +let print_cc = function + | CDecl -> string "__cdecl" + | StdCall -> string "__stdcall" + | FastCall -> string "__fastcall" + +let print_lident (idents, ident) = + separate_map dot string (idents @ [ ident ]) + +let print_program p decls = + separate_map (hardline ^^ hardline) p decls + +let print_files print_decl files = + separate_map hardline (fun (f, p) -> + string (String.uppercase f) ^^ colon ^^ jump (print_program print_decl p) + ) files + +let printf_of_pprint f = + fun buf t -> + PPrint.ToBuffer.compact buf (f t) + +let printf_of_pprint_pretty f = + fun buf t -> + PPrint.ToBuffer.pretty 0.95 Utils.twidth buf (f t) + +let pdoc buf d = + PPrint.ToBuffer.compact buf d + +let english_join s = + match List.rev s with + | [] -> empty + | [ x ] -> x + | last :: first -> + group (separate (comma ^^ break1) (List.rev first) ^/^ string "and" ^/^ last) diff --git a/projet/src/kremlin/Utils.ml b/projet/src/kremlin/Utils.ml new file mode 100644 index 0000000..1ac1256 --- /dev/null +++ b/projet/src/kremlin/Utils.ml @@ -0,0 +1,73 @@ +let try_finally f h = let result = + try + f () + with e -> + h (); + raise e + in + h (); + result + +let with_open_in file_path f = + let c = open_in_bin file_path in + try_finally (fun () -> + f c + ) (fun () -> + close_in c + ) + +let with_open_out file_path f = + let c = open_out file_path in + try_finally (fun () -> + f c + ) (fun () -> + close_out c + ) + + +let cp dst src = with_open_out dst (fun oc -> + with_open_in src (fun ic -> + let buf = Bytes.create 2048 in + while + let l = input ic buf 0 2048 in + if l > 0 then begin + output oc buf 0 l; + true + end else + false + do () done + )) + + +let read ic = + let buf = Buffer.create 4096 in + let s = String.create 2048 in + while begin + let l = input ic s 0 (String.length s) in + if l > 0 then begin + Buffer.add_string buf (String.sub s 0 l); + true + end else begin + false + end + end do () done; + Buffer.contents buf + +let file_get_contents f = + with_open_in f read + +(** Sniff the size of the terminal for optimal use of the width. *) +let theight, twidth = + let height, width = ref 0, ref 0 in + match + Scanf.sscanf (List.hd (Process.read_stdout "stty" [|"size"|])) "%d %d" (fun h w -> + height := h; + width := w); + !height, !width + with + | exception _ -> + 24, 80 + | 0, 0 -> + 24, 80 + | h, w -> + h, w diff --git a/projet/src/prologue.h b/projet/src/prologue.h new file mode 100644 index 0000000..239161f --- /dev/null +++ b/projet/src/prologue.h @@ -0,0 +1,83 @@ +/* A forward declaration of [block] -- see below. */ + +struct block; + +/* -------------------------------------------------------------------------- */ + +/* The type [univ] is the universal type of the values that we manipulate. + A value is either an integer or a pointer to a heap-allocated memory + block. A C union is used to represent this disjunction. There is no tag + to distinguish between the two alternatives! The source program had + better be well-typed. */ + +typedef union { + + /* Either this is an integer... */ + int literal; + + /* ... or this is a pointer to a block. */ + struct block* pointer; + +} univ; + +/* -------------------------------------------------------------------------- */ + +/* The struct [block] describes the uniform layout of a heap-allocated + memory block. A block begins with an integer tag and continues with a + sequence of fields of type [univ]. */ + +struct block { + + /* Every memory block begins with an integer tag. */ + int tag; + + /* Then, we have a sequence of fields, whose number depends on the tag. + This idiom is known in C99 as a flexible array member. */ + univ data[]; + +}; + +/* -------------------------------------------------------------------------- */ + +/* Basic operations on memory blocks. */ + +/* The macro [ALLOC(n)] allocates a block of [n] fields, and is used in a + context where an expression of type [univ] is expected. We use a C + "compound literal" containing a "designated initializer" to indicate + that we wish to choose the "pointer" side of the union. We implicitly + assume that the compiler inserts no padding between the "tag" and "data" + parts of a memory block, so [(n + 1) * sizeof(univ)] is enough space to + hold a block of [n] fields. */ + +#define ALLOC(n) \ + (univ) { .pointer = malloc((n + 1) * sizeof(univ)) } + +/* In the following macros, [u] has type [univ], so [u.pointer] has type + [struct block] and is (or should be) the address of a memory block. + [i] is a field number; the numbering of fields is 0-based. */ + +#define GET_TAG(u) \ + (u.pointer->tag) + +#define SET_TAG(u,t) \ + (u.pointer->tag = t) + +#define GET_FIELD(u,i) \ + (u.pointer->data[i]) + +#define SET_FIELD(u,i,v) \ + (u.pointer->data[i]=v) + +/* -------------------------------------------------------------------------- */ + +/* Basic operations on integers. */ + +/* The macro [FROM_INT(i)] converts the integer [i] to the type [univ]. */ + +#define FROM_INT(i) \ + (univ) { .literal = i } + +/* The macro [TO_INT(u)] converts [u] to the type [int]. */ + +#define TO_INT(u) \ + u.literal diff --git a/projet/src/test/.gitignore b/projet/src/test/.gitignore new file mode 100644 index 0000000..55b60e0 --- /dev/null +++ b/projet/src/test/.gitignore @@ -0,0 +1 @@ +test.native diff --git a/projet/src/test/.merlin b/projet/src/test/.merlin new file mode 100644 index 0000000..1a1fd25 --- /dev/null +++ b/projet/src/test/.merlin @@ -0,0 +1 @@ +B _build diff --git a/projet/src/test/Makefile b/projet/src/test/Makefile new file mode 100644 index 0000000..d66fadf --- /dev/null +++ b/projet/src/test/Makefile @@ -0,0 +1,21 @@ +OCAMLBUILD := ocamlbuild -use-ocamlfind -classic-display +TARGET := test.native +TESTS := ../tests + +.PHONY: all test expected clean + +all: + @ $(OCAMLBUILD) $(TARGET) + +test: all + @ $(MAKE) -C .. + @ ./$(TARGET) --verbosity 1 + +expected: all + @ $(MAKE) -C .. + @ ./$(TARGET) --verbosity 1 --create-expected + +clean: + @ rm -f *~ *.native *.byte + @ $(OCAMLBUILD) -clean + @ rm -f $(TESTS)/*.c $(TESTS)/*.exe $(TESTS)/*.out $(TESTS)/*.err $(TESTS)/*~ diff --git a/projet/src/test/_tags b/projet/src/test/_tags new file mode 100644 index 0000000..749aa28 --- /dev/null +++ b/projet/src/test/_tags @@ -0,0 +1,3 @@ +true: \ + warn(A-44), \ + package(unix) diff --git a/projet/src/test/auxiliary.ml b/projet/src/test/auxiliary.ml new file mode 100644 index 0000000..a7bd0c4 --- /dev/null +++ b/projet/src/test/auxiliary.ml @@ -0,0 +1,138 @@ +type filename = string +type command = string + +let has_suffix suffix name = + Filename.check_suffix name suffix + +let fail format = + Printf.kprintf failwith format + +let try_finally f h = + let result = try f() with e -> h(); raise e in + h(); result + +let with_process_code_result command (f : in_channel -> 'a) : int * 'a = + let ic = Unix.open_process_in command in + set_binary_mode_in ic false; + match f ic with + | exception e -> + ignore (Unix.close_process_in ic); raise e + | result -> + match Unix.close_process_in ic with + | Unix.WEXITED code -> + code, result + | Unix.WSIGNALED _ + | Unix.WSTOPPED _ -> + 99 (* arbitrary *), result + +let with_process_result command (f : in_channel -> 'a) : 'a = + let code, result = with_process_code_result command f in + if code = 0 then + result + else + fail "Command %S failed with exit code %d." command code + +let with_open_in filename (f : in_channel -> 'a) : 'a = + let ic = open_in filename in + try_finally + (fun () -> f ic) + (fun () -> close_in_noerr ic) + +let with_open_out filename (f : out_channel -> 'a) : 'a = + let oc = open_out filename in + try_finally + (fun () -> f oc) + (fun () -> close_out_noerr oc) + +let chunk_size = + 16384 + +let exhaust (ic : in_channel) : string = + let buffer = Buffer.create chunk_size in + let chunk = Bytes.create chunk_size in + let rec loop () = + let length = input ic chunk 0 chunk_size in + if length = 0 then + Buffer.contents buffer + else begin + Buffer.add_subbytes buffer chunk 0 length; + loop() + end + in + loop() + +let read_whole_file filename = + with_open_in filename exhaust + +let absolute_directory (path : string) : string = + try + let pwd = Sys.getcwd() in + Sys.chdir path; + let result = Sys.getcwd() in + Sys.chdir pwd; + result + with Sys_error _ -> + Printf.fprintf stderr "Error: the directory %s does not seem to exist.\n" path; + exit 1 + +let get_number_of_cores () = + try match Sys.os_type with + | "Win32" -> + int_of_string (Sys.getenv "NUMBER_OF_PROCESSORS") + | _ -> + with_process_result "getconf _NPROCESSORS_ONLN" (fun ic -> + let ib = Scanf.Scanning.from_channel ic in + Scanf.bscanf ib "%d" (fun n -> n) + ) + with + | Not_found + | Sys_error _ + | Failure _ + | Scanf.Scan_failure _ + | End_of_file + | Unix.Unix_error _ -> + 1 + +let silent command : command = + command ^ " >/dev/null 2>/dev/null" + +(* [groups eq xs] segments the list [xs] into a list of groups, where several + consecutive elements belong in the same group if they are equivalent in the + sense of the function [eq]. *) + +(* The auxiliary function [groups1] deals with the case where we have an open + group [group] of which [x] is a member. The auxiliary function [group0] + deals with the case where we have no open group. [groups] is the list of + closed groups found so far, and [ys] is the list of elements that remain to + be examined. *) + +let rec groups1 eq groups x group ys = + match ys with + | [] -> + group :: groups + | y :: ys -> + if eq x y then + groups1 eq groups x (y :: group) ys + else + groups0 eq (List.rev group :: groups) (y :: ys) + +and groups0 eq groups ys = + match ys with + | [] -> + groups + | y :: ys -> + groups1 eq groups y [y] ys + +let groups eq (xs : 'a list) : 'a list list = + List.rev (groups0 eq [] xs) + +(* [sep ss] separates the nonempty strings in the list [ss] with a space, + and concatenates everything, producing a single string. *) + +let sep (ss : string list) : string = + let ss = List.filter (fun s -> String.length s > 0) ss in + match ss with + | [] -> + "" + | s :: ss -> + List.fold_left (fun s1 s2 -> s1 ^ " " ^ s2) s ss diff --git a/projet/src/test/test.ml b/projet/src/test/test.ml new file mode 100644 index 0000000..fe98195 --- /dev/null +++ b/projet/src/test/test.ml @@ -0,0 +1,288 @@ +open Sys +open Array +open List +open Filename +open Printf +open Auxiliary + +(* -------------------------------------------------------------------------- *) + +(* Settings. *) + +let create_expected = + ref false + +let verbosity = + ref 0 + +let usage = + sprintf "Usage: %s\n" argv.(0) + +let spec = Arg.align [ + "--create-expected", Arg.Set create_expected, + " recreate the expected-output files"; + "--verbosity", Arg.Int ((:=) verbosity), + " set the verbosity level (0-2)"; +] + +let () = + Arg.parse spec (fun _ -> ()) usage + +let create_expected = + !create_expected + +let verbosity = + !verbosity + +(* -------------------------------------------------------------------------- *) + +(* Logging. *) + +(* 0 is minimal verbosity; + 1 shows some progress messages; + 2 is maximal verbosity. *) + +let log level format = + kprintf (fun s -> + if level <= verbosity then + print_string s + ) format + +(* Extend [fail] to display an information message along the way. + The message is immediately emitted by the worker, depending on + the verbosity level, whereas the failure message is sent back + to the master. *) + +let fail id format = + log 1 "[FAIL] %s\n%!" id; + fail format + +(* When issuing an external command, log it along the way. *) + +let command cmd = + log 2 "%s\n%!" cmd; + command cmd + +(* -------------------------------------------------------------------------- *) + +(* Paths. *) + +let root = + absolute_directory ".." + +let src = + root + +let good = + root ^ "/tests" + +let good_slash filename = + good ^ "/" ^ filename + +let joujou = + src ^ "/joujou" + +let gcc = + sprintf "gcc -O2 -I %s" src + +(* -------------------------------------------------------------------------- *) + +(* Test files. *) + +let thisfile = + "this input file" + +let lambda basename = + basename ^ ".lambda" + +(* -------------------------------------------------------------------------- *) + +(* Test inputs and outputs. *) + +(* A test input is a basename, without the .lambda extension. *) + +type input = + | PositiveTest of filename + +type inputs = input list + +let print_input = function + | PositiveTest basename -> + basename + +type outcome = + | OK + | Fail of string (* message *) + +let print_outcome = function + | OK -> + "" + | Fail msg -> + msg + +type output = + input * outcome + +type outputs = output list + +let print_output (input, outcome) = + printf "\n[FAIL] %s\n%s" + (print_input input) + (print_outcome outcome) + +(* -------------------------------------------------------------------------- *) + +(* Auxiliary functions. *) + +let check_expected directory id result expected = + let cmd = sep ["cd"; directory; "&&"; "cp"; "-f"; result; expected] in + let copy() = + if command cmd <> 0 then + fail id "Failed to create %s.\n" expected + in + (* If we are supposed to create the [expected] file, do so. *) + if create_expected then + copy() + (* Otherwise, check that the file [expected] exists. If it does not exist, + create it by renaming [result] to [expected], then fail and invite the + user to review the newly created file. *) + else if not (file_exists (directory ^ "/" ^ expected)) then begin + copy(); + let cmd = sep ["more"; directory ^ "/" ^ expected] in + fail id "The file %s did not exist.\n\ + I have just created it. Please review it.\n %s\n" + expected cmd + end + +(* -------------------------------------------------------------------------- *) + +(* Running a positive test. *) + +(* + Conventions: + The file %.c stores the output of joujou. + The file %.exe is the compiled program produced by gcc. + The file %.out stores the output of the compiled program. + The file %.exp stores the expected output. + The file %.err stores error output (at several stages). + *) + +let process_positive_test (base : string) : unit = + + (* Display an information message. *) + log 2 "Testing %s...