1.12.2 Equivalence of path induction and based path induction

1.12.2 Equivalence of path induction and based path induction

The two

induction

principles for the identity type introduced above are

equivalent

.
It is easy to see that path induction follows from the based path induction principle.
Indeed, let us assume the

premises

of path induction:

C

:

∏

x

,

y

:

A

(

x

=

A

y

)

→

𝒰

,

c

:

∏

x

:

A

C

⁢

(

x

,

x

,

𝗋𝖾𝖿𝗅

x

)

.

Now, given an

element

x

:

A

, we can instantiate both of the above, obtaining

C

′

:

∏

y

:

A

(

x

=

A

y

)

→

𝒰

,

C

′

:

≡

C

(

x

)

,

c

′

:

C

′

⁢

(

x

,

𝗋𝖾𝖿𝗅

x

)

,

c

′

:

≡

c

(

x

)

.

Clearly,

C

′

and

c

′

match the premises of based path induction and hence we can construct

g

:

∏

(

y

:

A

)

∏

(

p

:

x

=

y

)

C

′

⁢

(

y

,

p

)

with the defining equality

g

(

x

,

𝗋𝖾𝖿𝗅

x

)

:

≡

c

′

.

Now we observe that

g

’s

codomain

is equal to

C

⁢

(

x

,

y

,

p

)

.
Thus, discharging our

assumption

x

:

A

, we can derive a function

f

:

∏

(

x

,

y

:

A

)

∏

(

p

:

x

=

A

y

)

C

⁢

(

x

,

y

,

p

)

with the required judgmental equality

f

(

x

,

x

,

𝗋𝖾𝖿𝗅

x

)

≡

g

(

x

,

𝗋𝖾𝖿𝗅

x

)

:

≡

c

′

:

≡

c

(

x

)

.

Another proof of this fact is to observe that any such

f

can be obtained as an

instance

of

𝗂𝗇𝖽

=

A

′

so it suffices to define

𝗂𝗇𝖽

=

A

in terms of

𝗂𝗇𝖽

=

A

′

as

𝗂𝗇𝖽

=

A

(

C

,

c

,

x

,

y

,

p

)

:

≡

𝗂𝗇𝖽

=

A

′

(

x

,

C

(

x

)

,

c

(

x

)

,

y

,

p

)

.

The other direction is a bit trickier; it is not clear how we can use a particular instance of path induction to derive a particular instance of
based path induction. What we can do instead is to construct one instance of path induction which shows
all possible instantiations of based path induction at once.
Define

D

:

∏

x

,

y

:

A

(

x

=

A

y

)

→

𝒰

,

D

⁢

(

x

,

y

,

p

)

:

≡

∏

C

:

∏

(

z

:

A

)

(

x

=

A

z

)

→

𝒰

C

(

x

,

𝗋𝖾𝖿𝗅

x

)

→

C

(

y

,

p

)

.

Then we can construct the function

d

:

∏

x

:

A

D

⁢

(

x

,

x

,

𝗋𝖾𝖿𝗅

x

)

,

d

:

≡

λ

x

.

λ

C

.

λ

(

c

:

C

(

x

,

𝗋𝖾𝖿𝗅

x

)

)

.

c

and hence using path induction obtain

f

:

∏

(

x

,

y

:

A

)

∏

(

p

:

x

=

A

y

)

D

⁢

(

x

,

y

,

p

)

with

f

(

x

,

x

,

𝗋𝖾𝖿𝗅

x

)

:

≡

d

(

x

)

. Unfolding the definition of

D

, we can expand the type of

f

:

f

:

∏

(

x

,

y

:

A

)

∏

(

p

:

x

=

A

y

)

∏

(

C

:

∏

(

z

:

A

)

(

x

=

A

z

)

→

𝒰

)

C

⁢

(

x

,

𝗋𝖾𝖿𝗅

x

)

→

C

⁢

(

y

,

p

)

.

Now given

x

:

A

and

p

:

a

=

A

x

, we can derive the

conclusion

of based path induction:

f

⁢

(

a

,

x

,

p

,

C

,

c

)

:

C

⁢

(

x

,

p

)

.

Notice that we also obtain the correct definitional equality.

Another proof is to observe that any use of based path induction is an instance of

𝗂𝗇𝖽

=

A

′

and to define

𝗂𝗇𝖽

=

A

′

(

a

,

C

,

c

,

x

,

p

)

:

≡

𝗂𝗇𝖽

=

A

(

(

λ

⁢

x

,

y

.

λ

⁢

p

.

∏

(

C

:

∏

(

z

:

A

)

(

x

=

A

z

)

→

𝒰

)

C

⁢

(

x

,

𝗋𝖾𝖿𝗅

x

)

→

C

⁢

(

y

,

p

)

)

,

(

λ

x

.

λ

C

.

λ

d

.

d

)

,

a

,

x

,

p

,

C

,

c

)

Note that the construction given above uses

universes

. That is, if we want to
model

𝗂𝗇𝖽

=

A

′

with

C

:

∏

(

x

:

A

)

(

a

=

A

x

)

→

𝒰

i

, we need
to use

𝗂𝗇𝖽

=

A

with

D

:

∏

x

,

y

:

A

(

x

=

A

y

)

→

𝒰

i

+

1

since

D

quantifies over all

C

of the given type. While this is

compatible

with our definition of universes, it is also possible to
derive

𝗂𝗇𝖽

=

A

′

without using universes: we can show that

𝗂𝗇𝖽

=

A

entails Lemma 1 (

http://planetmath.org/23typefamiliesarefibrations#Thmlem1

),Lemma 3.11.8 (

http://planetmath.org/311contractibility#Thmprelem5

), and that these two principles imply

𝗂𝗇𝖽

=

A

′

directly.
We leave the details to the reader as

http://planetmath.org/node/87560

Exercise 1.7.

We can use either of the foregoing formulations of identity types
to establish that equality is an

equivalence relation

, that every function preserves equality and that every family respects equality. We leave the details to the next chapter, where this will be derived and explained in the

context

of

homotopy type theory

.

Title

1.12.2 Equivalence of path induction and based path induction

\metatable

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