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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.
Let x6 of type
ι
be given.
Let x7 of type
ι
be given.
Apply unknownprop_b456609235d152f08bccfce314d541d7c44f3716137c00b0ce21cf467ba83d17 with
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
)
)
,
λ x8 .
If_i
(
x8
=
4a7ef..
)
x0
(
If_i
(
x8
=
4ae4a..
4a7ef..
)
x1
(
If_i
(
x8
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
x2
(
If_i
(
x8
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
(
If_i
(
x8
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x4
(
If_i
(
x8
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
x5
(
If_i
(
x8
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
)
x6
x7
)
)
)
)
)
)
,
4ae4a..
4a7ef..
,
λ x8 x9 .
x9
=
x1
leaving 2 subgoals.
The subproof is completed by applying unknownprop_0d106f71868864661e83bc0a2d56eb37ce1d35c8eb0429b88d5f5c42054109b1.
Apply If_i_0 with
4ae4a..
4a7ef..
=
4a7ef..
,
x0
,
If_i
(
4ae4a..
4a7ef..
=
4ae4a..
4a7ef..
)
x1
(
If_i
(
4ae4a..
4a7ef..
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
x2
(
If_i
(
4ae4a..
4a7ef..
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
(
If_i
(
4ae4a..
4a7ef..
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x4
(
If_i
(
4ae4a..
4a7ef..
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
x5
(
If_i
(
4ae4a..
4a7ef..
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
)
x6
x7
)
)
)
)
)
,
λ x8 x9 .
x9
=
x1
leaving 2 subgoals.
The subproof is completed by applying unknownprop_24ff2ea632296eb0012bd83ffdc0e75761169422164b438efe0673b96d912be0.
Apply If_i_1 with
4ae4a..
4a7ef..
=
4ae4a..
4a7ef..
,
x1
,
If_i
(
4ae4a..
4a7ef..
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
x2
(
If_i
(
4ae4a..
4a7ef..
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
(
If_i
(
4ae4a..
4a7ef..
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x4
(
If_i
(
4ae4a..
4a7ef..
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
x5
(
If_i
(
4ae4a..
4a7ef..
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
)
x6
x7
)
)
)
)
.
Let x8 of type
ι
→
ι
→
ο
be given.
Assume H0:
x8
(
4ae4a..
4a7ef..
)
(
4ae4a..
4a7ef..
)
.
The subproof is completed by applying H0.
■