\n%!" base; + + (* A suggested command. *) + let more filename = + sep ["more"; good_slash filename] + in + + (* Run joujou. *) + let source = lambda base in + let c = base ^ ".c" in + let errors = base ^ ".err" in + let cmd = sep ( + "cd" :: good :: "&&" :: + joujou :: source :: sprintf ">%s" c :: sprintf "2>%s" errors :: [] + ) in + if command cmd <> 0 then + fail base "joujou fails on this source file.\n\ + To see the source file:\n %s\n\ + To see the error messages:\n %s\n" + (more source) (more errors); + + (* Run the C compiler. *) + let binary = base ^ ".exe" in + let errors = base ^ ".err" in + let cmd = sep ( + "cd" :: good :: "&&" :: + gcc :: c :: "-o" :: binary :: sprintf "2>%s" errors :: [] + ) in + if command cmd <> 0 then + fail base "The C compiler rejects the C file.\n\ + To see the C file:\n %s\n\ + To see the compiler's error messages:\n %s\n" + (more c) (more errors); + + (* Run the compiled program. *) + let out = base ^ ".out" in + let cmd = sep ( + "cd" :: good :: "&&" :: + sprintf "./%s" binary :: sprintf ">%s" out :: "2>&1" :: [] + ) in + if command cmd <> 0 then + fail base "The compiled program fails.\n\ + To see its output:\n %s\n" (more out); + + (* Check that the file [exp] exists. *) + let exp = base ^ ".exp" in + check_expected good base out exp; + + (* Check that the output coincides with what was expected. *) + let cmd = sep ("cd" :: good :: "&&" :: "diff" :: exp :: out :: []) in + if command (silent cmd) <> 0 then + fail base "joujou accepts this input file, but produces incorrect output.\n\ + To see the difference:\n (%s)\n" + cmd; + + (* Succeed. *) + log 1 "[OK] %s\n%!" base + +(* -------------------------------------------------------------------------- *) + +(* Running a test. *) + +let process input : output = + try + begin match input with + | PositiveTest basenames -> + process_positive_test basenames + end; + input, OK + with Failure msg -> + input, Fail msg + +(* -------------------------------------------------------------------------- *) + +(* [run] runs a bunch of tests in sequence. *) + +let run (inputs : inputs) : outputs = + flush stdout; flush stderr; + let outputs = map process inputs in + outputs + +(* -------------------------------------------------------------------------- *) + +(* Main. *) + +(* Menhir can accept several .mly files at once. By convention, if several + files have the same name up to a numeric suffix, then they belong in a + single group and should be fed together to Menhir. *) + +let inputs directory : filename list = + readdir directory + |> to_list + |> filter (has_suffix ".lambda") + |> map chop_extension + |> sort compare + +let positive : inputs = + inputs good + |> map (fun basename -> PositiveTest basename) + +let inputs = + positive + +let outputs : outputs = + printf "Preparing to run %d tests...\n%!" (length inputs); + run inputs + +let successful, failed = + partition (fun (_, o) -> o = OK) outputs + +let () = + printf "%d out of %d tests are successful.\n" + (length successful) (length inputs); + failed |> iter (fun (input, outcome) -> + printf "\n[FAIL] %s\n%s" (print_input input) (print_outcome outcome) + ) diff --git a/projet/src/tests/.gitignore b/projet/src/tests/.gitignore new file mode 100644 index 0000000..9038d29 --- /dev/null +++ b/projet/src/tests/.gitignore @@ -0,0 +1,4 @@ +*.err +*.out +*.exe +*.c diff --git a/projet/src/tests/bool.exp b/projet/src/tests/bool.exp new file mode 100644 index 0000000..829400e --- /dev/null +++ b/projet/src/tests/bool.exp @@ -0,0 +1,20 @@ +1 +0 +1 +0 +0 +24 +120 +8 +13 +5 +0 +1 +7 +1 +1 +2 +6 +24 +120 +42 diff --git a/projet/src/tests/bool.lambda b/projet/src/tests/bool.lambda new file mode 100644 index 0000000..6eaf5da --- /dev/null +++ b/projet/src/tests/bool.lambda @@ -0,0 +1,88 @@ +(* Thunks. *) +let force = fun x -> x 0 in +(* Church Booleans. *) +let false = fun x -> fun y -> y in +let true = fun x -> fun y -> x in +let k = true in +let cond = fun p -> fun x -> fun y -> p x y in +let forcecond = fun p -> fun x -> fun y -> force (p x y) in +let bool2int = fun b -> cond b 1 0 in +let _ = print (bool2int true) in +let _ = print (bool2int false) in +(* Church pairs. *) +let pair = fun x -> fun y -> fun p -> cond p x y in +let fst = fun p -> p true in +let snd = fun p -> p false in +(* Church naturals. *) +let zero = fun f -> fun x -> x in +let one = fun f -> fun x -> f x in +let plus = fun m -> fun n -> fun f -> fun x -> m f (n f x) in +let two = plus one one in +let three = plus one two in +let four = plus two two in +let five = plus four one in +let six = plus four two in +let succ = plus one in +let mult = fun m -> fun n -> fun f -> m (n f) in +let exp = fun m -> fun n -> n m in +let is_zero = fun n -> n (k false) true in +let convert = fun n -> n (fun x -> x + 1) 0 in +let _ = print (bool2int (is_zero zero)) in +let _ = print (bool2int (is_zero one)) in +let _ = print (bool2int (is_zero two)) in +(* Factorial, based on Church naturals. *) +let loop = fun p -> + let n = succ (fst p) in + pair n (mult n (snd p)) +in +let fact = fun n -> + snd (n loop (pair zero one)) +in +let _ = print (convert (fact four)) in +let _ = print (convert (fact five)) in +(* Fibonacci, based on Church naturals. *) +let loop = fun p -> + let fib1 = fst p in + pair (plus fib1 (snd p)) fib1 +in +let fib = fun n -> + snd (n loop (pair one one)) +in +let _ = print (convert (fib five)) in +let _ = print (convert (fib six)) in +(* Predecessor. *) +let loop = fun p -> + let pred = fst p in + pair (succ pred) pred +in +let pred = fun n -> + snd (n loop (pair zero zero)) +in +let _ = print (convert (pred six)) in +(* Comparison, yielding a Church Boolean. *) +let geq = fun m -> fun n -> + m pred n (k false) true +in +let _ = print (bool2int (geq four six)) in +let _ = print (bool2int (geq six one)) in +(* Iteration. *) +let iter = fun f -> fun n -> + n f (f one) +in +(* Ackermann's function. *) +let ack = fun n -> n iter succ n in +let _ = print (convert (ack two)) in +(* A fixpoint. *) +let fix = fun f -> (fun y -> f (fun z -> y y z)) (fun y -> f (fun z -> y y z)) in +(* Another version of fact. *) +let fact = fun n -> + fix (fun fact -> fun n -> forcecond (is_zero n) (fun _ -> one) (fun _ -> mult n (fact (pred n)))) n +in +let _ = print (convert (fact zero)) in +let _ = print (convert (fact one)) in +let _ = print (convert (fact two)) in +let _ = print (convert (fact three)) in +let _ = print (convert (fact four)) in +let _ = print (convert (fact five)) in + +print 42 diff --git a/projet/src/tests/church.exp b/projet/src/tests/church.exp new file mode 100755 index 0000000..d81cc07 --- /dev/null +++ b/projet/src/tests/church.exp @@ -0,0 +1 @@ +42 diff --git a/projet/src/tests/church.lambda b/projet/src/tests/church.lambda new file mode 100644 index 0000000..0cbd2e6 --- /dev/null +++ b/projet/src/tests/church.lambda @@ -0,0 +1,16 @@ +let i = fun x -> x in +let k = fun x -> fun y -> x in +let zero = fun f -> i in +let one = fun f -> fun x -> f x in +let plus = fun m -> fun n -> fun f -> fun x -> m f (n f x) in +let succ = plus one in +let mult = fun m -> fun n -> fun f -> m (n f) in +let exp = fun m -> fun n -> n m in +let two = succ one in +let three = succ two in +let six = mult two three in +let seven = succ six in +let twenty_one = mult three seven in +let forty_two = mult two twenty_one in +let convert = fun n -> n (fun x -> x + 1) 0 in +print (convert forty_two) diff --git a/projet/src/tests/hello.exp b/projet/src/tests/hello.exp new file mode 100755 index 0000000..d81cc07 --- /dev/null +++ b/projet/src/tests/hello.exp @@ -0,0 +1 @@ +42 diff --git a/projet/src/tests/hello.lambda b/projet/src/tests/hello.lambda new file mode 100644 index 0000000..86f387e --- /dev/null +++ b/projet/src/tests/hello.lambda @@ -0,0 +1 @@ +print (21 + 21) diff --git a/projet/src/tests/loop/loop.lambda b/projet/src/tests/loop/loop.lambda new file mode 100644 index 0000000..0629593 --- /dev/null +++ b/projet/src/tests/loop/loop.lambda @@ -0,0 +1,10 @@ +(* This program is supposed to never terminate. + This can actually work if the C compiler is + smart enough to recognize and optimize tail + calls. It works for me with clang but not with + gcc, for some reason. *) +let rec loop = fun x -> + let _ = print x in + loop (x + 1) +in +loop 0 diff --git a/projet/src/tests/printprint.exp b/projet/src/tests/printprint.exp new file mode 100755 index 0000000..daaac9e --- /dev/null +++ b/projet/src/tests/printprint.exp @@ -0,0 +1,2 @@ +42 +42 diff --git a/projet/src/tests/printprint.lambda b/projet/src/tests/printprint.lambda new file mode 100644 index 0000000..f0501dd --- /dev/null +++ b/projet/src/tests/printprint.lambda @@ -0,0 +1,2 @@ +(* Because [print] returns its argument, this program should print 42 twice. *) +print (print 42) diff --git a/projet/sujet.pdf b/projet/sujet.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b7f0a6d1ad0dcab0c9af9792d6320bc21e946fa8 GIT binary patch literal 112447 zcma%>Lz^(rlBCPFZQHhO+qP}^cyk#9x>se*_YEh8N( z6zTl($T}1=GXVpEy^$3Z4-XW*jH#Wuiv5O~woo3MDH>aGTO3HfbL#JkU`C;8MnG*YQMQu3!Gg_9)F)m*NXQbteph*^ETNP8Z?C$}qE_^!PPmgOeZ 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zHt7&|++kmIdXJ3F34}<&Cb8EE3DQ$vy5S+Nwil`Y&mDFbdp%yO-5?S;X6_1)Vkv$( z#jx;lNoD#s2nC&6ZA$SQ$jm!yANBeE+|o zIQ#!Vks3-s6IcdG8)Kj=kv1a{6C;NXEQ7MSJMceESO#?>Z6+cXBBrlFC0kq2*C6ZH z7E1qFVPgB$`wzDNPw$IC4$N+6jzpy_u8|#0g*Z&{z6@PA6RzwjAurNRsX zW~o6CVg?K(1fe1bB_RRLS^okmD94`MlbW+7`oZk+g1iV_tdAs!njr|2APgUs3qO{N z8kGZ1!uUN{0Y;S(9!dd$jS>7!0nQ02&?G+3TZ@mihuumDjNJfpjmRd<^Hmh?LXhS` zsGZ9DxEep^5B*6HfrK!0XAPJb)*rM21MpYb(TMBu9-@dotcZRxnO@cdVz_fghuCdtXgh%3*!@9o&W#< literal 0 HcmV?d00001 From 4de6bf8544761fb03cc804e17858959fe73bb9f3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Wed, 13 Dec 2017 14:08:29 +0100 Subject: [PATCH 78/81] README update. --- README.md | 7 ++++++- 1 file changed, 6 insertions(+), 1 deletion(-) diff --git a/README.md b/README.md index a131adb..f2a0187 100644 --- a/README.md +++ b/README.md @@ -65,6 +65,11 @@ effectful programming with monads and algebraic effects; tagless interpreters; programming with total functions; generic programming. We also show the limits of dependently-typed functional programming. +## Project + +The [programming project](project/) is now available! +The deadline is **Friday, February 16, 2018**. + ## Approximate syllabus ### Functional Programming: Under the Hood @@ -159,7 +164,7 @@ We also show the limits of dependently-typed functional programming. Two written exams (a partial exam on Friday Dec 1 and a final exam) and one programming project or several programming exercises are used to evaluate -the students that follow the full course. Only the partial exam will count +the students that follow the full course. Only the partial exam will count to grade students who split the course. Only course notes and hand written notes are allowed for the exams. From 36df95f02f5985c797f119cbf324e263374b4399 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Wed, 13 Dec 2017 14:09:19 +0100 Subject: [PATCH 79/81] Makefile. --- Makefile | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/Makefile b/Makefile index e22c835..98639b2 100644 --- a/Makefile +++ b/Makefile @@ -1,3 +1,9 @@ +.PHONY: html clean + +html: README.html + +clean: + rm -f README.html *~ README.html: README.md pandoc -s -f markdown -t html -o $@ $< From 00ebc635448519fddbe390bb62018d547b67cdc3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Wed, 13 Dec 2017 14:10:41 +0100 Subject: [PATCH 80/81] Rename projet to project. --- {projet => project}/src/.merlin | 0 {projet => project}/src/CPS.ml | 0 {projet => project}/src/CPS.mli | 0 {projet => project}/src/Cook.ml | 0 {projet => project}/src/Cook.mli | 0 {projet => project}/src/Defun.ml | 0 {projet => project}/src/Defun.mli | 0 {projet => project}/src/Error.ml | 0 {projet => project}/src/Error.mli | 0 {projet => project}/src/Finish.ml | 0 {projet => project}/src/Finish.mli | 0 {projet => project}/src/Lambda.ml | 0 {projet => project}/src/Lexer.mll | 0 {projet => project}/src/Main.ml | 0 {projet => project}/src/Makefile | 0 {projet => project}/src/Parser.mly | 0 {projet => project}/src/RawLambda.ml | 0 {projet => project}/src/Tail.ml | 0 {projet => project}/src/Top.ml | 0 {projet => project}/src/_tags | 0 {projet => project}/src/alphalib/Atom.ml | 0 {projet => project}/src/alphalib/Atom.mli | 0 {projet => project}/src/kremlin/C.ml | 0 {projet => project}/src/kremlin/Common.ml | 0 {projet => project}/src/kremlin/Constant.ml | 0 {projet => project}/src/kremlin/Options.ml | 0 {projet => project}/src/kremlin/PrintC.ml | 0 {projet => project}/src/kremlin/PrintCommon.ml | 0 {projet => project}/src/kremlin/Utils.ml | 0 {projet => project}/src/prologue.h | 0 {projet => project}/src/test/.gitignore | 0 {projet => project}/src/test/.merlin | 0 {projet => project}/src/test/Makefile | 0 {projet => project}/src/test/_tags | 0 {projet => project}/src/test/auxiliary.ml | 0 {projet => project}/src/test/test.ml | 0 {projet => project}/src/tests/.gitignore | 0 {projet => project}/src/tests/bool.exp | 0 {projet => project}/src/tests/bool.lambda | 0 {projet => project}/src/tests/church.exp | 0 {projet => project}/src/tests/church.lambda | 0 {projet => project}/src/tests/hello.exp | 0 {projet => project}/src/tests/hello.lambda | 0 {projet => project}/src/tests/loop/loop.lambda | 0 {projet => project}/src/tests/printprint.exp | 0 {projet => project}/src/tests/printprint.lambda | 0 {projet => project}/sujet.pdf | Bin 47 files changed, 0 insertions(+), 0 deletions(-) rename {projet => project}/src/.merlin (100%) rename {projet => project}/src/CPS.ml (100%) rename {projet => project}/src/CPS.mli (100%) rename {projet => project}/src/Cook.ml (100%) rename {projet => project}/src/Cook.mli (100%) rename {projet => project}/src/Defun.ml (100%) rename 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rename from projet/src/tests/bool.exp rename to project/src/tests/bool.exp diff --git a/projet/src/tests/bool.lambda b/project/src/tests/bool.lambda similarity index 100% rename from projet/src/tests/bool.lambda rename to project/src/tests/bool.lambda diff --git a/projet/src/tests/church.exp b/project/src/tests/church.exp similarity index 100% rename from projet/src/tests/church.exp rename to project/src/tests/church.exp diff --git a/projet/src/tests/church.lambda b/project/src/tests/church.lambda similarity index 100% rename from projet/src/tests/church.lambda rename to project/src/tests/church.lambda diff --git a/projet/src/tests/hello.exp b/project/src/tests/hello.exp similarity index 100% rename from projet/src/tests/hello.exp rename to project/src/tests/hello.exp diff --git a/projet/src/tests/hello.lambda b/project/src/tests/hello.lambda similarity index 100% rename from projet/src/tests/hello.lambda rename to project/src/tests/hello.lambda diff --git a/projet/src/tests/loop/loop.lambda b/project/src/tests/loop/loop.lambda similarity index 100% rename from projet/src/tests/loop/loop.lambda rename to project/src/tests/loop/loop.lambda diff --git a/projet/src/tests/printprint.exp b/project/src/tests/printprint.exp similarity index 100% rename from projet/src/tests/printprint.exp rename to project/src/tests/printprint.exp diff --git a/projet/src/tests/printprint.lambda b/project/src/tests/printprint.lambda similarity index 100% rename from projet/src/tests/printprint.lambda rename to project/src/tests/printprint.lambda diff --git a/projet/sujet.pdf b/project/sujet.pdf similarity index 100% rename from projet/sujet.pdf rename to project/sujet.pdf From 407f13862bd8e84d504fe36c8ce7eb05319ea38d Mon Sep 17 00:00:00 2001 From: REMY Didier Date: Tue, 19 Dec 2017 09:55:02 +0100 Subject: [PATCH 81/81] Partiel 2017 --- README.