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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.
Apply unknownprop_b456609235d152f08bccfce314d541d7c44f3716137c00b0ce21cf467ba83d17 with
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
)
,
λ x7 .
If_i
(
x7
=
4a7ef..
)
x0
(
If_i
(
x7
=
4ae4a..
4a7ef..
)
x1
(
If_i
(
x7
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
x2
(
If_i
(
x7
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
(
If_i
(
x7
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x4
(
If_i
(
x7
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
x5
x6
)
)
)
)
)
,
4ae4a..
(
4ae4a..
4a7ef..
)
,
λ x7 x8 .
x8
=
x2
leaving 2 subgoals.
The subproof is completed by applying unknownprop_bcc68e14c852aec011dc79b8ee401d729519c729ef73436787d031f53d57be8a.
Apply If_i_0 with
4ae4a..
(
4ae4a..
4a7ef..
)
=
4a7ef..
,
x0
,
If_i
(
4ae4a..
(
4ae4a..
4a7ef..
)
=
4ae4a..
4a7ef..
)
x1
(
If_i
(
4ae4a..
(
4ae4a..
4a7ef..
)
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
x2
(
If_i
(
4ae4a..
(
4ae4a..
4a7ef..
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
(
If_i
(
4ae4a..
(
4ae4a..
4a7ef..
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x4
(
If_i
(
4ae4a..
(
4ae4a..
4a7ef..
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
x5
x6
)
)
)
)
,
λ x7 x8 .
x8
=
x2
leaving 2 subgoals.
The subproof is completed by applying unknownprop_38b5b92bfd131ef0428a8ee212f746419349b56c1e9077ab24b3c25647caace1.
Apply If_i_0 with
4ae4a..
(
4ae4a..
4a7ef..
)
=
4ae4a..
4a7ef..
,
x1
,
If_i
(
4ae4a..
(
4ae4a..
4a7ef..
)
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
x2
(
If_i
(
4ae4a..
(
4ae4a..
4a7ef..
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
(
If_i
(
4ae4a..
(
4ae4a..
4a7ef..
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x4
(
If_i
(
4ae4a..
(
4ae4a..
4a7ef..
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
x5
x6
)
)
)
,
λ x7 x8 .
x8
=
x2
leaving 2 subgoals.
The subproof is completed by applying unknownprop_e0e602efce8c3fbbb791d74669e968feb76e9d28b29f3878aea98e1fe1efe584.
Apply If_i_1 with
4ae4a..
(
4ae4a..
4a7ef..
)
=
4ae4a..
(
4ae4a..
4a7ef..
)
,
x2
,
If_i
(
4ae4a..
(
4ae4a..
4a7ef..
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
(
If_i
(
4ae4a..
(
4ae4a..
4a7ef..
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x4
(
If_i
(
4ae4a..
(
4ae4a..
4a7ef..
)
=
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
x5
x6
)
)
.
Let x7 of type
ι
→
ι
→
ο
be given.
Assume H0:
x7
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
.
The subproof is completed by applying H0.
■