A general definition ofDTT's

HOTTESTOctober 9,2011

Andrej Bauerjointwork with

Philipp HaselwarterPeter Lefanu Lumsdaine

What are type theories

A DTT is a formal system

syntax expressions types

binding dependencies

judgmentsderivation

Methods for dealing with such formalisms

concrete syntax words obsolete

abstract syntax well foundedtreesreasonable t

intrinsic syntax objects ofmetalevel formal systemlogical framework Q1

Tandtypetheory

Steps1 Mathematical preliminaries2 Raw syntaxRawtypetheories definition

4 Acceptable TT

5 Well founded TT

6 Resultsmetatheoremsexamples Martin Lovian

1 A type 1 A Bsemantics

Ptt A 1 t asa implementation

MITT84 MITTExamples

c c HottbookTTuniversesalatarshi

Counter examples

cubical TT

T D tthinear

Raw syntax

Free bound variables Pick some standardway e.g deBruijnindices

T Xi Ai Xi A Xu An

8 Xi't KiB xn D shape

Ifl positionso D n O n D

Signature describe symbols of TTarity Syntactic dances

IT Ty Etty o Hy ty.tnA Hm Hyol lty.nl Hmm

app Ctm Illy ol ly.tl AmolHm017 t 21 0

Symbols eHm'T ITH B

5 Alds its ix xAXA listIAI

Atype x ATBHtype y tely Bly f A B et I AGx3BG Ky's.ely TICA BAD

w app A B e e

f N WVeck131 KythterolyA

signature

Raw type theoriesT Craw typetheory over a signature 2

family of rulespremisesWhat is a rule r

1 Atype x ATB type1 ITCA l type conclusion

R info about metavariablesa familyof premises judgements pfitytieexprtzy.lyconclusion judgement T t J to BKarol

K

A N Ltg El 2 2,2B t Hy Htmol pf te Ctm I metavariables as

a tin symbols localtotherule

A111

Tt A typeTt B TM Tt A B uniqueness of tyringT t t A1 t t B

u Eka ta Lw

Ella type

u El u thecode of the universeEl decodes codes to types

a Elfuitui.EICui.fm1 A type in Elica type

List A type A B

1 ListAListB

1Antyle 1Artie ii AalBakitype XiAitBaki lyle

1 An Aa Xi Ant B Bakr congruencerule

1 IT An Blet IT xi.az Baki

D Dms

Ttt A T't Atype

### Substitution elimination**Theorem:** Suppose `T` is a congruous type theory. If `T` derives a judgement `Γ ⊢ J`, then it also derives `J` without any applications of the substitution rules.### Derivability of presuppositions**Theorem:** Suppose `T` is presuppository. If `T` derives a judgement `Γ ⊢ J`, then it also derives its presuppositions.### Uniqueness of typing**Theorem:** Suppose `T` is tight. If `T` derives `Γ ⊢ A type`, `Γ ⊢ B type`, `Γ ⊢ t : A`,and `Γ ⊢ t : B`, then it also derives `Γ ⊢ A ≡ B`.

D DD 1Atype B My introInversion principles

1 TAB type 1 TAB type