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/README.md b/README.md index eebc21b..3957978 100644 --- a/README.md +++ b/README.md @@ -159,6 +159,8 @@ Only course notes and hand written notes are allowed for the exams. Although the course has changed, you may still have a look at previous exams available with solutions: +- mid-term exam 2017-2018: + [Call-by-name. Extensible records](http://gallium.inria.fr/~remy/mpri/exams/partiel-2017-2018.pdf) - mid-term exam 2016-2017: [Record concatenation](http://gallium.inria.fr/~remy/mpri/exams/partiel-2016-2017.pdf) - mid-term exam 2015-2016:

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00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Tue, 26 Sep 2017 16:50:45 +0200 Subject: [PATCH 34/81] Added Even.v. --- README.md | 7 ++-- coq/Even.v | 98 ++++++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 102 insertions(+), 3 deletions(-) create mode 100644 coq/Even.v diff --git a/README.md b/README.md index afd64aa..271a809 100644 --- a/README.md +++ b/README.md @@ -69,11 +69,12 @@ We also show the limits of dependently-typed functional programming. Syntax and operational semantics, on paper and on a machine ([slides 01a](slides/fpottier-01a.pdf)) ([slides 01b](slides/fpottier-01b.pdf)) - ([Coq demo](coq/DemoSyntaxReduction.v)) - ([OCaml solution to Newton-Raphson exercise](ocaml/NewtonRaphson.ml)). + ([OCaml solution to Newton-Raphson exercise](ocaml/NewtonRaphson.ml)) + ([Even.v](coq/Even.v)). * (29/09/2017) From a small-step semantics down to an efficient interpreter, - in several stages. + in several stages + ([Coq demo](coq/DemoSyntaxReduction.v)). * (06/10/2017) Compiling away first-class functions: closure conversion, defunctionalization. * (13/10/2017) Compiling away the call stack: the CPS transformation. * (20/10/2017) Equational reasoning and program optimizations. diff --git a/coq/Even.v b/coq/Even.v new file mode 100644 index 0000000..c68de79 --- /dev/null +++ b/coq/Even.v @@ -0,0 +1,98 @@ +(* 22/09/2017. Someone asked during the course whether [~ (even 1)] can be + proved, and if so, how. Here are several solutions, courtesy of + Pierre-Evariste Dagand. *) + +Inductive even: nat -> Prop := +| even_O: + even 0 +| even_SS: + forall n, even n -> even (S (S n)). + +(* 1. The shortest proof uses the tactic [inversion] to deconstruct the + hypothesis [even 1], that is, to perform case analysis. The tactic + automatically finds that this case is impossible, so the proof is + finished. *) + +Lemma even1_v1: + even 1 -> False. +Proof. + inversion 1. + (* In case you wish the see the proof term: *) + (* Show Proof. *) +Qed. + +(* For most practical purposes, the above proof *script* is good enough, and + is most concise. However, those who wish to understand what they are doing + may prefer to write a proof *term* by hand, in the Calculus of Inductive + Constructions, instead of letting [inversion] construct a (possibly + needlessly complicated) proof term. *) + +(* 2. Generalizing with equality. *) + +Lemma even1_v2': + forall n, even n -> n = 1 -> False. +Proof. + exact (fun n t => + match t with + | even_O => + fun (q: 0 = 1) => + match q with (* IMPOSSIBLE *) end + | even_SS n _ => + fun (q : S (S n) = 1) => + match q with (* IMPOSSIBLE *) end + end + ). +Qed. + +Lemma even1_v2: + even 1 -> False. +Proof. + eauto using even1_v2'. +Qed. + +(* 3. Type-theoretically, through a large elimination. *) + +Lemma even1_v3': + forall n, + even n -> + match n with + | 0 => True + | 1 => False + | S (S _) => True + end. +Proof. + exact (fun n t => + match t with + | even_O => I + | even_SS _ _ => I + end + ). +Qed. + +Lemma even1_v3: + even 1 -> False. +Proof. + apply even1_v3'. +Qed. + +(* 3'. Same technique, using a clever [match ... in ... return]. *) + +Lemma even1_v4': + even 1 -> False. +Proof. + exact (fun t => + match t in even n + return ( + match n with + | 0 => True + | 1 => False + | S (S _) => True + end + (* BUG: we need the following (pointless) type annotation *) + : Prop) + with + | even_O => I + | even_SS _ _ => I + end + ). +Qed. From 213264633fad4ea1dd48feaae460a7e6502a0234 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Thu, 28 Sep 2017 10:36:07 +0200 Subject: [PATCH 35/81] Expose more Coq files. --- coq/AutosubstExtra.v | 169 +++++++++ coq/Autosubst_IsRen.v | 87 +++++ coq/LambdaCalculusBigStep.v | 534 +++++++++++++++++++++++++++ coq/LambdaCalculusFreeVars.v | 152 ++++++++ coq/LambdaCalculusInterpreter.v | 226 ++++++++++++ coq/LambdaCalculusReduction.v | 619 ++++++++++++++++++++++++++++++++ coq/LambdaCalculusSyntax.v | 192 ++++++++++ coq/LambdaCalculusValues.v | 179 +++++++++ coq/Option.v | 120 +++++++ coq/Sequences.v | 321 +++++++++++++++++ 10 files changed, 2599 insertions(+) create mode 100644 coq/AutosubstExtra.v create mode 100644 coq/Autosubst_IsRen.v create mode 100644 coq/LambdaCalculusBigStep.v create mode 100644 coq/LambdaCalculusFreeVars.v create mode 100644 coq/LambdaCalculusInterpreter.v create mode 100644 coq/LambdaCalculusReduction.v create mode 100644 coq/LambdaCalculusSyntax.v create mode 100644 coq/LambdaCalculusValues.v create mode 100644 coq/Option.v create mode 100644 coq/Sequences.v diff --git a/coq/AutosubstExtra.v b/coq/AutosubstExtra.v new file mode 100644 index 0000000..af2ad35 --- /dev/null +++ b/coq/AutosubstExtra.v @@ -0,0 +1,169 @@ +Require Import Omega. +Require Import Autosubst.Autosubst. + +(* -------------------------------------------------------------------------- *) + +(* A more recognizable notation for lifting. *) + +Notation lift i t := (t.[ren(+i)]). + +(* -------------------------------------------------------------------------- *) + +Section Extras. + +Context A `{Ids A, Rename A, Subst A, SubstLemmas A}. + +Lemma up_ren: + forall xi, + ren (upren xi) = up (ren xi). +Proof. + intros. autosubst. +Qed. + +Lemma upn_ren: + forall i xi, + ren (iterate upren i xi) = upn i (ren xi). +Proof. + induction i; intros. + { reflexivity. } + { rewrite <- fold_up_upn. rewrite <- IHi. asimpl. reflexivity. } +Qed. + +Lemma plus_upn: (* close to [up_liftn] *) + forall i sigma, + (+i) >>> upn i sigma = sigma >> ren (+i). +Proof. + induction i; intros. + { rewrite iterate_0. autosubst. } + { rewrite iterate_S. asimpl. rewrite IHi. autosubst. } +Qed. + +Lemma up_sigma_up_ren: + forall t i sigma, + t.[up sigma].[up (ren (+i))] = t.[up (ren (+i))].[upn (1 + i) sigma]. +Proof. + intros. asimpl. rewrite plus_upn. asimpl. reflexivity. +Qed. + +Lemma upn_k_sigma_x: + forall k sigma x, + x < k -> + upn k sigma x = ids x. +Proof. + induction k; intros; asimpl. + { omega. } + { destruct x; asimpl. + { eauto. } + { rewrite IHk by omega. autosubst. } + } +Qed. + +Lemma push_substitution_last: + forall t v i, + t.[v .: ren (+i)] = t.[up (ren (+i))].[v/]. +Proof. + intros. autosubst. +Qed. + +Lemma push_substitution_last_up_hoist: + forall t v i j, + t.[up (v .: ren (+i))].[up (ren (+j))] = + t.[up (up (ren (+j + i)))].[up (lift j v .: ids)]. +Proof. + intros. autosubst. +Qed. + +Lemma lift_lift: + forall i j t, + lift i (lift j t) = lift (i + j) t. +Proof. + intros. autosubst. +Qed. + +Lemma lift_upn: + forall t i sigma, + (lift i t).[upn i sigma] = lift i t.[sigma]. +Proof. + intros. asimpl. + erewrite plus_upn. + reflexivity. +Qed. + +Lemma lift_up: + forall t sigma, + (lift 1 t).[up sigma] = lift 1 t.[sigma]. +Proof. + intros. change up with (upn 1). eapply lift_upn. +Qed. + +Lemma up_sigma_f: + forall (sigma : var -> A) (f : A -> A), + f (ids 0) = ids 0 -> + (forall i t, lift i (f t) = f (lift i t)) -> + up (sigma >>> f) = up sigma >>> f. +Proof. + intros. f_ext. intros [|x]; asimpl; eauto. +Qed. + +Lemma upn_sigma_f: + forall (sigma : var -> A) (f : A -> A), + f (ids 0) = ids 0 -> + (forall i t, lift i (f t) = f (lift i t)) -> + forall i, + upn i (sigma >>> f) = upn i sigma >>> f. +Proof. + induction i; intros. + { reflexivity. } + { do 2 rewrite <- fold_up_upn. rewrite IHi. erewrite up_sigma_f by eauto. reflexivity. } +Qed. + +Lemma upn_theta_sigma_ids: + forall theta sigma i, + theta >> sigma = ids -> + upn i theta >> upn i sigma = ids. +Proof. + intros theta sigma i Hid. + rewrite up_comp_n. + rewrite Hid. + rewrite up_id_n. + reflexivity. +Qed. + +Lemma up_theta_sigma_ids: + forall theta sigma, + theta >> sigma = ids -> + up theta >> up sigma = ids. +Proof. + change up with (upn 1). eauto using upn_theta_sigma_ids. +Qed. + +Lemma scons_scomp: + forall (T : A) Gamma theta, + T.[theta] .: (Gamma >> theta) = (T .: Gamma) >> theta. +Proof. + intros. autosubst. +Qed. + +(* BUG: the two sides of this equation are distinct, yet they are + printed identically. *) + +Goal + forall v f, + v .: (ids >>> f) = (v .: ids) >>> f. +Proof. + intros. + Fail reflexivity. +Abort. + +End Extras. + +(* This incantation means that [eauto with autosubst] can use the tactic + [autosubst] to prove an equality. *) + +Hint Extern 1 (_ = _) => autosubst : autosubst. + +(* This incantation means that [eauto with autosubst] can use the lemmas + whose names are listed here. This is useful when an equality involves + metavariables, so the tactic [autosubst] fails. *) + +Hint Resolve scons_scomp : autosubst. diff --git a/coq/Autosubst_IsRen.v b/coq/Autosubst_IsRen.v new file mode 100644 index 0000000..152782b --- /dev/null +++ b/coq/Autosubst_IsRen.v @@ -0,0 +1,87 @@ +Require Import Coq.Logic.ClassicalUniqueChoice. +Require Import Autosubst.Autosubst. +Require Import AutosubstExtra. + +Section Lemmas. + +Context {A} `{Ids A, Rename A, Subst A, SubstLemmas A}. + +(* The predicate [is_ren sigma] means that the substitution [sigma] is in fact + a renaming [ren xi]. *) + +(* When stating a lemma that involves a renaming, it is preferable to use a + substitution [sigma], together with a hypothesis [is_ren sigma], rather + than request that [sigma] be of the form [ren xi]. This allows us to use + [obvious] to check that [sigma] is a renaming, whereas we would otherwise + have to manually rewrite [sigma] to the form [ren xi]. *) + +Definition is_ren sigma := + exists xi, sigma = ren xi. + +(* One way of proving that [sigma] is a renaming is to prove that [sigma] maps + every variable [x] to a variable [y]. *) + +Lemma prove_is_ren: + forall sigma, + (forall x y, ids x = ids y -> x = y) -> + (forall x, exists y, sigma x = ids y) -> + is_ren sigma. +Proof. + (* This proof uses the axiom of unique choice. If one is willing to use + the stronger axiom of choice, then one can remove the hypothesis that + [ids] is injective. *) + intros ? Hinj Hxy. + assert (Hxi: exists xi : var -> var, forall x, sigma x = ids (xi x)). + { eapply unique_choice with (R := fun x y => sigma x = ids y). + intros x. destruct (Hxy x) as [ y Heqy ]. exists y. + split. + { assumption. } + { intros x' Heqx'. eapply Hinj. congruence. } + } + destruct Hxi as [ xi ? ]. + exists xi. + f_ext; intros x. eauto. +Qed. + +(* Applying [up] or [upn i] to a renaming produces a renaming. *) + +Lemma up_is_ren: + forall sigma, + is_ren sigma -> + is_ren (up sigma). +Proof. + intros ? [ xi ? ]. subst. exists (upren xi). + erewrite <- up_ren by eauto with typeclass_instances. reflexivity. +Qed. + +Lemma upn_is_ren: + forall sigma i, + is_ren sigma -> + is_ren (upn i sigma). +Proof. + intros ? ? [ xi ? ]. subst. exists (iterate upren i xi). + erewrite <- upn_ren by eauto with typeclass_instances. reflexivity. +Qed. + +(* Composing two renamings yields a renaming. *) + +Lemma comp_is_ren: + forall sigma1 sigma2, + is_ren sigma1 -> + is_ren sigma2 -> + is_ren (sigma1 >> sigma2). +Proof. + intros ? ? [ xi1 ? ] [ xi2 ? ]. subst. exists (xi1 >>> xi2). autosubst. +Qed. + +Lemma is_ren_ids: + is_ren ids. +Proof. + exists id. autosubst. +Qed. + +End Lemmas. + +Hint Unfold is_ren : is_ren obvious. + +Hint Resolve up_is_ren upn_is_ren comp_is_ren is_ren_ids : is_ren obvious. diff --git a/coq/LambdaCalculusBigStep.v b/coq/LambdaCalculusBigStep.v new file mode 100644 index 0000000..0e1c4c6 --- /dev/null +++ b/coq/LambdaCalculusBigStep.v @@ -0,0 +1,534 @@ +Require Import List. +Require Import MyTactics. +Require Import Sequences. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusFreeVars. +Require Import LambdaCalculusValues. +Require Import LambdaCalculusReduction. + +(* -------------------------------------------------------------------------- *) +(* -------------------------------------------------------------------------- *) + +(* A big-step call-by-value semantics. *) + +Inductive bigcbv : term -> term -> Prop := +| BigcbvValue: + forall v, + is_value v -> + bigcbv v v +| BigcbvApp: + forall t1 t2 u1 v2 v, + bigcbv t1 (Lam u1) -> + bigcbv t2 v2 -> + bigcbv u1.