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Proofgold Proof
pf
Let x0 of type
ι
be given.
Apply and3I with
∀ x1 .
prim1
x1
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x2 .
bij
x0
x0
(
λ x3 .
f482f..
x2
x3
)
)
)
⟶
∀ x2 .
prim1
x2
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x3 .
bij
x0
x0
(
λ x4 .
f482f..
x3
x4
)
)
)
⟶
prim1
(
(
λ x3 x4 .
0fc90..
x0
(
λ x5 .
f482f..
x4
(
f482f..
x3
x5
)
)
)
x1
x2
)
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x3 .
bij
x0
x0
(
λ x4 .
f482f..
x3
x4
)
)
)
,
∀ x1 .
prim1
x1
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x2 .
bij
x0
x0
(
λ x3 .
f482f..
x2
x3
)
)
)
⟶
∀ x2 .
prim1
x2
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x3 .
bij
x0
x0
(
λ x4 .
f482f..
x3
x4
)
)
)
⟶
∀ x3 .
prim1
x3
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x4 .
bij
x0
x0
(
λ x5 .
f482f..
x4
x5
)
)
)
⟶
(
λ x4 x5 .
0fc90..
x0
(
λ x6 .
f482f..
x5
(
f482f..
x4
x6
)
)
)
x1
(
(
λ x4 x5 .
0fc90..
x0
(
λ x6 .
f482f..
x5
(
f482f..
x4
x6
)
)
)
x2
x3
)
=
(
λ x4 x5 .
0fc90..
x0
(
λ x6 .
f482f..
x5
(
f482f..
x4
x6
)
)
)
(
(
λ x4 x5 .
0fc90..
x0
(
λ x6 .
f482f..
x5
(
f482f..
x4
x6
)
)
)
x1
x2
)
x3
,
∃ x1 .
and
(
prim1
x1
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x2 .
bij
x0
x0
(
λ x3 .
f482f..
x2
x3
)
)
)
)
(
and
(
∀ x2 .
prim1
x2
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x3 .
bij
x0
x0
(
λ x4 .
f482f..
x3
x4
)
)
)
⟶
and
(
(
λ x3 x4 .
0fc90..
x0
(
λ x5 .
f482f..
x4
(
f482f..
x3
x5
)
)
)
x1
x2
=
x2
)
(
(
λ x3 x4 .
0fc90..
x0
(
λ x5 .
f482f..
x4
(
f482f..
x3
x5
)
)
)
x2
x1
=
x2
)
)
(
∀ x2 .
prim1
x2
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x3 .
bij
x0
...
...
)
)
⟶
∃ x3 .
and
(
prim1
x3
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x4 .
bij
x0
x0
(
λ x5 .
f482f..
x4
x5
)
)
)
)
(
and
(
(
λ x4 x5 .
0fc90..
x0
(
λ x6 .
f482f..
x5
(
f482f..
x4
x6
)
)
)
x2
x3
=
x1
)
(
(
λ x4 x5 .
0fc90..
x0
(
λ x6 .
f482f..
x5
(
f482f..
x4
x6
)
)
)
x3
x2
=
x1
)
)
)
)
leaving 3 subgoals.
...
...
...
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