That depends on what presentation of Type Theory you are using. Your remarks apply to the extrinsic approach from the last millennium. More recent presentation of Type Theory built in substitution and weakening and use an intrinsic approach which avoids talking about preterms you don’t really care about.

https://dl.acm.org/doi/10.1145/2837614.2837638

Cheers,
Thorsten

From: homotopytypetheory@googlegroups.com <homotopytypetheory@googlegroups.com> on behalf of andrej.bauer@andrej.com <andrej.bauer@andrej.com>
Date: Tuesday, 15 November 2022 at 22:39
To: Homotopy Type Theory <homotopytypetheory@googlegroups.com>
Subject: Re: [HoTT] Question about the formal rules of cohesive homotopy type theory
>  Does this also include the structural rules of type theory such as the substitution and weakening rules?

I would just like to point out that substutition and weakening typically are not part of the rules. They are shown to be admissible. In this spirit, the question should have been: what is the precise version of substitution and weakening (which is a special case of substitution) that is admissible in cohesive type theory?

With kind regards,

Andrej

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