[v2/] v -> + bigcbv (App t1 t2) v +| BigcbvLet: + forall t1 t2 v1 v, + bigcbv t1 v1 -> + bigcbv t2.[v1/] v -> + bigcbv (Let t1 t2) v +. + +Hint Constructors bigcbv : bigcbv. + +(* The tactic [invert_bigcbv] looks for a hypothesis of the form [bigcbv t v] + and inverts it. *) + +Ltac invert_bigcbv := + pick bigcbv invert; + try solve [ false; eauto 3 with obvious ]. + +(* -------------------------------------------------------------------------- *) + +(* If [bigcbv t v] holds, then [v] must be a value. *) + +Lemma bigcbv_is_value: + forall t v, + bigcbv t v -> + is_value v. +Proof. + induction 1; eauto. +Qed. + +Hint Resolve bigcbv_is_value : is_value obvious. + +(* -------------------------------------------------------------------------- *) + +(* If [t] evaluates to [v] according to the big-step semantics, + then [t] reduces to [v] according to the small-step semantics. *) + +Lemma bigcbv_star_cbv: + forall t v, + bigcbv t v -> + star cbv t v. +Proof. + (* A detailed proof: *) + induction 1. + (* BigcbvValue *) + { eapply star_refl. } + (* BigcbvApp *) + { eapply star_trans. obvious. + eapply star_trans. obvious. + eapply star_step. obvious. + eauto. } + (* BigcbvLet *) + { eapply star_trans. obvious. + eapply star_step. obvious. + eauto. } +Restart. + (* A much shorter proof: *) + induction 1; eauto 6 with sequences obvious. +Qed. + +(* Conversely, if [t] reduces to [v] in the small-step semantics, + then [t] evaluates to [v] in the big-step semantics. *) + +Lemma cbv_bigcbv_bigcbv: + forall t1 t2, + cbv t1 t2 -> + forall v, + bigcbv t2 v -> + bigcbv t1 v. +Proof. + (* A detailed proof: *) + induction 1; intros; subst; try solve [ false; tauto ]. + (* BetaV *) + { econstructor; eauto with bigcbv. } + (* LetV *) + { econstructor; eauto with bigcbv. } + (* AppL *) + { invert_bigcbv. eauto with bigcbv. } + (* AppR *) + { invert_bigcbv. eauto with bigcbv. } + (* LetL *) + { invert_bigcbv. eauto with bigcbv. } +Restart. + (* A shorter proof: *) + induction 1; intros; subst; try solve [ false; tauto + | econstructor; eauto with bigcbv + | invert_bigcbv; eauto with bigcbv + ]. +Qed. + +Lemma star_cbv_bigcbv: + forall t v, + star cbv t v -> + is_value v -> + bigcbv t v. +Proof. + induction 1; eauto using cbv_bigcbv_bigcbv with bigcbv. +Qed. + +(* In conclusion, we have the following equivalence: *) + +Theorem star_cbv_bigcbv_eq: + forall t v, + (star cbv t v /\ is_value v) <-> bigcbv t v. +Proof. + split; intros; unpack; + eauto using star_cbv_bigcbv, bigcbv_star_cbv with is_value. +Qed. + +(* -------------------------------------------------------------------------- *) +(* -------------------------------------------------------------------------- *) + +(* A big-step call-by-value semantics with explicit environments. *) + +(* A closure is a pair of a term and an environment. A cvalue [cv] must be a + closure, as we have no other forms of values. An environment [e] is a list + of cvalues. *) + +(* We break the mutual induction between [cvalue] and [cenv] by inlining the + definition of [cenv] into the definition of [cvalue]. *) + +(* CVALUE *) +Inductive cvalue := +| Clo: {bind term} -> list cvalue -> cvalue. + +Definition cenv := + list cvalue. +(* CVALUE *) + +(* This dummy cvalue is passed below as an argument to [nth], but is really + irrelevant, as the condition [x < length e] ensures that the dummy cvalue + is never used. *) + +Definition dummy_cvalue : cvalue := + Clo (Var 0) nil. + +(* The judgement [ebigcbv e t cv] means that, under the environment [e], the + term [t] evaluates to [cv]. *) + +Inductive ebigcbv : cenv -> term -> cvalue -> Prop := +| EBigcbvVar: + forall e x cv, + (* The variable [x] must be in the domain of [e]. *) + x < length e -> + (* This allows us to safely look up [e] at [x]. *) + cv = nth x e dummy_cvalue -> + ebigcbv e (Var x) cv +| EBigcbvLam: + forall e t, + (* The free variables of [t] must be less than or equal to [length e]. *) + (forall cv, fv (length (cv :: e)) t) -> + (* This allows us to build a closure that is indeed closed. *) + ebigcbv e (Lam t) (Clo t e) +| EBigcbvApp: + forall e e' t1 t2 u1 cv2 cv, + (* Evaluate [t1] to a closure, *) + ebigcbv e t1 (Clo u1 e') -> + (* evaluate [t2] to a value, *) + ebigcbv e t2 cv2 -> + (* and evaluate the function body, in a suitable environment. *) + ebigcbv (cv2 :: e') u1 cv -> + ebigcbv e (App t1 t2) cv +| EBigcbvLet: + forall e t1 t2 cv1 cv, + (* Evaluate [t1] to a value, *) + ebigcbv e t1 cv1 -> + (* and evaluate [t2] under a suitable environment. *) + ebigcbv (cv1 :: e) t2 cv -> + ebigcbv e (Let t1 t2) cv +. + +Hint Constructors ebigcbv : ebigcbv. + +(* -------------------------------------------------------------------------- *) + +(* To explain what the above semantics means, and to prove that it is + equivalent to the big-step semantics, we first define a function that + decodes a cvalue into a value. *) + +(* Ideally, the function [decode] should be defined by the equation: + + decode (Clo t e) = (Lam t).[decode_cenv e] + + where the auxiliary function [decode_cenv] maps [decode] over the + environment [e], and converts the result to a substitution, that + is, a function of type [term -> term]: + + decode_cenv e = fun x => nth x (map decode e) (decode dummy_cvalue) + + The definitions below are a bit more awkward (as Coq does not support + mutual induction very well), but mean the same thing. *) + +Definition dummy_value : term := + Lam (Var 0). + +Fixpoint decode (cv : cvalue) : term := + match cv with + | Clo t e => + (Lam t).[fun x => nth x (map decode e) dummy_value] + end. + (* I am not even sure why this definition is accepted by Coq? *) + +Definition decode_cenv e := + fun x => nth x (map decode e) (decode dummy_cvalue). + +(* The first equation in the above comment is satisfied, as shown by + the following lemma. The second equation in the above comment is + satisfied too, by definition. *) + +Lemma decode_eq: + forall t e, + decode (Clo t e) = (Lam t).[decode_cenv e]. +Proof. + reflexivity. +Qed. + +(* This equation is useful, too. *) + +Lemma decode_cenv_Var_eq: + forall x e, + (Var x).[decode_cenv e] = decode (nth x e dummy_cvalue). +Proof. + intros. asimpl. unfold decode_cenv. rewrite map_nth. eauto. +Qed. + +(* The tactic [decode] rewrites using the above two equations. *) + +Hint Rewrite decode_eq decode_cenv_Var_eq : decode. +Ltac decode := autorewrite with decode in *. + +(* We make [decode] opaque, so its definition is not unfolded by Coq. *) + +Opaque decode. + +(* [decode cv] is always a value. *) + +Lemma is_value_decode: + forall cv, + is_value (decode cv). +Proof. + intros. destruct cv. decode. asimpl. tauto. +Qed. + +Lemma is_value_decode_cenv: + forall x e, + is_value (Var x).[decode_cenv e]. +Proof. + intros. decode. eauto using is_value_decode. +Qed. + +Local Hint Resolve is_value_decode is_value_decode_cenv : is_value obvious. + +(* A composition of two substitutions is the same thing as one substitution. *) + +Lemma decode_cenv_cons: + forall t e cv, + t.[up (decode_cenv e)].[decode cv/] = t.[decode_cenv (cv :: e)]. +Proof. + intros. autosubst. (* wonderful *) +Qed. + +(* The tactic [nonvalue_eq_decode] closes a subgoal when there is a hypothesis + of the form [_ = decode_cenv e x], where the left-hand side of the equation + clearly is not a value. This is a contradiction. *) + +Ltac nonvalue_eq_decode := + match goal with + | heq: _ = decode_cenv ?e ?x |- _ => + assert (hv: is_value (decode_cenv e x)); [ + solve [ obvious ] + | rewrite <- heq in hv; false; is_value ] + end. + +(* -------------------------------------------------------------------------- *) + +(* If [t] evaluates to [cv] under environment [e], + then, in the big-step semantics, + the term [t.[decode_cenv e]] evaluates to the value [decode cv]. *) + +Lemma ebigcbv_bigcbv: + forall e t cv, + ebigcbv e t cv -> + bigcbv t.[decode_cenv e] (decode cv). +Proof. + induction 1; intros; subst. + (* EBigcbvVar *) + { decode. econstructor. obvious. } + (* EBigcbvLam *) + { econstructor. obvious. } + (* EBigcbvApp *) + { decode. + asimpl. econstructor; eauto. + asimpl. eauto. } + (* EBigcbvLet *) + { asimpl. econstructor; eauto. + asimpl. eauto. } +Qed. + +(* A simplified corollary, where [t] is closed and is therefore evaluated + under the empty environment. *) + +Lemma ebigcbv_bigcbv_nil: + forall t cv, + ebigcbv nil t cv -> + closed t -> + bigcbv t (decode cv). +Proof. + intros. + replace t with t.[decode_cenv nil] by eauto using closed_unaffected. + eauto using ebigcbv_bigcbv. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* The converse statement: if in the big-step semantics, the term + [t.[decode_cenv e]] evaluates to the value [decode cv], then [t] evaluates + to [cv] under environment [e]. *) + +(* This proof does not work (and, in fact, the statement is wrong). A failed + proof attempt reveals two problems... *) + +Lemma bigcbv_ebigcbv_failed: + forall e t cv, + bigcbv t.[decode_cenv e] (decode cv) -> + ebigcbv e t cv. +Proof. + inversion 1; intros; subst. + (* BigcbvValue *) + { (* [t] must be a variable or a lambda-abstraction. *) + destruct t; [ | | false; is_value | false; is_value ]. + (* Case: [t] is a variable. *) + { econstructor. + (* Here, we have two subgoals, neither of which can be proved. *) + { (* Problem 1: we are unable to prove [x < length e]. In order + to establish such a property, we would have to express the + hypothesis that the free variables of [t] are in the domain + of the environment [e]. And, for this hypothesis to be + inductive, we have to further require every closure in + the environment [e] to satisfy a similar condition. *) + admit. } + { (* Problem 2: we have [decode cv1 = decode cv2], and the goal + is [cv1 = cv2]. This goal cannot be proved, as the function + [decode] is not injective: multiple closures represent the + same lambda-abstraction. *) + decode. + admit. } +Abort. + +(* -------------------------------------------------------------------------- *) + +(* In order to fix the above failed statement, we need to express the + following well-formedness invariant: whenever a closure [Clo t e] is + constructed, the free variables of the term [t] are in the domain of the + environment [env], and, recursively, every value in [e] is well-formed. *) + +Inductive wf_cvalue : cvalue -> Prop := +| WfCvalue: + forall t e, + (forall cv, fv (length (cv :: e)) t) -> + wf_cenv e -> + wf_cvalue (Clo t e) + +with wf_cenv : cenv -> Prop := +| WfCenv: + forall e, + Forall wf_cvalue e -> + wf_cenv e. + +(* The following trivial lemmas (which repeat the definition) are provided as + hints for [eauto with wf_cvalue]. *) + +Lemma use_wf_cvalue_1: + forall t e cv, + wf_cvalue (Clo t e) -> + fv (length (cv :: e)) t. +Proof. + inversion 1; eauto. +Qed. + +Lemma use_wf_cvalue_2: + forall t e, + wf_cvalue (Clo t e) -> + wf_cenv e. +Proof. + inversion 1; eauto. +Qed. + +Lemma prove_wf_cenv_nil: + wf_cenv nil. +Proof. + econstructor. eauto. +Qed. + +Lemma prove_wf_cenv_cons: + forall cv e, + wf_cvalue cv -> + wf_cenv e -> + wf_cenv (cv :: e). +Proof. + inversion 2; intros; subst. + econstructor. + econstructor; eauto. +Qed. + +Hint Resolve + use_wf_cvalue_1 use_wf_cvalue_2 prove_wf_cenv_nil prove_wf_cenv_cons +: wf_cvalue. + +(* The following lemma states that the invariant is preserved by [ebigcbv]. + That is, if the term [t] is successfully evaluated in the well-formed + environment [e] to a cvalue [cv], then [cv] is well-formed. *) + +Lemma ebigcbv_wf_cvalue: + forall e t cv, + ebigcbv e t cv -> + wf_cenv e -> + wf_cvalue cv. +Proof. + induction 1; intros; subst. + (* EBigcbvVar *) + { pick wf_cenv invert. rewrite Forall_forall in *. eauto using nth_In. } + (* EBigcbvLam *) + { econstructor; eauto. } + (* EBigcbvApp *) + { eauto 6 with wf_cvalue. } + (* EBigcbvLet *) + { eauto 6 with wf_cvalue. } +Qed. + +Hint Resolve ebigcbv_wf_cvalue : wf_cvalue. + +(* -------------------------------------------------------------------------- *) + +(* We can now make an amended statement: if in the big-step semantics, the + term [t.[decode_cenv e]] evaluates to a value [v], if the environment [e] + is well-formed, and if the free variables of [t] are in the domain of [e], + then under environment [e] the term [t] evaluates to some cvalue [cv] such + that [decode cv] is [v]. *) + +Lemma bigcbv_ebigcbv: + forall te v, + bigcbv te v -> + forall t e, + te = t.[decode_cenv e] -> + fv (length e) t -> + wf_cenv e -> + exists cv, + ebigcbv e t cv /\ decode cv = v. +Proof. + induction 1; intros; subst. + (* BigcbvValue *) + { (* [t] must be a variable or a lambda-abstraction. *) + destruct t; [ | | false; is_value | false; is_value ]; fv. + (* Case: [t] is a variable. *) + { eexists; split. econstructor; eauto. decode. eauto. } + (* Case: [t] is a lambda-abstraction. *) + { eexists; split. econstructor; eauto. reflexivity. } + } + (* BigcbvApp *) + { (* [t] must be an application. *) + destruct t; + match goal with h: _ = _ |- _ => asimpl in h end; + [ nonvalue_eq_decode | false; congruence | | false; congruence ]. + fv. unpack. + (* The equality [App _ _ = App _ _] can be simplified. *) + injections. subst. + (* Exploit two of the induction hypotheses (forward). *) + edestruct IHbigcbv1; eauto. unpack. clear IHbigcbv1. + edestruct IHbigcbv2; eauto. unpack. clear IHbigcbv2. + (* If [decode cv] is [Lam _], then [cv] must be a closure. This should + be a lemma. Because (here) every cvalue is a closure, it is trivial. *) + match goal with h: decode ?cv = Lam _ |- _ => + destruct cv as [ t' e' ]; + rewrite decode_eq in h; + asimpl in h; + injections; subst + end. + (* Now, exploit the third induction hypothesis (forward). *) + edestruct IHbigcbv3; eauto using decode_cenv_cons with wf_cvalue. + unpack. clear IHbigcbv3. + (* The result follows. *) + eauto with ebigcbv. + } + (* BigcbvLet *) + { (* [t] must be a [Let] construct. *) + destruct t; + match goal with h: _ = _ |- _ => asimpl in h end; + [ nonvalue_eq_decode | false; congruence | false; congruence | ]. + fv. unpack. + injections. subst. + (* Exploit the induction hypotheses (forward). *) + edestruct IHbigcbv1; eauto. unpack. clear IHbigcbv1. + edestruct IHbigcbv2; eauto using decode_cenv_cons with wf_cvalue. + unpack. clear IHbigcbv2. + (* The result follows. *) + eauto with ebigcbv. + } +Qed. + +(* A simplified corollary, where [t] is closed and is therefore evaluated + under the empty environment. *) + +Lemma bigcbv_ebigcbv_nil: + forall t v, + bigcbv t v -> + closed t -> + exists cv, + ebigcbv nil t cv /\ decode cv = v. +Proof. + intros. eapply bigcbv_ebigcbv; eauto with wf_cvalue. + rewrite closed_unaffected by eauto. reflexivity. +Qed. diff --git a/coq/LambdaCalculusFreeVars.v b/coq/LambdaCalculusFreeVars.v new file mode 100644 index 0000000..1d5fdbb --- /dev/null +++ b/coq/LambdaCalculusFreeVars.v @@ -0,0 +1,152 @@ +Require Import MyTactics. +Require Import LambdaCalculusSyntax. + +(* This technical lemma states that the renaming [lift 1] is injective. *) + +Lemma lift_inj_Var: + forall t x, + lift 1 t = Var (S x) <-> t = Var x. +Proof. + split; intros. + { eauto using lift_inj. } + { subst. eauto. } +Qed. + +(* -------------------------------------------------------------------------- *) + +(* The predicate [fv k t] means that the free variables of the term [t] are + contained in the semi-open interval [0..k). *) + +Definition fv k t := + t.