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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
∀ x1 .
prim1
x1
x0
⟶
and
(
cad8f..
x1
)
(
prim1
(
f482f..
x1
4a7ef..
)
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
.
Claim L1:
aae7a..
(
a4c2a..
x0
(
λ x1 .
prim1
(
aae7a..
4a7ef..
(
f482f..
x1
(
4ae4a..
4a7ef..
)
)
)
x0
)
(
λ x1 .
f482f..
x1
(
4ae4a..
4a7ef..
)
)
)
(
a4c2a..
x0
(
λ x1 .
prim1
(
aae7a..
(
4ae4a..
4a7ef..
)
(
f482f..
x1
(
4ae4a..
4a7ef..
)
)
)
x0
)
(
λ x1 .
f482f..
x1
(
4ae4a..
4a7ef..
)
)
)
=
x0
Apply set_ext with
aae7a..
(
a4c2a..
x0
(
λ x1 .
prim1
(
aae7a..
4a7ef..
(
f482f..
x1
(
4ae4a..
4a7ef..
)
)
)
x0
)
(
λ x1 .
f482f..
x1
(
4ae4a..
4a7ef..
)
)
)
(
a4c2a..
x0
(
λ x1 .
prim1
(
aae7a..
(
4ae4a..
4a7ef..
)
(
f482f..
x1
(
4ae4a..
4a7ef..
)
)
)
x0
)
(
λ x1 .
f482f..
x1
(
4ae4a..
4a7ef..
)
)
)
,
x0
leaving 2 subgoals.
Let x1 of type
ι
be given.
Assume H1:
prim1
x1
(
aae7a..
(
a4c2a..
x0
(
λ x2 .
prim1
(
aae7a..
4a7ef..
(
f482f..
x2
(
4ae4a..
4a7ef..
)
)
)
x0
)
(
λ x2 .
f482f..
x2
(
4ae4a..
4a7ef..
)
)
)
(
a4c2a..
x0
(
λ x2 .
prim1
(
aae7a..
(
4ae4a..
4a7ef..
)
(
f482f..
x2
(
4ae4a..
4a7ef..
)
)
)
x0
)
(
λ x2 .
f482f..
x2
(
4ae4a..
4a7ef..
)
)
)
)
.
Apply unknownprop_583e189228469f510dae093aa816b0d084f1acaf0341e7deab9d9a676d1b11ef with
a4c2a..
x0
(
λ x2 .
prim1
(
aae7a..
4a7ef..
(
f482f..
x2
(
4ae4a..
4a7ef..
)
)
)
x0
)
(
λ x2 .
f482f..
x2
(
4ae4a..
4a7ef..
)
)
,
a4c2a..
x0
(
λ x2 .
prim1
(
aae7a..
(
4ae4a..
4a7ef..
)
(
f482f..
x2
(
4ae4a..
4a7ef..
)
)
)
x0
)
(
λ x2 .
f482f..
x2
(
4ae4a..
4a7ef..
)
)
,
x1
,
prim1
x1
x0
leaving 3 subgoals.
The subproof is completed by applying H1.
Assume H2:
∃ x2 .
and
(
prim1
x2
(
a4c2a..
x0
(
λ x3 .
prim1
(
aae7a..
4a7ef..
(
f482f..
x3
(
4ae4a..
4a7ef..
)
)
)
x0
)
(
λ x3 .
f482f..
x3
(
4ae4a..
4a7ef..
)
)
)
)
(
x1
=
aae7a..
4a7ef..
x2
)
.
Apply exandE_i with
λ x2 .
prim1
x2
(
a4c2a..
x0
(
λ x3 .
prim1
(
aae7a..
4a7ef..
(
f482f..
x3
(
4ae4a..
4a7ef..
)
)
)
x0
)
(
λ x3 .
f482f..
x3
(
4ae4a..
4a7ef..
)
)
)
,
λ x2 .
x1
=
aae7a..
4a7ef..
x2
,
prim1
x1
x0
leaving 2 subgoals.
The subproof is completed by applying H2.
Let x2 of type
ι
be given.
Assume H3:
prim1
x2
(
a4c2a..
...
...
...
)
.
...
...
...
Apply L1 with
λ x1 x2 .
cad8f..
x1
.
The subproof is completed by applying unknownprop_74e7cf1e344053cc0ec428569910c022764085f38ffd3769b306f1021b002573 with
a4c2a..
x0
(
λ x1 .
prim1
(
aae7a..
4a7ef..
(
f482f..
x1
(
4ae4a..
4a7ef..
)
)
)
x0
)
(
λ x1 .
f482f..
x1
(
4ae4a..
4a7ef..
)
)
,
a4c2a..
x0
(
λ x1 .
prim1
(
aae7a..
(
4ae4a..
4a7ef..
)
(
f482f..
x1
(
4ae4a..
4a7ef..
)
)
)
x0
)
(
λ x1 .
f482f..
x1
(
4ae4a..
4a7ef..
)
)
.
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