Preservation under Substitution with Telescopes

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
2
down vote

favorite












In the simply typed lambda calculus, one can show the following result, known as "preservation under substitution":



  • If $Gamma vdash v : tau_1$ and $(x : tau_1) vdash t : tau_2$,
    then $Gamma vdash [v/x]t : tau_2 $.

However, the proof of this relies on the property of permutation, that we can rearrange contexts and it will preserve typing of terms.



I'm wondering, can we prove a similar property for dependently typed languages? The problem is that, here, permutation may not hold, since telescopes are used in place of environments, and the types themselves may refer to variables. Moreover, since the types and terms overlap, we have to substitute in $Gamma$ and $tau_2$ as well.



Does anyone have a good reference to proving such a preservation property for a dependently typed language? Are there tricks that are used to avoid the permutations?






reference-request pl.programming-languages type-theory lambda-calculus dependent-type







share|cite|improve this question

























    up vote
    2
    down vote

    favorite












    In the simply typed lambda calculus, one can show the following result, known as "preservation under substitution":



    • If $Gamma vdash v : tau_1$ and $(x : tau_1) vdash t : tau_2$,
      then $Gamma vdash [v/x]t : tau_2 $.

    However, the proof of this relies on the property of permutation, that we can rearrange contexts and it will preserve typing of terms.



    I'm wondering, can we prove a similar property for dependently typed languages? The problem is that, here, permutation may not hold, since telescopes are used in place of environments, and the types themselves may refer to variables. Moreover, since the types and terms overlap, we have to substitute in $Gamma$ and $tau_2$ as well.



    Does anyone have a good reference to proving such a preservation property for a dependently typed language? Are there tricks that are used to avoid the permutations?






    reference-request pl.programming-languages type-theory lambda-calculus dependent-type







    share|cite|improve this question























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      In the simply typed lambda calculus, one can show the following result, known as "preservation under substitution":



      • If $Gamma vdash v : tau_1$ and $(x : tau_1) vdash t : tau_2$,
        then $Gamma vdash [v/x]t : tau_2 $.

      However, the proof of this relies on the property of permutation, that we can rearrange contexts and it will preserve typing of terms.



      I'm wondering, can we prove a similar property for dependently typed languages? The problem is that, here, permutation may not hold, since telescopes are used in place of environments, and the types themselves may refer to variables. Moreover, since the types and terms overlap, we have to substitute in $Gamma$ and $tau_2$ as well.



      Does anyone have a good reference to proving such a preservation property for a dependently typed language? Are there tricks that are used to avoid the permutations?






      reference-request pl.programming-languages type-theory lambda-calculus dependent-type







      share|cite|improve this question













      In the simply typed lambda calculus, one can show the following result, known as "preservation under substitution":



      • If $Gamma vdash v : tau_1$ and $(x : tau_1) vdash t : tau_2$,
        then $Gamma vdash [v/x]t : tau_2 $.

      However, the proof of this relies on the property of permutation, that we can rearrange contexts and it will preserve typing of terms.



      I'm wondering, can we prove a similar property for dependently typed languages? The problem is that, here, permutation may not hold, since telescopes are used in place of environments, and the types themselves may refer to variables. Moreover, since the types and terms overlap, we have to substitute in $Gamma$ and $tau_2$ as well.



      Does anyone have a good reference to proving such a preservation property for a dependently typed language? Are there tricks that are used to avoid the permutations?






      reference-request pl.programming-languages type-theory lambda-calculus dependent-type




      reference-request pl.programming-languages type-theory lambda-calculus dependent-type






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 2 hours ago









      jmite

      705423




      705423




















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          3
          down vote













          The property, which I would call "typing of substitution" should hold in any type theory, and is not dependent on the exchange property (which I assume is what you mean by permutation)



          The key is that you need to generalize the inductive hypothesis to when the variable in t appears in a context. So for a dependent type theory you prove



          • If $Gamma vdash t_1 : tau_1$ and $Gamma, x : tau_1, Delta vdash t_2 : tau_2$ then $Gamma,Delta[t_1/x] vdash t_2[t_1/x] : tau_2[t_1/x]$





          share|cite|improve this answer




















            Your Answer




            StackExchange.ifUsing("editor", function ()
            return StackExchange.using("mathjaxEditing", function ()
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            );
            );
            , "mathjax-editing");

            StackExchange.ready(function()
            var channelOptions =
            tags: "".split(" "),
            id: "114"
            ;
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function()
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled)
            StackExchange.using("snippets", function()
            createEditor();
            );

            else
            createEditor();

            );

            function createEditor()
            StackExchange.prepareEditor(
            heartbeatType: 'answer',
            convertImagesToLinks: false,
            noModals: false,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: null,
            bindNavPrevention: true,
            postfix: "",
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            );



            );













             

            draft saved


            draft discarded


















            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcstheory.stackexchange.com%2fquestions%2f41816%2fpreservation-under-substitution-with-telescopes%23new-answer', 'question_page');

            );