[upn k (ren (+1))] = t. + +(* The predicate [fv] is characterized by the following lemmas. *) + +Lemma fv_Var_eq: + forall k x, + fv k (Var x) <-> x < k. +Proof. + unfold fv. asimpl. induction k; intros. + (* Base case. *) + { asimpl. split; intros; false. + { unfold ids, Ids_term in *. injections. omega. } + { omega. } + } + (* Step. *) + { destruct x; asimpl. + { split; intros. { omega. } { reflexivity. } } + rewrite lift_inj_Var. rewrite IHk. omega. } +Qed. + +Lemma fv_Lam_eq: + forall k t, + fv k (Lam t) <-> fv (S k) t. +Proof. + unfold fv. intros. asimpl. split; intros. + { injections. eauto. } + { unpack. congruence. } +Qed. + +Lemma fv_App_eq: + forall k t1 t2, + fv k (App t1 t2) <-> fv k t1 /\ fv k t2. +Proof. + unfold fv. intros. asimpl. split; intros. + { injections. eauto. } + { unpack. congruence. } +Qed. + +Lemma fv_Let_eq: + forall k t1 t2, + fv k (Let t1 t2) <-> fv k t1 /\ fv (S k) t2. +Proof. + unfold fv. intros. asimpl. split; intros. + { injections. eauto. } + { unpack. congruence. } +Qed. + +Hint Rewrite fv_Var_eq fv_Lam_eq fv_App_eq fv_Let_eq : fv. + +(* The tactic [fv] uses the above lemmas as rewriting rules. *) + +Ltac fv := + autorewrite with fv in *. + +(* -------------------------------------------------------------------------- *) + +(* The predicate [closed t] means that the term [t] is closed, that is, [t] + has no free variables. *) + +Definition closed := + fv 0. + +(* [closed t] is equivalent to [lift 1 t = t]. *) + +Lemma closed_eq: + forall t, + closed t <-> lift 1 t = t. +Proof. + unfold closed, fv. asimpl. tauto. +Qed. + +(* The following lemmas allow decomposing a closedness hypothesis. + Because [closed] is not an inductive notion, there is no lemma + for [Lam] and for the right-hand side of [Let]. *) + +Lemma closed_Var: + forall x, + ~ closed (Var x). +Proof. + unfold closed; intros; fv. omega. +Qed. + +Lemma closed_AppL: + forall t1 t2, + closed (App t1 t2) -> + closed t1. +Proof. + unfold closed; intros; fv. tauto. +Qed. + +Lemma closed_AppR: + forall t1 t2, + closed (App t1 t2) -> + closed t2. +Proof. + unfold closed; intros; fv. tauto. +Qed. + +Lemma closed_LetL: + forall t1 t2, + closed (Let t1 t2) -> + closed t1. +Proof. + unfold closed; intros; fv. tauto. +Qed. + +Hint Resolve closed_Var closed_AppL closed_AppR closed_LetL : closed. + +(* -------------------------------------------------------------------------- *) + +(* If the free variables of the term [t] are below [k], then [t] is unaffected + by a substitution of the form [upn k sigma]. *) + +Lemma fv_unaffected: + forall t k sigma, + fv k t -> + t.[upn k sigma] = t. +Proof. + induction t; intros; fv; unpack; asimpl; + try solve [ eauto using upn_k_sigma_x with typeclass_instances + | f_equal; eauto ]. +Qed. + +(* If the term [t] is closed, then [t] is unaffected by any substitution. *) + +Lemma closed_unaffected: + forall t sigma, + closed t -> + t.[sigma] = t. +Proof. + intros. eapply fv_unaffected with (k := 0). eauto. +Qed. diff --git a/coq/LambdaCalculusInterpreter.v b/coq/LambdaCalculusInterpreter.v new file mode 100644 index 0000000..5796683 --- /dev/null +++ b/coq/LambdaCalculusInterpreter.v @@ -0,0 +1,226 @@ +Require Import Option. +Require Import List. +Require Import MyTactics. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusFreeVars. +Require Import LambdaCalculusBigStep. + +(* We now wish to define an interpreter for the lambda-calculus. In other + words, whereas [ebigcbv] is a relation, we now wish to define a function + [interpret] whose graph is the relation [ebigcbv]. *) + +(* At the moment, our lambda-calculus is pure (every value is a function) + so the interpreter cannot encounter a runtime error. *) + +(* -------------------------------------------------------------------------- *) + +(* We might naively wish to write the following code, which Coq rejects, + because this function is not obviously terminating. (Exercise: which + recursive call is the culprit?) Indeed, an interpreter for the untyped + lambda-calculus does not always terminate: there are lambda-terms whose + evaluation diverges. (Exercise: exhibit a term that reduces to itself + in one or more reduction steps. Prove in Coq that this is the case.) *) + +(* FAILINTERPRET *) +Fail Fixpoint interpret (e : cenv) (t : term) : cvalue := + match t with + | Var x => + nth x e dummy_cvalue + (* dummy is used when x is out of range *) + | Lam t => + Clo t e + | App t1 t2 => + let cv1 := interpret e t1 in + let cv2 := interpret e t2 in + match cv1 with Clo u1 e' => + interpret (cv2 :: e') u1 + end +(* FAILINTERPRET *) + | Let t1 t2 => + let cv1 := interpret e t1 in + interpret (cv1 :: e) t2 +(* FAILINTERPRET *) + end. +(* FAILINTERPRET *) + +(* -------------------------------------------------------------------------- *) + +(* There are several potential solutions to the above problem. One solution + would be to write code in (some implementation of) the partiality monad. + (See Dagand's lectures.) The solution proposed here is to parameterize + the function [interpret] with a natural integer [n], which serves as an + amount of "fuel" (or "effort") that we are willing to invest before we + give up. Thus, termination becomes obvious. The downside is that the + interpreter can now fail (which means "not enough fuel"). Fortunately, + given enough fuel, every terminating term can be evaluated. *) + +(* For Coq to accept the following definition, the fuel [n] must decrease at + every recursive call. We might wish to be more precise and somehow explain + that [n] needs to decrease only at the third recursive call in the case of + [App]lications. That would require defining a lexicographic ordering on the + pair [n, t], arguing that this ordering is well-founded, and defining + [interpret] by well-founded recursion. This can be done in Coq but is more + complicated, so (here) not worth the trouble. *) + +(* FIXINTERPRET *) +Fixpoint interpret (n : nat) e t : option cvalue := + match n with + | 0 => None (* not enough fuel *) + | S n => + match t with + | Var x => Some (nth x e dummy_cvalue) + | Lam t => Some (Clo t e) + | App t1 t2 => + interpret n e t1 >>= fun cv1 => + interpret n e t2 >>= fun cv2 => + match cv1 with Clo u1 e' => + interpret n (cv2 :: e') u1 + end +(* FIXINTERPRET *) + | Let t1 t2 => + interpret n e t1 >>= fun cv1 => + interpret n (cv1 :: e) t2 +(* FIXINTERPRET *) + end end. +(* FIXINTERPRET *) + +(* -------------------------------------------------------------------------- *) + +(* The interpreter is correct with respect to the big-step semantics. *) + +Lemma interpret_ebigcbv: + forall n e t cv, + interpret n e t = Some cv -> + fv (length e) t -> + wf_cenv e -> + ebigcbv e t cv. +Proof. + (* The definition of [interpret] is by induction on [n], so this proof + must be by induction on [n] as well. *) + induction n; destruct t; simpl; intros; + fv; unpack; injections; subst; + try solve [ congruence ]. + (* Var *) + { econstructor; eauto. } + (* Lam *) + { econstructor; eauto. } + (* App *) + { repeat invert_bind_Some. + (* Every cvalue is a closure. Name the components of the closure + obtained by interpreting [t1]. *) + match goal with h: interpret _ _ t1 = Some ?cv |- _ => + destruct cv as [ t' e' ] + end. + (* The goal follows. *) + econstructor; eauto 11 with wf_cvalue. } + (* Let *) + { invert_bind_Some. + econstructor; eauto with wf_cvalue. } +Qed. + +(* A simplified corollary, where [t] is closed and is therefore evaluated + under the empty environment, and where we conclude with a [bigcbv] + judgement. *) + +Lemma interpret_bigcbv_nil: + forall n t cv, + interpret n nil t = Some cv -> + closed t -> + bigcbv t (decode cv). +Proof. + eauto using ebigcbv_bigcbv_nil, interpret_ebigcbv with wf_cvalue. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* The interpreter is monotonic with respect to the amount of fuel that is + provided: the more fuel, the better (that is, the more defined the result). *) + +Lemma interpret_monotonic: + forall n1 n2 e t, + n1 <= n2 -> + less_defined (interpret n1 e t) (interpret n2 e t). +Proof. + (* This series of tactics get rid of the easy cases: *) + induction n1; destruct t; simpl; intros; + (* [less_defined None _] is always true. *) + eauto with less_defined; + (* If [S n1 <= n2], then [n2] must be a successor. *) + (destruct n2; [ omega |]); simpl; + (* [less_defined] is reflexive. *) + eauto with less_defined. + + (* Two more complex cases remain, namely [App] and [Let]. Probably + the proof could be further automated, but I did not try. *) + (* App *) + { eapply prove_less_defined_bind. + { eauto using le_S_n. } + { intros _ [ t' e' ]. (* destruct the closure produced by [t1] *) + eapply prove_less_defined_bind; eauto using le_S_n. } + } + (* Let *) + { eauto 6 using le_S_n with less_defined. } +Qed. + +(* A reformulation. *) + +Lemma interpret_monotonic_corollary: + forall n1 n2 e t cv, + interpret n1 e t = Some cv -> + n1 <= n2 -> + interpret n2 e t = Some cv. +Proof. + generalize interpret_monotonic. unfold less_defined. eauto. +Qed. + +(* -------------------------------------------------------------------------- *) + +(* The interpreter is complete with respect to the big-step semantics + [ebigcbv]. That is, given enough fuel, and given a term whose value is + [cv], it will compute [cv]. *) + +Lemma ebigcbv_interpret: + forall e t cv, + ebigcbv e t cv -> + exists n, + interpret n e t = Some cv. +Proof. + (* We can see, in the proof, that the necessary amount of fuel, [n], is + the height of the derivation of the judgement [ebigcbv e t cv]. + Indeed, at every [App] or [Let] node, we count 1 plus the maximum + amount of fuel required by our children. *) + induction 1; intros; subst. + (* EBigcbvVar *) + { exists 1. eauto. } + (* EBigcbvLam *) + { exists 1. eauto. } + (* EBigcbvApp *) + { destruct IHebigcbv1 as [ n1 ? ]. + destruct IHebigcbv2 as [ n2 ? ]. + destruct IHebigcbv3 as [ n3 ? ]. + eexists (S (max (max n1 n2) n3)). simpl. + eauto 6 using prove_bind_Some, interpret_monotonic_corollary with omega. } + (* EBigcbvLet *) + { destruct IHebigcbv1 as [ n1 ? ]. + destruct IHebigcbv2 as [ n2 ? ]. + eexists (S (max n1 n2)). simpl. + eauto using prove_bind_Some, interpret_monotonic_corollary with omega. } +Qed. + +(* The interpreter is complete with respect to the big-step semantics + [bigcbv]. That is, given enough fuel, and given a term [t] whose value is + [v], it will compute a cvalue [cv] which decodes to [v]. We state this in + the case where [t] is closed and is therefore evaluated under the empty + environment. *) + +Lemma bigcbv_interpret_nil: + forall t v, + bigcbv t v -> + closed t -> + exists n cv, + interpret n nil t = Some cv /\ decode cv = v. +Proof. + intros. + edestruct bigcbv_ebigcbv_nil; eauto. unpack. + edestruct ebigcbv_interpret; eauto. +Qed. diff --git a/coq/LambdaCalculusReduction.v b/coq/LambdaCalculusReduction.v new file mode 100644 index 0000000..0830f95 --- /dev/null +++ b/coq/LambdaCalculusReduction.v @@ -0,0 +1,619 @@ +Require Import MyTactics. +Require Import Sequences. +Require Import LambdaCalculusSyntax. +Require Import LambdaCalculusValues. +Require Import LambdaCalculusFreeVars. + +(* We give a symbolic name to each reduction rule. *) + +Inductive rule := +| RuleBetaV (* reduction of a beta-v redex: (\x.t) v *) +| RuleLetV (* reduction of a let-v redex: let x = v in t *) +| RuleBeta (* reduction of a beta redex: (\x.t) u *) +| RuleLet (* reduction of a let redex: let x = u in t *) +| RuleParBetaV (* reduction of a beta-v redex and reduction in both sides *) +| RuleParLetV (* reduction of a let redex and reduction in both sides *) +| RuleVar (* no reduction *) +| RuleLam (* reduction in [Lam _] *) +| RuleAppL (* reduction in [App _ u] *) +| RuleAppVR (* reduction in [App v _], if [v] is a value *) +| RuleAppLR (* reduction in both sides of [App _ _] *) +| RuleLetL (* reduction in [Let _ u] *) +| RuleLetR (* reduction in [Let t _] *) +| RuleLetLR (* reduction in both sides of [Let _ _] *). + +(* A mask is a set of rules. *) + +Definition mask := + rule -> Prop. + +(* A generic small-step reduction semantics, parameterized with a mask. *) + +Inductive red (mask : mask) : term -> term -> Prop := +| RedBetaV: + forall t v u, + mask RuleBetaV -> + is_value v -> + t.[v/] = u -> + red mask (App (Lam t) v) u +| RedLetV: + forall t v u, + mask RuleLetV -> + is_value v -> + t.[v/] = u -> + red mask (Let v t) u +| RedBeta: + forall t1 t2 u, + mask RuleBeta -> + t1.[t2/] = u -> + red mask (App (Lam t1) t2) u +| RedLet: + forall t1 t2 u, + mask RuleLet -> + t2.