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Proofgold Proof
pf
Let x0 of type
ι
be given.
Let x1 of type
ι
be given.
Assume H0:
0b8ef..
x0
=
6c5f4..
x1
.
Claim L1:
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x3
x3
)
)
(
prim0
(
prim0
x2
x2
)
(
prim0
x3
x3
)
)
)
)
)
(
prim1
(
λ x2 .
x2
)
)
=
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x3
x2
)
)
(
prim0
(
prim0
x3
x3
)
(
prim0
x2
x3
)
)
)
)
)
(
prim1
(
λ x2 .
x2
)
)
⟶
∀ x2 : ο .
x2
Assume H1:
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x3
x3
)
)
(
prim0
(
prim0
x2
x2
)
(
prim0
x3
x3
)
)
)
)
)
(
prim1
(
λ x2 .
x2
)
)
=
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x3
x2
)
)
(
prim0
(
prim0
x3
x3
)
(
prim0
x2
x3
)
)
)
)
)
(
prim1
(
λ x2 .
x2
)
)
.
Apply unknownprop_ffff67c8af57907058601518623f56f08b1980b5dd3f4aa47f3f04cf81b40e90 with
236c6..
,
236c6..
.
Apply unknownprop_3b55668ebff4b5090f31d28781617fd36508e9eaa3458c4a604deaa4db096497 with
prim0
236c6..
236c6..
,
prim0
236c6..
236c6..
,
prim0
236c6..
236c6..
,
236c6..
.
Apply unknownprop_3b55668ebff4b5090f31d28781617fd36508e9eaa3458c4a604deaa4db096497 with
prim0
(
prim0
236c6..
236c6..
)
236c6..
,
prim0
(
prim0
236c6..
236c6..
)
(
prim0
236c6..
236c6..
)
,
prim0
(
prim0
236c6..
236c6..
)
236c6..
,
prim0
(
prim0
236c6..
236c6..
)
236c6..
.
Apply unknownprop_d095413e3d6561ecf3858ab767f11a6b26abbe476371cd3ff75f5d610cbf6aa9 with
prim0
(
prim0
(
prim0
236c6..
236c6..
)
236c6..
)
(
prim0
(
prim0
236c6..
236c6..
)
(
prim0
236c6..
236c6..
)
)
,
prim0
(
prim0
236c6..
236c6..
)
(
prim0
(
prim0
236c6..
236c6..
)
(
prim0
236c6..
236c6..
)
)
,
prim0
(
prim0
(
prim0
236c6..
236c6..
)
236c6..
)
(
prim0
(
prim0
236c6..
236c6..
)
236c6..
)
,
prim0
(
prim0
(
prim0
236c6..
236c6..
)
(
prim0
236c6..
236c6..
)
)
(
prim0
236c6..
(
prim0
236c6..
236c6..
)
)
.
Apply unknownprop_ede725485070ff6bd749ebc14706bb968573533ce94967136e2d0850c2f5f853 with
λ x2 .
prim0
(
prim0
(
prim0
x2
236c6..
)
(
prim0
x2
x2
)
)
(
prim0
(
prim0
236c6..
236c6..
)
(
prim0
x2
x2
)
)
,
λ x2 .
prim0
(
prim0
(
prim0
x2
236c6..
)
(
prim0
x2
...
)
)
...
,
...
.
...
Apply L1.
Apply unknownprop_d095413e3d6561ecf3858ab767f11a6b26abbe476371cd3ff75f5d610cbf6aa9 with
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x3
x3
)
)
(
prim0
(
prim0
x2
x2
)
(
prim0
x3
x3
)
)
)
)
)
(
prim1
(
λ x2 .
x2
)
)
,
x0
,
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x3
x2
)
)
(
prim0
(
prim0
x3
x3
)
(
prim0
x2
x3
)
)
)
)
)
(
prim1
(
λ x2 .
x2
)
)
,
x1
.
The subproof is completed by applying H0.
■