            Post as a guest

















            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            3
            down vote













            The property, which I would call "typing of substitution" should hold in any type theory, and is not dependent on the exchange property (which I assume is what you mean by permutation)



            The key is that you need to generalize the inductive hypothesis to when the variable in t appears in a context. So for a dependent type theory you prove



            • If $Gamma vdash t_1 : tau_1$ and $Gamma, x : tau_1, Delta vdash t_2 : tau_2$ then $Gamma,Delta[t_1/x] vdash t_2[t_1/x] : tau_2[t_1/x]$





            share|cite|improve this answer
























              up vote
              3
              down vote













              The property, which I would call "typing of substitution" should hold in any type theory, and is not dependent on the exchange property (which I assume is what you mean by permutation)



              The key is that you need to generalize the inductive hypothesis to when the variable in t appears in a context. So for a dependent type theory you prove



              • If $Gamma vdash t_1 : tau_1$ and $Gamma, x : tau_1, Delta vdash t_2 : tau_2$ then $Gamma,Delta[t_1/x] vdash t_2[t_1/x] : tau_2[t_1/x]$





              share|cite|improve this answer






















                up vote
                3
                down vote










                up vote
                3
                down vote









                The property, which I would call "typing of substitution" should hold in any type theory, and is not dependent on the exchange property (which I assume is what you mean by permutation)



                The key is that you need to generalize the inductive hypothesis to when the variable in t appears in a context. So for a dependent type theory you prove



                • If $Gamma vdash t_1 : tau_1$ and $Gamma, x : tau_1, Delta vdash t_2 : tau_2$ then $Gamma,Delta[t_1/x] vdash t_2[t_1/x] : tau_2[t_1/x]$





                share|cite|improve this answer












                The property, which I would call "typing of substitution" should hold in any type theory, and is not dependent on the exchange property (which I assume is what you mean by permutation)



                The key is that you need to generalize the inductive hypothesis to when the variable in t appears in a context. So for a dependent type theory you prove



                • If $Gamma vdash t_1 : tau_1$ and $Gamma, x : tau_1, Delta vdash t_2 : tau_2$ then $Gamma,Delta[t_1/x] vdash t_2[t_1/x] : tau_2[t_1/x]$






                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 1 hour ago









                Max New

                630416




                630416



























                     

                    draft saved


                    draft discarded















































                     


                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function ()
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcstheory.stackexchange.com%2fquestions%2f41816%2fpreservation-under-substitution-with-telescopes%23new-answer', 'question_page');

                    );

                    Post as a guest
































































                    Comments

                    Post a Comment

                    Popular posts from this blog

                    Long meetings (6-7 hours a day): Being “babysat” by supervisor

                    Image
                    Clash Royale CLAN TAG #URR8PPP .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty margin-bottom:0; up vote 31 down vote favorite 4 I hold a PhD in mechanical engineering with a 7 year background in self-guided, self-paced research including 2 years as a graduate level course instructor and a subject matter expert for various engineering projects in a university setting in the United States. For the last 10-11 months, I am working in a research industry in France where my task is to debug C++ spaghetti code written over a period of several years by my current supervisor. It is a team of 2: just I and my supervisor. Diurnally, I am posed with a "bug of the day" to get out of this C++ code. My supervisor generally gives me about 15-30 minutes to solve it and then accosts me about whether or not the task is completed and then rinses and repeats this until the end of the day. Off-late, my supervisor has been holding 6-7 hour long "meetin
                    Read more

                    Is the Concept of Multiple Fantasy Races Scientifically Flawed? [closed]

                    Image
                    Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite Many fantasy worlds have multiple humanoid species (elves, orcs, humans, etc) living in the same world, in addition to that, they often have the ability to interbreed. And I can't help but question their believability... fantasy-races reproduction interspecies-relations share | improve this question edited Aug 9 at 14:46 Michael Kjörling ♦ 21.1k 10 85 161 asked Aug 9 at 13:13 Chlodio 118 6 closed as unclear what you're asking by MichaelK, Gryphon, John, Vincent, Frostfyre Aug 9 at 15:25 Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question. 3 Stack Exchange exist to
                    Read more

                    Confectionery

                    Image
                    Prepared foods made with a lot of sugar or other cabohydrates "Sweetmeat" redirects here. For the racehorse, see Sweetmeat (horse). Not to be confused with Sweetbread. This Kransekake is a traditional Scandinavian baker's confection. Confectionery is the art of making confections , which are food items that are rich in sugar and carbohydrates. Exact definitions are difficult. [1] In general, though, confectionery is divided into two broad and somewhat overlapping categories, bakers' confections and sugar confections. [2] Bakers' confectionery, also called flour confections , includes principally sweet pastries, cakes, and similar baked goods. Sugar confectionery includes candies ( sweets in British English), candied nuts, chocolates, chewing gum, bubble gum, pastillage, and other confections that are made primarily of sugar. In some cases, chocolate confections (confections made of chocolate) are treated as a separate category, as are sugar-free versions of
                    Read more