[t1/] = u -> + red mask (Let t1 t2) u +| RedParBetaV: + forall t1 v1 t2 v2 u, + mask RuleParBetaV -> + is_value v1 -> + red mask t1 t2 -> + red mask v1 v2 -> + t2.[v2/] = u -> + red mask (App (Lam t1) v1) u +| RedParLetV: + forall t1 t2 v1 v2 u, + mask RuleParLetV -> + is_value v1 -> + red mask t1 t2 -> + red mask v1 v2 -> + t2.[v2/] = u -> + red mask (Let v1 t1) u +| RedVar: + forall x, + mask RuleVar -> + red mask (Var x) (Var x) +| RedLam: + forall t1 t2, + mask RuleLam -> + red mask t1 t2 -> + red mask (Lam t1) (Lam t2) +| RedAppL: + forall t1 t2 u, + mask RuleAppL -> + red mask t1 t2 -> + red mask (App t1 u) (App t2 u) +| RedAppVR: + forall v u1 u2, + mask RuleAppVR -> + is_value v -> + red mask u1 u2 -> + red mask (App v u1) (App v u2) +| RedAppLR: + forall t1 t2 u1 u2, + mask RuleAppLR -> + red mask t1 t2 -> + red mask u1 u2 -> + red mask (App t1 u1) (App t2 u2) +| RedLetL: + forall t1 t2 u, + mask RuleLetL -> + red mask t1 t2 -> + red mask (Let t1 u) (Let t2 u) +| RedLetR: + forall t u1 u2, + mask RuleLetR -> + red mask u1 u2 -> + red mask (Let t u1) (Let t u2) +| RedLetLR: + forall t1 t2 u1 u2, + mask RuleLetLR -> + red mask t1 t2 -> + red mask u1 u2 -> + red mask (Let t1 u1) (Let t2 u2) +. + +Hint Constructors red : red obvious. + +(* The following mask defines the call-by-value reduction semantics. *) + +Definition cbv_mask rule := + match rule with + | RuleBetaV (* reduction of a beta-v redex: (\x.t) v *) + | RuleLetV (* reduction of a let-v redex: let x = v in t *) + | RuleAppL (* reduction in [App _ u] *) + | RuleAppVR (* reduction in [App v _], if [v] is a value *) + | RuleLetL (* reduction in [Let _ u] *) + => True + | _ => False + end. + +Notation cbv := (red cbv_mask). + +(* The following mask defines the call-by-name reduction semantics. *) + +Definition cbn_mask rule := + match rule with + | RuleBeta (* reduction of a beta redex: (\x.t) v *) + | RuleLet (* reduction of a let redex: let x = v in t *) + | RuleAppL (* reduction in [App _ u] *) + => True + | _ => False + end. + +Notation cbn := (red cbn_mask). + +(* The parallel by-value reduction semantics allows beta-v reductions under + arbitrary contexts, including under lambda-abstractions. Furthermore, it + allows parallel reductions (and allows no reduction at all). *) + +Definition pcbv_mask rule := + match rule with + | RuleParBetaV (* reduction of a beta redex and reduction in both sides *) + | RuleParLetV (* reduction of a let redex and reduction in both sides *) + | RuleVar (* no reduction *) + | RuleLam (* reduction in [Lam _] *) + | RuleAppLR (* reduction in both sides of [App _ _] *) + | RuleLetLR (* reduction in both sides of [Let _ _] *) + => True + | _ => False + end. + +Notation pcbv := (red pcbv_mask). + +(* The tactic [obvious] should be able to prove goals of the form + [red mask t1 t2], where [mask] is a known mask. *) + +Hint Extern 1 (cbv_mask _) => (simpl; tauto) : red obvious. +Hint Extern 1 (cbn_mask _) => (simpl; tauto) : red obvious. +Hint Extern 1 (pcbv_mask _) => (simpl; tauto) : red obvious. + +Goal cbv (Let (App (Lam (Var 0)) (Var 0)) (Var 0)) (Let (Var 0) (Var 0)). +Proof. obvious. Qed. + +Goal cbv (Let (Var 0) (Var 0)) (Var 0). +Proof. obvious. Qed. + +Goal cbn (Let (Var 0) (Var 0)) (Var 0). +Proof. obvious. Qed. + +Goal + let id := Lam (Var 0) in + let t := (Let (Lam (Var 0)) (Var 0)) in + cbn (App id t) t. +Proof. simpl. obvious. Qed. + +Goal pcbv (App (App (Lam (Var 0)) (Var 0)) (App (Lam (Var 0)) (Var 0))) + (App (Var 0) (Var 0)). +Proof. + eauto 8 with obvious. +Qed. + +(* The tactic [step] applies to a goal of the form [star (red mask) t1 t2]. It + causes the term [t1] to take one step of reduction towards [t1'], turning + the goal into [star (red mask) t1' t2]. *) + +Ltac step := + eapply star_step; [ obvious |]. + +(* The tactic [finished] applies to a goal of the form [star (red mask) t1 t2]. + It turns the goal into [t1 = t2]. *) + +Ltac finished := + eapply star_refl_eq. + +(* The tactic [invert_cbv] inverts a hypothesis of the form [cbv t1 t2]. *) + +Ltac invert_cbv := + pick (red cbv_mask) invert; + try solve [ false; eauto 3 with obvious ]. + +Ltac invert_star_cbv := + pick (star cbv) invert. + +Ltac invert_cbn := + pick (red cbn_mask) invert; + try solve [ false; eauto 3 with obvious ]. + +(* If the following four rules are enabled, then reduction is reflexive. *) + +Lemma red_refl: + forall mask : mask, + mask RuleVar -> + mask RuleLam -> + mask RuleAppLR -> + mask RuleLetLR -> + forall t, + red mask t t. +Proof. + induction t; eauto with red. +Qed. + +(* [RuleBetaV] and [RuleLetV] are special cases of [RuleParBetaV] and + [RuleParLetV], hence are admissible in parallel call-by-value reduction. *) + +Lemma pcbv_RedBetaV: + forall t v u, + is_value v -> + t.[v/] = u -> + pcbv (App (Lam t) v) u. +Proof. + eauto using red_refl with obvious. +Qed. + +Lemma pcbv_RedLetV: + forall t v u, + is_value v -> + t.[v/] = u -> + pcbv (Let v t) u. +Proof. + eauto using red_refl with obvious. +Qed. + +(* Sequences of reduction, [star cbv], can be carried out under a context. *) + +Lemma star_cbv_AppL: + forall t1 t2 u, + star cbv t1 t2 -> + star cbv (App t1 u) (App t2 u). +Proof. + induction 1; eauto with sequences obvious. +Qed. + +Lemma star_pcbv_AppL: + forall t1 t2 u, + star pcbv t1 t2 -> + star pcbv (App t1 u) (App t2 u). +Proof. + induction 1; eauto using red_refl with sequences obvious. +Qed. + +Lemma plus_pcbv_AppL: + forall t1 t2 u, + plus pcbv t1 t2 -> + plus pcbv (App t1 u) (App t2 u). +Proof. + induction 1. + econstructor; [ | eauto using star_pcbv_AppL ]. + eapply RedAppLR; eauto using red_refl with obvious. +Qed. + +Lemma star_cbv_AppR: + forall t u1 u2, + is_value t -> + star cbv u1 u2 -> + star cbv (App t u1) (App t u2). +Proof. + induction 2; eauto with sequences obvious. +Qed. + +Hint Resolve star_cbv_AppL star_pcbv_AppL plus_pcbv_AppL star_cbv_AppR : red obvious. + +Lemma star_cbv_AppLR: + forall t1 t2 u1 u2, + star cbv t1 t2 -> + star cbv u1 u2 -> + is_value t2 -> + star cbv (App t1 u1) (App t2 u2). +Proof. + eauto with sequences obvious. +Qed. + +Lemma star_cbv_LetL: + forall t1 t2 u, + star cbv t1 t2 -> + star cbv (Let t1 u) (Let t2 u). +Proof. + induction 1; eauto with sequences obvious. +Qed. + +Hint Resolve star_cbv_AppLR star_cbv_LetL : red obvious. + +(* Reduction commutes with substitutions of values for variables. (This + includes renamings.) This is true of every reduction strategy, with + the proviso that if [RuleVar] is enabled, then [RuleLam], [RuleAppLR] + and [RuleLetLR] must be enabled as well, so that reduction is reflexive. *) + +Lemma red_subst: + forall mask : mask, + (mask RuleVar -> mask RuleLam) -> + (mask RuleVar -> mask RuleAppLR) -> + (mask RuleVar -> mask RuleLetLR) -> + forall t1 t2, + red mask t1 t2 -> + forall sigma, + is_value_subst sigma -> + red mask t1.[sigma] t2.[sigma]. +Proof. + induction 4; simpl; intros; subst; + try solve [ econstructor; solve [ eauto with is_value | autosubst ]]. + (* Case: [Var] *) + { eauto using red_refl. } +Qed. + +Lemma star_red_subst: + forall mask : mask, + (mask RuleVar -> mask RuleLam) -> + (mask RuleVar -> mask RuleAppLR) -> + (mask RuleVar -> mask RuleLetLR) -> + forall t1 t2 sigma, + star (red mask) t1 t2 -> + is_value_subst sigma -> + star (red mask) t1.[sigma] t2.[sigma]. +Proof. + induction 4; eauto using red_subst with sequences. +Qed. + +(* Call-by-value reduction is contained in parallel call-by-value. *) + +Lemma cbv_subset_pcbv: + forall t1 t2, + cbv t1 t2 -> + pcbv t1 t2. +Proof. + induction 1; try solve [ tauto ]; eauto using red_refl with red. +Qed. + +(* Under call-by-value, values do not reduce. *) + +Lemma values_do_not_reduce: + forall t1 t2, + cbv t1 t2 -> + ~ is_value t1. +Proof. + inversion 1; is_value. +Qed. + +Hint Resolve values_do_not_reduce : is_value obvious. + +Hint Extern 1 (False) => (eapply values_do_not_reduce) : is_value obvious. + +Lemma is_value_irred: + forall v, + is_value v -> + irred cbv v. +Proof. + intros. intro. obvious. +Qed. + +Hint Resolve is_value_irred : irred obvious. + +(* Under every strategy, the property of being a value is preserved by + reduction. *) + +Lemma values_are_stable: + forall mask v1 v2, + red mask v1 v2 -> + is_value v1 -> + is_value v2. +Proof. + induction 1; is_value. +Qed. + +Lemma nonvalues_are_stable: + forall mask v1 v2, + red mask v1 v2 -> + ~ is_value v2 -> + ~ is_value v1. +Proof. + induction 1; is_value. +Qed. + +Hint Resolve values_are_stable nonvalues_are_stable : is_value obvious. + +(* [cbv] is deterministic. *) + +Lemma cbv_deterministic: + forall t t1, + cbv t t1 -> + forall t2, + cbv t t2 -> + t1 = t2. +Proof. + (* Induction over [cbv t t1]. *) + induction 1; try solve [ tauto ]; intros; subst; + (* Invert the second hypothesis, [cbv t t2]. The fact that values do not + reduce is used to eliminate some cases. *) + invert_cbv; + (* The result follows. *) + f_equal; eauto. +Qed. + +(* Inversion lemmas for [irred]. *) + +Lemma invert_irred_cbv_App_1: + forall t u, + irred cbv (App t u) -> + irred cbv t. +Proof. + intros. eapply irred_irred; obvious. +Qed. + +Lemma invert_irred_cbv_App_2: + forall t u, + irred cbv (App t u) -> + is_value t -> + irred cbv u. +Proof. + intros. eapply irred_irred; obvious. +Qed. + +Lemma invert_irred_cbv_App_3: + forall t u, + irred cbv (App t u) -> + is_value t -> + is_value u -> + forall t', t <> Lam t'. +Proof. + intros ? ? Hirred. repeat intro. subst. + eapply Hirred. obvious. +Qed. + +Lemma invert_irred_cbv_Let_1: + forall t u, + irred cbv (Let t u) -> + irred cbv t. +Proof. + intros. eapply irred_irred; obvious. +Qed. + +Lemma invert_irred_cbv_Let_2: + forall t u, + irred cbv (Let t u) -> + ~ is_value t. +Proof. + intros ? ? Hirred ?. eapply Hirred. obvious. +Qed. + +Hint Resolve + invert_irred_cbv_App_1 + invert_irred_cbv_App_2 + invert_irred_cbv_Let_1 + invert_irred_cbv_Let_2 +: irred. + +(* An analysis of irreducible terms for call-by-value reduction. A stuck + term is either an application [v1 v2] where [v1] is not a function or + a stuck term in an evaluation context. *) + +Inductive stuck : term -> Prop := +| StuckApp: + forall v1 v2, + is_value v1 -> + is_value v2 -> + (forall t, v1 <> Lam t) -> + stuck (App v1 v2) +| StuckAppL: + forall t u, + stuck t -> + stuck (App t u) +| StuckAppR: + forall v u, + is_value v -> + stuck u -> + stuck (App v u) +| StuckLetL: + forall t u, + stuck t -> + stuck (Let t u). + +Hint Constructors stuck : stuck. + +(* To go wrong is to reduce to a stuck term. *) + +Definition goes_wrong t := + exists t', star cbv t t' /\ stuck t'. + +(* A stuck term cannot be a value. *) + +Lemma stuck_nonvalue: + forall t, + stuck t -> + ~ is_value t. +Proof. + induction 1; is_value. +Qed. + +(* Every stuck term is irreducible. *) + +Ltac prove_irred_cbv := + do 2 intro; invert_cbv. + +Lemma stuck_irred: + forall t, + stuck t -> + irred cbv t. +Proof. + induction 1; prove_irred_cbv; try solve [ + eauto using irreducible_terms_do_not_reduce + | eapply stuck_nonvalue; obvious + ]. + (* StuckApp *) + { congruence. } +Qed. + +Hint Resolve stuck_irred : irred obvious. + +(* Every irreducible term either is a value or is stuck. *) + +Lemma irred_cbv_is_value_or_stuck: + forall t, + irred cbv t -> + is_value t \/ stuck t. +Proof. + induction t; intro Hirred; + try solve [ is_value ]; right. + (* App *) + { assert (H1: irred cbv t1). { eauto with irred. } + destruct (IHt1 H1); [| eauto with stuck ]. + assert (H2: irred cbv t2). { eauto with irred. } + destruct (IHt2 H2); [| eauto with stuck ]. + eapply StuckApp; eauto using invert_irred_cbv_App_3. } + (* Let *) + { assert (H: irred cbv t). { eauto with irred. } + destruct (IHt H); [| eauto with stuck ]. + assert (~ is_value t). { eauto with irred. } + tauto. } +Qed. + +(* The converse is true as well. *) + +Lemma irred_cbv_characterization: + forall t, + irred cbv t <-> + is_value t \/ stuck t. +Proof. + split. + { eauto using irred_cbv_is_value_or_stuck. } + { intuition eauto with irred. } +Qed. + +(* A closed value must be a lambda-abstraction. *) + +Lemma closed_value: + forall v, + is_value v -> + closed v -> + exists t, v = Lam t. +Proof. + intros. destruct v as [| t | | ]; try solve [ false; is_value ]. + { false. eapply closed_Var. eauto. } + { exists t. eauto. } +Qed. + +(* A stuck term cannot be closed. *) + +Lemma stuck_closed: + forall t, + stuck t -> + closed t -> + False. +Proof. + induction 1; intros; eauto with closed. + (* StuckApp *) + { assert (ht1: exists t1, v1 = Lam t1). + { eauto using closed_value with closed. } + destruct ht1 as (?&?). subst v1. congruence. } +Qed. + +(* Under call-by-value, every closed term either reduces or is a value. *) + +Lemma cbv_progress: + forall t, + closed t -> + is_value t \/ exists u, cbv t u. +Local Ltac ih IH := + destruct IH as [| [ ? ? ]]; [ eauto with closed | | obvious ]. +Proof. + (* We give a direct proof, but the result also follows from + [irred_cbv_is_value_or_stuck] and [stuck_closed]. *) + induction t as [| | t1 IHt1 t2 IHt2 | t1 IHt1 t2 IHt2 ]; + try solve [ left; obvious ]; right. + (* App *) + { ih IHt1. + ih IHt2. + destruct (closed_value t1) as [ u1 ? ]; eauto with closed; subst t1. + obvious. + } + (* Let *) + { ih IHt1. obvious. } +Qed. diff --git a/coq/LambdaCalculusSyntax.v b/coq/LambdaCalculusSyntax.v new file mode 100644 index 0000000..56d1898 --- /dev/null +++ b/coq/LambdaCalculusSyntax.v @@ -0,0 +1,192 @@ +Require Import Coq.Wellfounded.Inverse_Image. +Require Import MyTactics. +Require Export Autosubst.Autosubst. +Require Export AutosubstExtra. +Require Export Autosubst_IsRen. + +(* The syntax of the lambda-calculus. *) + +Inductive term := +| Var (x : var) +| Lam (t : {bind term}) +| App (t1 t2 : term) +| Let (t1 : term) (t2 : {bind term}) +. + +Instance Ids_term : Ids term. derive. Defined. +Instance Rename_term : Rename term. derive. Defined. +Instance Subst_term : Subst term. derive. Defined. +Instance SubstLemmas_term : SubstLemmas term. derive. Qed. + +(* If the image of [t] through a substitution is a variable, then [t] must + itself be a variable. *) + +Lemma subst_is_var: + forall t sigma x, + t.