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Proofgold Proof
pf
Let x0 of type
ι
be given.
Let x1 of type
ι
→
ι
be given.
Let x2 of type
ι
→
ι
→
ο
be given.
Let x3 of type
ι
be given.
Assume H0:
prim1
x3
(
38062..
x0
x1
x2
)
.
Apply unknownprop_e4362c04e65a765de9cf61045b78be0adc0f9e51a17754420e1088df0891ff67 with
0fc90..
x0
(
λ x4 .
x1
x4
)
,
λ x4 .
x2
(
f482f..
x4
4a7ef..
)
(
f482f..
x4
(
4ae4a..
4a7ef..
)
)
,
x3
,
∃ x4 .
and
(
prim1
x4
x0
)
(
∃ x5 .
and
(
prim1
x5
(
x1
x4
)
)
(
and
(
x3
=
0fc90..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
(
λ x6 .
If_i
(
x6
=
4a7ef..
)
x4
x5
)
)
(
x2
x4
x5
)
)
)
leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H1:
prim1
x3
(
0fc90..
x0
(
λ x4 .
x1
x4
)
)
.
Assume H2:
x2
(
f482f..
x3
4a7ef..
)
(
f482f..
x3
(
4ae4a..
4a7ef..
)
)
.
Apply unknownprop_b7e2d2f0cdd97ab9c73648950725dee9c8306169301c5c70b54b64bd81f587fb with
x0
,
x1
,
x3
,
∃ x4 .
and
(
prim1
x4
x0
)
(
∃ x5 .
and
(
prim1
x5
(
x1
x4
)
)
(
and
(
x3
=
0fc90..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
(
λ x6 .
If_i
(
x6
=
4a7ef..
)
x4
x5
)
)
(
x2
x4
x5
)
)
)
leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type
ι
be given.
Assume H3:
(
λ x5 .
and
(
prim1
x5
x0
)
(
∃ x6 .
and
(
prim1
x6
(
x1
x5
)
)
(
x3
=
0fc90..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
(
λ x7 .
If_i
(
x7
=
4a7ef..
)
x5
x6
)
)
)
)
x4
.
Apply H3 with
∃ x5 .
and
(
prim1
x5
x0
)
(
∃ x6 .
and
(
prim1
x6
(
x1
x5
)
)
(
and
(
x3
=
0fc90..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
(
λ x7 .
If_i
(
x7
=
4a7ef..
)
x5
x6
)
)
(
x2
x5
x6
)
)
)
.
Assume H4:
prim1
x4
x0
.
Assume H5:
∃ x5 .
and
(
prim1
x5
(
x1
x4
)
)
(
x3
=
0fc90..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
(
λ x6 .
If_i
(
x6
=
4a7ef..
)
x4
x5
)
)
.
Apply H5 with
∃ x5 .
and
(
prim1
x5
x0
)
(
∃ x6 .
and
(
prim1
x6
(
x1
x5
)
)
(
and
(
x3
=
0fc90..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
(
λ x7 .
If_i
(
x7
=
4a7ef..
)
x5
x6
)
)
(
x2
x5
x6
)
)
)
.
Let x5 of type
ι
be given.
Assume H6:
(
λ x6 .
and
(
prim1
x6
(
x1
x4
)
)
(
x3
=
0fc90..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
(
λ x7 .
If_i
(
x7
=
4a7ef..
)
x4
x6
)
)
)
x5
.
Apply H6 with
∃ x6 .
and
(
prim1
x6
x0
)
(
∃ x7 .
and
(
prim1
x7
(
x1
x6
)
)
(
and
(
x3
=
0fc90..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
(
λ x8 .
If_i
(
x8
=
4a7ef..
)
x6
x7
)
)
(
x2
x6
x7
)
)
)
.
Assume H7:
prim1
x5
(
x1
x4
)
.
Assume H8:
x3
=
0fc90..
...
...
.
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