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Proofgold Proof
pf
Claim L0:
∀ x0 .
80242..
x0
⟶
∀ x1 .
80242..
x1
⟶
∀ x2 x3 :
ι →
ι → ι
.
(
∀ x4 .
prim1
x4
(
56ded..
(
e4431..
x0
)
)
⟶
∀ x5 .
80242..
x5
⟶
x2
x4
x5
=
x3
x4
x5
)
⟶
(
∀ x4 .
prim1
x4
(
56ded..
(
e4431..
x1
)
)
⟶
x2
x0
x4
=
x3
x0
x4
)
⟶
(
λ x4 x5 .
λ x6 :
ι →
ι → ι
.
02a50..
(
0ac37..
(
94f9e..
(
23e07..
x4
)
(
λ x7 .
x6
x7
x5
)
)
(
94f9e..
(
23e07..
x5
)
(
λ x7 .
x6
x4
x7
)
)
)
(
0ac37..
(
94f9e..
(
5246e..
x4
)
(
λ x7 .
x6
x7
x5
)
)
(
94f9e..
(
5246e..
x5
)
(
λ x7 .
x6
x4
x7
)
)
)
)
x0
x1
x2
=
(
λ x4 x5 .
λ x6 :
ι →
ι → ι
.
02a50..
(
0ac37..
(
94f9e..
(
23e07..
x4
)
(
λ x7 .
x6
x7
x5
)
)
(
94f9e..
(
23e07..
x5
)
(
λ x7 .
x6
x4
x7
)
)
)
(
0ac37..
(
94f9e..
(
5246e..
x4
)
(
λ x7 .
x6
x7
x5
)
)
(
94f9e..
(
5246e..
x5
)
(
λ x7 .
x6
x4
x7
)
)
)
)
x0
x1
x3
Let x0 of type
ι
be given.
Assume H0:
80242..
x0
.
Let x1 of type
ι
be given.
Assume H1:
80242..
x1
.
Let x2 of type
ι
→
ι
→
ι
be given.
Let x3 of type
ι
→
ι
→
ι
be given.
Assume H2:
∀ x4 .
prim1
x4
(
56ded..
(
e4431..
x0
)
)
⟶
∀ x5 .
80242..
x5
⟶
x2
x4
x5
=
x3
x4
x5
.
Assume H3:
∀ x4 .
prim1
x4
(
56ded..
(
e4431..
x1
)
)
⟶
x2
x0
x4
=
x3
x0
x4
.
Claim L4:
...
...
Claim L5:
...
...
Claim L6:
...
...
Claim L7:
...
...
Apply L4 with
λ x4 x5 .
02a50..
(
0ac37..
x5
(
94f9e..
(
23e07..
x1
)
(
λ x6 .
x2
x0
x6
)
)
)
(
0ac37..
(
94f9e..
(
5246e..
x0
)
(
λ x6 .
x2
x6
x1
)
)
(
94f9e..
(
5246e..
x1
)
(
λ x6 .
x2
x0
x6
)
)
)
=
02a50..
(
0ac37..
(
94f9e..
(
23e07..
x0
)
(
λ x6 .
x3
x6
x1
)
)
(
94f9e..
(
23e07..
x1
)
(
λ x6 .
x3
x0
x6
)
)
)
(
0ac37..
(
94f9e..
(
5246e..
x0
)
(
λ x6 .
x3
x6
x1
)
)
(
94f9e..
(
5246e..
x1
)
(
λ x6 .
x3
x0
x6
)
)
)
.
Apply L5 with
λ x4 x5 .
02a50..
(
0ac37..
(
94f9e..
(
23e07..
x0
)
(
λ x6 .
x3
x6
x1
)
)
x5
)
(
0ac37..
(
94f9e..
(
5246e..
x0
)
...
)
...
)
=
...
.
...
Apply unknownprop_0d078a833e20a13be15691da098ab7e638a79d34f684fe40cbe56431440f9e3e with
λ x0 x1 .
λ x2 :
ι →
ι → ι
.
02a50..
(
0ac37..
(
94f9e..
(
23e07..
x0
)
(
λ x3 .
x2
x3
x1
)
)
(
94f9e..
(
23e07..
x1
)
(
λ x3 .
x2
x0
x3
)
)
)
(
0ac37..
(
94f9e..
(
5246e..
x0
)
(
λ x3 .
x2
x3
x1
)
)
(
94f9e..
(
5246e..
x1
)
(
λ x3 .
x2
x0
x3
)
)
)
.
The subproof is completed by applying L0.
■