[sigma] = ids x -> + exists y, + t = ids y. +Proof. + intros ? ? ? Heq. destruct t; compute in Heq; solve [ eauto | congruence ]. +Qed. + +(* The identity substitution [ids] is injective. *) + +Lemma inj_ids: + forall x y, + ids x = ids y -> + x = y. +Proof. + intros ? ? Heq. compute in Heq. congruence. +Qed. + +(* If the composition of two substitutions [sigma1] and [sigma2] is the + identity substitution, then [sigma1] must be a renaming. *) + +Lemma ids_implies_is_ren: + forall sigma1 sigma2, + sigma1 >> sigma2 = ids -> + is_ren sigma1. +Proof. + intros ? ? Hid. + eapply prove_is_ren; [ eapply inj_ids | intros x ]. + eapply subst_is_var with (sigma := sigma2) (x := x). + rewrite <- Hid. reflexivity. +Qed. + +Hint Resolve ids_implies_is_ren : is_ren obvious. + +(* The size of a term. *) + +Fixpoint size (t : term) : nat := + match t with + | Var _ => 0 + | Lam t => 1 + size t + | App t1 t2 + | Let t1 t2 => 1 + size t1 + size t2 + end. + +(* The size of a term is preserved by renaming. *) + +Lemma size_renaming: + forall t sigma, + is_ren sigma -> + size t.[sigma] = size t. +Proof. + induction t; intros sigma Hsigma; asimpl; + repeat (match goal with h: forall sigma, _ |- _ => rewrite h by obvious; clear h end); + try reflexivity. + (* [Var] *) + { destruct Hsigma as [ xi ? ]. subst. reflexivity. } +Qed. + +(* The [size] function imposes a well-founded ordering on terms. *) + +Lemma smaller_wf: + well_founded (fun t1 t2 => size t1 < size t2). +Proof. + eauto using wf_inverse_image, lt_wf. +Qed. + +(* The tactic [size_induction] facilitates proofs by induction on the + size of a term. The following lemmas are used in the construction + of this tactic. *) + +Lemma size_induction_intro: + forall (P : term -> Prop), + (forall n t, size t < n -> P t) -> + (forall t, P t). +Proof. + eauto. (* just instantiate [n] with [size t + 1] *) +Defined. + +Lemma size_induction_induction: + forall (P : term -> Prop), + (forall n, (forall t, size t < n -> P t) -> (forall t, size t < S n -> P t)) -> + (forall n t, size t < n -> P t). +Proof. + intros P IH. induction n; intros. + { false. eapply Nat.nlt_0_r. eauto. } + { eauto. } +Defined. + +Lemma size_induction: + forall (P : term -> Prop), + (forall n, (forall t, size t < n -> P t) -> (forall t, size t < S n -> P t)) -> + (forall t, P t). +Proof. + intros P IH. + eapply size_induction_intro. + eapply size_induction_induction. + eauto. +Defined. + +Ltac size_induction := + (* We assume the goal is of the form [forall t, P t]. *) + intro t; pattern t; + match goal with |- ?P t => + simpl; eapply (@size_induction P); clear + end; + intros n IH t Htn. + (* The goal should now be of the form [P t] + with a hypothesis [IH: (forall t, size t < n -> P t] + and a hypothesis [Htn: size t < S n]. *) + +(* The tactic [size] proves goals of the form [size t < n]. The tactic + [obvious] is also extended to prove such goals. *) + +Hint Extern 1 (size ?t.[?sigma] < ?n) => + rewrite size_renaming by obvious + : size obvious. + +Hint Extern 1 (size ?t < ?n) => + simpl in *; omega + : size obvious. + +Ltac size := + eauto with size. + +(* The following is a direct proof of [smaller_wf]. We do not use any + preexisting lemmas, and end the proof with [Defined] instead of [Qed], + so as to make the proof term transparent. Also, we avoid the tactic + [omega], which produces huge proof terms. This allows Coq to compute + with functions that are defined by well-founded recursion. *) + +Lemma smaller_wf_transparent: + well_founded (fun t1 t2 => size t1 < size t2). +Proof. + unfold well_founded. size_induction. + constructor; intros u Hu. + eapply IH. eapply lt_S_n. eapply le_lt_trans; eauto. +Defined. + +(* The following lemmas can be useful in situations where the tactic + [asimpl] performs too much simplification. *) + +Lemma subst_var: + forall x sigma, + (Var x).[sigma] = sigma x. +Proof. + autosubst. +Qed. + +Lemma subst_lam: + forall t sigma, + (Lam t).[sigma] = Lam (t.[up sigma]). +Proof. + autosubst. +Qed. + +Lemma subst_app: + forall t1 t2 sigma, + (App t1 t2).[sigma] = App t1.[sigma] t2.[sigma]. +Proof. + autosubst. +Qed. + +Lemma subst_let: + forall t1 t2 sigma, + (Let t1 t2).[sigma] = Let t1.[sigma] t2.[up sigma]. +Proof. + autosubst. +Qed. diff --git a/coq/LambdaCalculusValues.v b/coq/LambdaCalculusValues.v new file mode 100644 index 0000000..dc6f8dd --- /dev/null +++ b/coq/LambdaCalculusValues.v @@ -0,0 +1,179 @@ +Require Import MyTactics. +Require Import LambdaCalculusSyntax. + +(* The syntactic subcategory of values is decidable. *) + +Definition if_value {A} (t : term) (a1 a2 : A) : A := + match t with + | Var _ | Lam _ => a1 + | _ => a2 + end. + +Definition is_value (t : term) := + if_value t True False. + +Hint Extern 1 (is_value _) => (simpl; tauto) : is_value obvious. + +(* A term either is or is not a value. *) + +Lemma value_or_nonvalue: + forall t, + is_value t \/ ~ is_value t. +Proof. + destruct t; simpl; eauto. +Qed. + +(* Simplification rules for [if_value]. *) + +Lemma if_value_value: + forall A v (a1 a2 : A), + is_value v -> + if_value v a1 a2 = a1. +Proof. + destruct v; simpl; tauto. +Qed. + +Lemma if_value_nonvalue: + forall A t (a1 a2 : A), + ~ is_value t -> + if_value t a1 a2 = a2. +Proof. + destruct t; simpl; tauto. +Qed. + +(* The syntactic subcategory of values is preserved by renamings. *) + +Lemma is_value_renaming: + forall v xi, + is_value v -> + is_value v.[ren xi]. +Proof. + destruct v; simpl; eauto. +Qed. + +Lemma is_nonvalue_renaming: + forall v xi, + ~ is_value v -> + ~ is_value v.[ren xi]. +Proof. + destruct v; simpl; eauto. +Qed. + +Hint Resolve is_value_renaming is_nonvalue_renaming : is_value obvious. + +Ltac is_value := + eauto with is_value. + +(* The tactic [not_a_value] can be used to prove that the current case + is impossible, because we have a hypothesis of the form [~ is_value v], + where [v] clearly is a value. *) + +Ltac not_a_value := + solve [ false; is_value ]. + +Ltac if_value := + repeat first [ rewrite if_value_value by is_value | + rewrite if_value_nonvalue by is_value ]. + +(* The tactic [value_or_nonvalue t] creates two cases: either [t] is a value, + or it isn't. *) + +Ltac value_or_nonvalue t := + destruct (value_or_nonvalue t); + if_value. + +(* The tactic [value_or_app_or_let] creates three cases: either [t] is a value, + or it is an application, or it is a [let] construct. *) + +Ltac value_or_app_or_let t := + value_or_nonvalue t; [| + destruct t as [ ? | ? | t1 t2 | t1 t2 ]; [ not_a_value | not_a_value | |] + (* In principle, we should not fix the names [t1] and [t2] here, + as it might cause name clashes. *) + ]. + +(* The predicate [is_value_subst sigma] holds if every term in the + codomain of the substitution [sigma] is a value. *) + +Definition is_value_subst (sigma : var -> term) := + forall x, is_value (sigma x). + +Lemma is_value_subst_ids: + is_value_subst ids. +Proof. + intros x. is_value. +Qed. + +Lemma is_value_subst_cons: + forall v sigma, + is_value v -> + is_value_subst sigma -> + is_value_subst (v .: sigma). +Proof. + intros. intros [|x]; is_value. +Qed. + +Definition is_value_subst_up: + forall sigma, + is_value_subst sigma -> + is_value_subst (up sigma). +Proof. + intros sigma h. intros [|x]; asimpl. + { simpl. eauto. } + { is_value. } +Qed. + +Definition is_value_subst_upn: + forall sigma i, + is_value_subst sigma -> + is_value_subst (upn i sigma). +Proof. + induction i; intros; asimpl. + { eauto. } + { rewrite <- fold_up_upn. eauto using is_value_subst_up. } +Qed. + +Lemma is_value_subst_renaming: + forall sigma i, + is_value_subst sigma -> + is_value_subst (sigma >> ren (+i)). +Proof. + intros. intro x. asimpl. is_value. +Qed. + +Hint Resolve is_value_subst_up is_value_subst_upn is_value_subst_renaming + : is_value obvious. + +Lemma values_are_preserved_by_value_substitutions: + forall v sigma, + is_value v -> + is_value_subst sigma -> + is_value v.[sigma]. +Proof. + destruct v; simpl; intros; eauto with is_value. +Qed. + +Lemma nonvalues_are_preserved_by_substitutions: + forall t sigma, + ~ is_value t -> + ~ is_value t.[sigma]. +Proof. + destruct t; simpl; tauto. +Qed. + +Hint Resolve + is_value_subst_ids + is_value_subst_cons + values_are_preserved_by_value_substitutions + nonvalues_are_preserved_by_substitutions +: is_value obvious. + +Lemma is_ren_is_value_subst: + forall sigma, + is_ren sigma -> + is_value_subst sigma. +Proof. + intros ? [ xi ? ]. subst. eauto with is_value. +Qed. + +Hint Resolve is_ren_is_value_subst : is_value obvious. diff --git a/coq/Option.v b/coq/Option.v new file mode 100644 index 0000000..5ed2a2e --- /dev/null +++ b/coq/Option.v @@ -0,0 +1,120 @@ +(* -------------------------------------------------------------------------- *) + +(* The [bind] combinator of the option monad. *) + +Definition bind {A B} (f : option A) (k : A -> option B) : option B := + match f with + | None => + None + | Some a => + k a + end. + +(* The standard syntactic sugar for [bind]. [f >>= k] can be read as ``first + do [f]; then, if successful, with the result of [f], do [k]''. *) + +Notation "f >>= k" := (bind f k) (at level 55). + +(* -------------------------------------------------------------------------- *) + +(* These lemmas help prove an equation of the form [f >>= k = b]. *) + +Lemma prove_bind_None: + forall {A B} {f} {k : A -> option B}, + f = None -> + f >>= k = None. +Proof. + intros. subst. eauto. +Qed. + +Lemma prove_bind_Some: + forall {A B} {f} {k : A -> option B} {a b}, + f = Some a -> + k a = b -> + f >>= k = b. +Proof. + intros. subst. eauto. +Qed. + +(* This lemma helps exploit an equation of the form [f >>= k = Some b]. *) + +Lemma invert_bind_Some: + forall {A B} {f} {k : A -> option B} {b}, + f >>= k = Some b -> + exists a, f = Some a /\ k a = Some b. +Proof. + destruct f; simpl; intros. + { eauto. } + { congruence. } +Qed. + +Ltac invert_bind_Some := + match goal with + h: ?f >>= _ = Some _ |- _ => + let heq := fresh in + generalize (invert_bind_Some h); clear h; intros (?&?&h) + end. + +(* -------------------------------------------------------------------------- *) + +(* The standard ordering on options, where [None] is less defined then + everything, and every element of the form [Some a] is less defined + than itself only. *) + +Definition less_defined {A} (o1 o2 : option A) := + forall a, o1 = Some a -> o2 = Some a. + +(* -------------------------------------------------------------------------- *) + +(* This lemma exploits an assertion of the form [less_defined (Some _) _]. *) + +Lemma invert_less_defined_Some: + forall {A} {a : A} {o : option A}, + less_defined (Some a) o -> + Some a = o. +Proof. + unfold less_defined. intros. symmetry. eauto. +Qed. + +Ltac invert_less_defined := + match goal with + | h: less_defined (Some _) _ |- _ => + generalize (invert_less_defined_Some h); clear h; intro h + end. + +(* -------------------------------------------------------------------------- *) + +(* These lemmas help prove assertions of the form [less_defined _ _]. *) + +Lemma prove_less_defined_None: + forall {A} {o : option A}, + less_defined None o. +Proof. + unfold less_defined. intros. congruence. +Qed. + +Lemma reflexive_less_defined: + forall {A} {o : option A}, + less_defined o o. +Proof. + unfold less_defined. eauto. +Qed. + +Local Hint Extern 1 (_ <> _) => congruence : congruence. + +Lemma prove_less_defined_bind: + forall {A B} {f1 f2 : option A} {k1 k2 : A -> option B}, + less_defined f1 f2 -> + (f1 <> None -> forall a, less_defined (k1 a) (k2 a)) -> + less_defined (f1 >>= k1) (f2 >>= k2). +Proof. + intros. destruct f1; simpl in *. + (* Case: [f1] is [Some _]. *) + { invert_less_defined. subst. simpl. eauto with congruence. } + (* Case: [f1] is [None]. *) + { eauto using prove_less_defined_None. } +Qed. + +Hint Resolve + prove_less_defined_None reflexive_less_defined prove_less_defined_bind +: less_defined. diff --git a/coq/Sequences.v b/coq/Sequences.v new file mode 100644 index 0000000..fb5a3ca --- /dev/null +++ b/coq/Sequences.v @@ -0,0 +1,321 @@ +(** A library of relation operators defining sequences of transitions + and useful properties about them. Originally by Xavier Leroy, with + some improvements and additions by François Pottier. *) + +Set Implicit Arguments. + +Section SEQUENCES. + +Variable A: Type. + +Implicit Types R S : A -> A -> Prop. +Implicit Types P : A -> Prop. + +(** Zero, one or several transitions: reflexive, transitive closure of [R]. *) + +Inductive star R : A -> A -> Prop := +| star_refl: + forall a, + star R a a +| star_step: + forall a b c, + R a b -> star R b c -> star R a c. + +Hint Constructors star. + +Lemma star_refl_eq: + forall R a b, a = b -> star R a b. +Proof. + intros. subst. eauto. +Qed. + +Lemma star_one: + forall R a b, R a b -> star R a b. +Proof. + eauto. +Qed. + +Lemma star_trans: + forall R a b, star R a b -> + forall c, star R b c -> star R a c. +Proof. + induction 1; eauto. +Qed. + +Lemma star_covariant: + forall R S, + (forall a b, R a b -> S a b) -> + (forall a b, star R a b -> star S a b). +Proof. + induction 2; eauto. +Qed. + +(* If [R] preserves some property [P], then [star R] preserves [P]. *) + +Lemma star_implication: + forall P R, + (forall a1 a2, R a1 a2 -> P a1 -> P a2) -> + (forall a1 a2, star R a1 a2 -> P a1 -> P a2). +Proof. + induction 2; eauto. +Qed. + +(* The same implication holds in the reverse direction (right to left). *) + +Lemma star_implication_reversed: + forall P R, + (forall a1 a2, R a1 a2 -> P a2 -> P a1) -> + (forall a1 a2, star R a1 a2 -> P a2 -> P a1). +Proof. + induction 2; eauto. +Qed. + +(** One or several transitions: transitive closure of [R]. *) + +Inductive plus R: A -> A -> Prop := +| plus_left: + forall a b c, + R a b -> star R b c -> plus R a c. + +Hint Constructors plus. + +Lemma plus_one: + forall R a b, R a b -> plus R a b. +Proof. + eauto. +Qed. + +Lemma plus_star: + forall R a b, plus R a b -> star R a b. +Proof. + inversion 1; eauto. +Qed. + +Lemma plus_covariant: + forall R S, + (forall a b, R a b -> S a b) -> + (forall a b, plus R a b -> plus S a b). +Proof. + induction 2; eauto using star_covariant. +Qed. + +(* A direct induction principle for [plus]: when [plus R a b] holds, + either there is just one step, or there is one, followed with more. *) + +Lemma plus_ind_direct: + forall R P : A -> A -> Prop, + (forall a b, R a b -> P a b) -> + (forall a b c, R a b -> plus R b c -> P b c -> P a c) -> + forall a b, plus R a b -> P a b. +Proof. + intros ? ? Hone Hmore a c Hplus. destruct Hplus as [ ? b ? hR hRStar ]. + generalize b c hRStar a hR. + clear b c hRStar a hR. + induction 1; eauto. +Qed. + +Lemma plus_star_trans: + forall R a b c, plus R a b -> star R b c -> plus R a c. +Proof. + inversion 1; eauto using star_trans. +Qed. + +Lemma star_plus_trans: + forall R a b c, star R a b -> plus R b c -> plus R a c. +Proof. + inversion 1; inversion 1; eauto using star_trans. +Qed. + +Lemma plus_trans: + forall R a b c, plus R a b -> plus R b c -> plus R a c. +Proof. + eauto using plus_star_trans, plus_star. +Qed. + +Lemma plus_right: + forall R a b c, star R a b -> R b c -> plus R a c. +Proof. + eauto using star_plus_trans. +Qed. + +(** Absence of transitions. *) + +Definition irred R a := + forall b, ~ R a b. + +Definition halts R a := + exists b, star R a b /\ irred R b. + +(** Infinitely many transitions. *) + +CoInductive infseq R : A -> Prop := +| infseq_step: + forall a b, + R a b -> infseq R b -> infseq R a. + +(** Properties of [irred]. *) + +Lemma irred_irred: + forall R t1 u1, + irred R t1 -> + (forall u2, R u1 u2 -> exists t2, R t1 t2) -> + irred R u1. +Proof. + unfold irred. intros ? ? ? Hirred Himpl u2 Hu2. + destruct (Himpl _ Hu2) as [ t2 Ht2 ]. + eapply Hirred. eauto. +Qed. + +Lemma irreducible_terms_do_not_reduce: + forall R a b, irred R a -> R a b -> False. +Proof. + unfold irred, not. eauto. +Qed. + +(** Coinduction principles to show the existence of infinite sequences. *) + +Lemma infseq_coinduction_principle: + forall R P, + (forall a, P a -> exists b, R a b /\ P b) -> + forall a, P a -> infseq R a. +Proof. + intros ? ? Hstep. cofix COINDHYP; intros a hPa. + destruct (Hstep a hPa) as (?