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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
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