Cwfs as semantics for dependent type theory
Syntax
Semantics
Γ
ctx
[
[
Γ
]
]
:
Con
γ
:
Δ
⇒
Γ
[
[
γ
]
]
:
Subst
[
[
Δ
]
]
[
[
Γ
]
]
Γ
⊢
A
type
[
[
A
]
]
:
Ty
[
[
Γ
]
]
Γ
⊢
a
:
A
[
[
a
]
]
:
Tm
[
[
Γ
]
]
[
[
A
]
]
Γ
,
x
:
A
ctx
[
[
Γ
,
x
:
A
]
]
:≡
[
[
Γ
]
]
▹
[
[
A
]
]
:
Con
Γ
,
x
:
A
⇒
Γ
p
:≡
(
pq
id
)
.1
:
Subst
(
[
[
Γ
]
]
▹
[
[
A
]
]
)
[
[
Γ
]
]
Γ
,
x
:
A
⊢
x
:
A
q
:≡
(
pq
id
)
.2
:
Tm
(
Γ
▹
A
)
A
[
p
]
Note: a category with families doesn't necessarily imply
any
type formers!