&?&?). + econstructor; eauto. +Qed. + +Lemma infseq_coinduction_principle_2: + forall R P a, + P a -> + (forall a, P a -> exists b, plus R a b /\ P b) -> + infseq R a. +Proof. + intros ? ? ? ? Hinv. + apply infseq_coinduction_principle with + (P := fun a => exists b, star R a b /\ P b). + (* Proof that the invariant is preserved. *) + { clear dependent a. + intros a (b&hStar&hPb). + inversion hStar; subst. + { destruct (Hinv b hPb) as [c [hPlus ?]]. + inversion hPlus; subst. eauto. } + { eauto. } + } + (* Proof that the invariant initially holds. *) + { eauto. } +Qed. + +Lemma infseq_plus: + forall R a, + infseq (plus R) a -> + infseq R a. +Proof. + intros. eapply infseq_coinduction_principle_2 + with (P := fun a => infseq (plus R) a). + { eauto. } + clear dependent a. intros a hInfSeq. + destruct hInfSeq. eauto. +Qed. + +(** An example of use of [infseq_coinduction_principle]. *) + +Lemma infseq_alternate_characterization: + forall R a, + (forall b, star R a b -> exists c, R b c) -> + infseq R a. +Proof. + intros R. apply infseq_coinduction_principle. + intros a Hinv. destruct (Hinv a); eauto. +Qed. + +Lemma infseq_covariant: + forall R S, + (forall a b, R a b -> S a b) -> + forall a, infseq R a -> infseq S a. +Proof. + intros. eapply infseq_coinduction_principle + with (P := fun a => infseq R a); [| eauto ]. + clear dependent a. intros a hInfSeq. + destruct hInfSeq. eauto. +Qed. + +(** A sequence either is infinite or stops on an irreducible term. + This property needs excluded middle from classical logic. *) + +Require Import Classical. + +Lemma infseq_or_finseq: + forall R a, + infseq R a \/ halts R a. +Proof. + intros. + destruct (classic (forall b, star R a b -> exists c, R b c)). + { left. eauto using infseq_alternate_characterization. } + { right. + destruct (not_all_ex_not _ _ H) as [b Hb]. + destruct (imply_to_and _ _ Hb). + unfold halts, irred, not. eauto. } +Qed. + +(** Additional properties for deterministic transition relations. *) + +Section DETERMINISM. + +Variable R : A -> A -> Prop. + +Hypothesis R_determ: forall a b c, R a b -> R a c -> b = c. + +Ltac R_determ := + match goal with h1: R ?a ?b1, h2: R ?a ?b2 |- _ => + assert (b1 = b2); [ eauto | subst ] + end. + +(** Uniqueness of transition sequences. *) + +Lemma star_star_inv: + forall a b, star R a b -> forall c, star R a c -> star R b c \/ star R c b. +Proof. + induction 1; inversion 1; intros; subst; try R_determ; eauto. +Qed. + +Lemma finseq_unique: + forall a b b', + star R a b -> irred R b -> + star R a b' -> irred R b' -> + b = b'. +Proof. + unfold irred, not. + intros ? ? ? Hab Hirred Hab' Hirred'. + destruct (star_star_inv Hab Hab') as [ Hbb' | Hbb' ]; + inversion Hbb'; subst; + solve [ eauto | elimtype False; eauto ]. +Qed. + +Lemma infseq_star_inv: + forall a b, star R a b -> infseq R a -> infseq R b. +Proof. + induction 1; inversion 1; intros; try R_determ; eauto. +Qed. + +Lemma infseq_finseq_excl: + forall a b, + star R a b -> irred R b -> infseq R a -> False. +Proof. + unfold irred, not. intros. + assert (h: infseq R b). { eauto using infseq_star_inv. } + inversion h. eauto. +Qed. + +Lemma infseq_halts_excl: + forall a, + halts R a -> infseq R a -> False. +Proof. + intros ? (?&?&?). eauto using infseq_finseq_excl. +Qed. + +End DETERMINISM. + +End SEQUENCES. + +Hint Resolve star_refl star_step star_one star_trans plus_left plus_one + plus_star plus_star_trans star_plus_trans plus_right: sequences. From b69e9ec88fb509ec5168c9eab7d25f6941e838b3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Thu, 28 Sep 2017 10:36:55 +0200 Subject: [PATCH 36/81] Add coq/README.md. --- coq/README.md | 17 +++++++++++++++++ 1 file changed, 17 insertions(+) create mode 100644 coq/README.md diff --git a/coq/README.md b/coq/README.md new file mode 100644 index 0000000..1afe19c --- /dev/null +++ b/coq/README.md @@ -0,0 +1,17 @@ +This code has been tested with Coq 8.5pl3. + +For now, this code requires my slightly patched version of Autosubst. +To install this library, proceed as follows: + +``` + git clone git@github.com:fpottier/autosubst.git + cd autosubst + make && make install +``` + +You can then compile the Coq code as follows: + +``` + make _CoqProject + make -j4 +``` From 548479586f2aa3b676c2b56da90b6dc716a7e8f8 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Thu, 28 Sep 2017 15:15:51 +0200 Subject: [PATCH 37/81] Remove marks in Coq files. --- coq/LambdaCalculusBigStep.v | 4 +--- coq/LambdaCalculusInterpreter.v | 8 -------- 2 files changed, 1 insertion(+), 11 deletions(-) diff --git a/coq/LambdaCalculusBigStep.v b/coq/LambdaCalculusBigStep.v index 0e1c4c6..53bd9ba 100644 --- a/coq/LambdaCalculusBigStep.v +++ b/coq/LambdaCalculusBigStep.v @@ -121,7 +121,7 @@ Qed. (* In conclusion, we have the following equivalence: *) -Theorem star_cbv_bigcbv_eq: +Lemma star_cbv_bigcbv_eq: forall t v, (star cbv t v /\ is_value v) <-> bigcbv t v. Proof. @@ -141,13 +141,11 @@ Qed. (* We break the mutual induction between [cvalue] and [cenv] by inlining the definition of [cenv] into the definition of [cvalue]. *) -(* CVALUE *) Inductive cvalue := | Clo: {bind term} -> list cvalue -> cvalue. Definition cenv := list cvalue. -(* CVALUE *) (* This dummy cvalue is passed below as an argument to [nth], but is really irrelevant, as the condition [x < length e] ensures that the dummy cvalue diff --git a/coq/LambdaCalculusInterpreter.v b/coq/LambdaCalculusInterpreter.v index 5796683..372fc71 100644 --- a/coq/LambdaCalculusInterpreter.v +++ b/coq/LambdaCalculusInterpreter.v @@ -21,7 +21,6 @@ Require Import LambdaCalculusBigStep. evaluation diverges. (Exercise: exhibit a term that reduces to itself in one or more reduction steps. Prove in Coq that this is the case.) *) -(* FAILINTERPRET *) Fail Fixpoint interpret (e : cenv) (t : term) : cvalue := match t with | Var x => @@ -35,13 +34,10 @@ Fail Fixpoint interpret (e : cenv) (t : term) : cvalue := match cv1 with Clo u1 e' => interpret (cv2 :: e') u1 end -(* FAILINTERPRET *) | Let t1 t2 => let cv1 := interpret e t1 in interpret (cv1 :: e) t2 -(* FAILINTERPRET *) end. -(* FAILINTERPRET *) (* -------------------------------------------------------------------------- *) @@ -62,7 +58,6 @@ Fail Fixpoint interpret (e : cenv) (t : term) : cvalue := [interpret] by well-founded recursion. This can be done in Coq but is more complicated, so (here) not worth the trouble. *) -(* FIXINTERPRET *) Fixpoint interpret (n : nat) e t : option cvalue := match n with | 0 => None (* not enough fuel *) @@ -76,13 +71,10 @@ Fixpoint interpret (n : nat) e t : option cvalue := match cv1 with Clo u1 e' => interpret n (cv2 :: e') u1 end -(* FIXINTERPRET *) | Let t1 t2 => interpret n e t1 >>= fun cv1 => interpret n (cv1 :: e) t2 -(* FIXINTERPRET *) end end. -(* FIXINTERPRET *) (* -------------------------------------------------------------------------- *) From fb3e42ac0d963eb7997aa787c52ee24e797c60cd Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pottier?= Date: Fri, 29 Sep 2017 11:21:20 +0200 Subject: [PATCH 38/81] New slides. New file ocaml/Lambda.ml. --- ocaml/Lambda.ml | 175 ++++++++++++++++++++++++++++++++++++++++ slides/fpottier-00.pdf | Bin 290079 -> 290070 bytes slides/fpottier-01a.pdf | Bin 162993 -> 162993 bytes slides/fpottier-01b.pdf | Bin 415997 -> 415997 bytes slides/fpottier-02.pdf | Bin 0 -> 622663 bytes 5 files changed, 175 insertions(+) create mode 100644 ocaml/Lambda.ml create mode 100644 slides/fpottier-02.pdf diff --git a/ocaml/Lambda.ml b/ocaml/Lambda.ml new file mode 100644 index 0000000..4184eb4 --- /dev/null +++ b/ocaml/Lambda.ml @@ -0,0 +1,175 @@ +(* -------------------------------------------------------------------------- *) + +(* The type of lambda-terms, in de Bruijn's representation. *) + +type var = int (* a de Bruijn index *) +type term = + | Var of var + | Lam of (* bind: *) term + | App of term * term + | Let of (* bind: *) term * term + +(* -------------------------------------------------------------------------- *) + +(* [lift_ i k] represents the substitution [upn i (ren (+k))]. Its effect is to + add [k] to every variable that occurs free in [t] and whose index is at + least [i]. *) + +let rec lift_ i k (t : term) : term = + match t with + | Var x -> + if x < i then t else Var (x + k) + | Lam t -> + Lam (lift_ (i + 1) k t) + | App (t1, t2) -> + App (lift_ i k t1, lift_ i k t2) + | Let (t1, t2) -> + Let (lift_ i k t1, lift_ (i + 1) k t2) + +(* [lift k t] adds [k] to every variable that occurs free in [t]. *) + +let lift k t = + lift_ 0 k t + +(* [subst i sigma] represents the substitution [upn i sigma]. *) + +let rec subst_ i (sigma : var -> term) (t : term) : term = + match t with + | Var x -> + if x < i then t else lift i (sigma (x - i)) + | Lam t -> + Lam (subst_ (i + 1) sigma t) + | App (t1, t2) -> + App (subst_ i sigma t1, subst_ i sigma t2) + | Let (t1, t2) -> + Let (subst_ i sigma t1, subst_ (i + 1) sigma t2) + +(* [subst sigma t] applies the substitution [sigma] to the term [t]. *) + +let subst sigma t = + subst_ 0 sigma t + +(* A singleton substitution [u .: ids]. *) + +let singleton (u : term) : var -> term = + function 0 -> u | x -> Var (x - 1) + +(* -------------------------------------------------------------------------- *) + +(* Recognizing a value. *) + +let is_value = function + | Var _ + | Lam _ -> + true + | App _ -> + false + | Let _ -> + false + +(* -------------------------------------------------------------------------- *) + +(* An auxiliary function: the [map] function for the type [_ option]. *) + +(* We name this function [in_context] because we use it below to perform + reduction under an evaluation context. *) + +let in_context f ox = + match ox with + | None -> None + | Some x -> Some (f x) + +(* -------------------------------------------------------------------------- *) + +(* Stepping in call-by-value. *) + +(* This is a naive, direct implementation of the call-by-value reduction + relation, following Plotkin's formulation. The function [step t] returns + [Some t'] if and only if the relation [cbv t t'] holds, and returns [None] + if no such term [t'] exists. *) + +let rec step (t : term) : term option = + match t with + | Lam _ | Var _ -> None + (* Plotkin's BetaV *) + | App (Lam t, v) when is_value v -> + Some (subst (singleton v) t) + (* Plotkin's AppL *) + | App (t, u) when not (is_value t) -> + in_context (fun t' -> App (t', u)) (step t) + (* Plotkin's AppVR *) + | App (v, u) when is_value v -> + in_context (fun u' -> App (v, u')) (step u) + (* All cases covered already, but OCaml cannot see it. *) + | App (_, _) -> + assert false + | Let (t, u) when not (is_value t) -> + in_context (fun t' -> Let (t', u)) (step t) + | Let (v, u) when is_value v -> + Some (subst (singleton v) u) + | Let (_, _) -> + assert false + +let rec eval (t : term) : term = + match step t with + | None -> + t + | Some t' -> + eval t' + +(* -------------------------------------------------------------------------- *) + +(* A naive, substitution-based big-step interpreter. *) + +exception RuntimeError +let rec eval (t : term) : term = + match t with + | Lam _ | Var _ -> t + | Let (t1, t2) -> + let v1 = eval t1 in + eval (subst (singleton v1) t2) + | App (t1, t2) -> + let v1 = eval t1 in + let v2 = eval t2 in + match v1 with + | Lam u1 -> eval (subst (singleton v2) u1) + | _ -> raise RuntimeError + +(* -------------------------------------------------------------------------- *) + +(* A term printer. *) + +open Printf + +let rec print f = function + | Var x -> + fprintf f "(Var %d)" x + | Lam t -> + fprintf f "(Lam %a)" print t + | App (t1, t2) -> + fprintf f "(App %a %a)" print t1 print t2 + | Let (t1, t2) -> + fprintf f "(Let %a %a)" print t1 print t2 + +let print t = + fprintf stdout "%a\n" print t + +(* -------------------------------------------------------------------------- *) + +(* Test. *) + +let id = + Lam (Var 0) + +let idid = + App (id, id) + +let () = + match step idid with + | None -> + assert 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