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Proofgold Proof
pf
Let x0 of type
ι
be given.
Let x1 of type
ι
be given.
Let x2 of type
ι
be given.
Let x3 of type
ι
be given.
Let x4 of type
ι
be given.
Let x5 of type
ι
be given.
Apply unknownprop_b456609235d152f08bccfce314d541d7c44f3716137c00b0ce21cf467ba83d17 with
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
,
λ x6 .
If_i
(
x6
=
4a7ef..
)
x0
(
If_i
(
x6
=
4ae4a..
4a7ef..
)
x1
(
If_i
(
x6
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
x2
(
If_i
(
x6
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
(
If_i
(
x6
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x4
x5
)
)
)
)
,
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
,
λ x6 x7 .
x7
=
x4
leaving 2 subgoals.
The subproof is completed by applying unknownprop_43f6998a149d8de21bb8a03189dc42a81b07339f977b1333c98547b472647d3c.
Apply If_i_0 with
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
4a7ef..
,
x0
,
If_i
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
4ae4a..
4a7ef..
)
x1
(
If_i
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
x2
(
If_i
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
(
If_i
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x4
x5
)
)
)
,
λ x6 x7 .
x7
=
x4
leaving 2 subgoals.
The subproof is completed by applying unknownprop_ad3d8dc286672dabbf45d75a5bddff9246ae99a1b47ead9979d5568de89957bd.
Apply If_i_0 with
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
4ae4a..
4a7ef..
,
x1
,
If_i
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
x2
(
If_i
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
(
If_i
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x4
x5
)
)
,
λ x6 x7 .
x7
=
x4
leaving 2 subgoals.
The subproof is completed by applying unknownprop_ac927f2ad247eb36bc322e3579f4429fc5e423f90a2de7867227e2cfdd51eb3e.
Apply If_i_0 with
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
4ae4a..
(
4ae4a..
4a7ef..
)
,
x2
,
If_i
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
(
If_i
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x4
x5
)
,
λ x6 x7 .
x7
=
x4
leaving 2 subgoals.
The subproof is completed by applying unknownprop_c89656f2288520ece9bcd1dc797dc051c5cdc0455848926e17e58cdafcf5fb33.
Apply If_i_0 with
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
,
x3
,
If_i
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x4
x5
,
λ x6 x7 .
x7
=
x4
leaving 2 subgoals.
The subproof is completed by applying unknownprop_37ab4afe9499820a5a47721f837a6c564315dfb9c67b9507741e6969ad940e64.
Apply If_i_1 with
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
,
x4
,
x5
.
Let x6 of type
ι
→
ι
→
ο
be given.
Assume H0:
x6
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
.
The subproof is completed by applying H0.
■