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Proofgold Proof
pf
Let x0 of type
ι
be given.
Let x1 of type
ι
be given.
Assume H0:
02b90..
x0
x1
.
Apply H0 with
and
(
and
(
and
(
and
(
80242..
(
02a50..
x0
x1
)
)
(
prim1
(
e4431..
(
02a50..
x0
x1
)
)
(
4ae4a..
(
0ac37..
(
a842e..
x0
(
λ x2 .
4ae4a..
(
e4431..
x2
)
)
)
(
a842e..
x1
(
λ x2 .
4ae4a..
(
e4431..
x2
)
)
)
)
)
)
)
(
∀ x2 .
prim1
x2
x0
⟶
099f3..
x2
(
02a50..
x0
x1
)
)
)
(
∀ x2 .
prim1
x2
x1
⟶
099f3..
(
02a50..
x0
x1
)
x2
)
)
(
∀ x2 .
80242..
x2
⟶
(
∀ x3 .
prim1
x3
x0
⟶
099f3..
x3
x2
)
⟶
(
∀ x3 .
prim1
x3
x1
⟶
099f3..
x2
x3
)
⟶
and
(
Subq
(
e4431..
(
02a50..
x0
x1
)
)
(
e4431..
x2
)
)
(
SNoEq_
(
e4431..
(
02a50..
x0
x1
)
)
(
02a50..
x0
x1
)
x2
)
)
.
Assume H1:
and
(
∀ x2 .
prim1
x2
x0
⟶
80242..
x2
)
(
∀ x2 .
prim1
x2
x1
⟶
80242..
x2
)
.
Apply H1 with
(
∀ x2 .
prim1
x2
x0
⟶
∀ x3 .
prim1
x3
x1
⟶
099f3..
x2
x3
)
⟶
and
(
and
(
and
(
and
(
80242..
(
02a50..
x0
x1
)
)
(
prim1
(
e4431..
(
02a50..
x0
x1
)
)
(
4ae4a..
(
0ac37..
(
a842e..
x0
(
λ x2 .
4ae4a..
(
e4431..
x2
)
)
)
(
a842e..
x1
(
λ x2 .
4ae4a..
(
e4431..
x2
)
)
)
)
)
)
)
(
∀ x2 .
prim1
x2
x0
⟶
099f3..
x2
(
02a50..
x0
x1
)
)
)
(
∀ x2 .
prim1
x2
x1
⟶
099f3..
(
02a50..
x0
x1
)
x2
)
)
(
∀ x2 .
80242..
x2
⟶
(
∀ x3 .
prim1
x3
x0
⟶
099f3..
x3
x2
)
⟶
(
∀ x3 .
prim1
x3
x1
⟶
099f3..
x2
x3
)
⟶
and
(
Subq
(
e4431..
(
02a50..
x0
x1
)
)
(
e4431..
x2
)
)
(
SNoEq_
(
e4431..
(
02a50..
x0
x1
)
)
(
02a50..
x0
x1
)
x2
)
)
.
Assume H2:
∀ x2 .
prim1
x2
x0
⟶
80242..
x2
.
Assume H3:
∀ x2 .
prim1
x2
x1
⟶
80242..
x2
.
Assume H4:
∀ x2 .
prim1
x2
x0
⟶
∀ x3 .
prim1
x3
x1
⟶
099f3..
x2
x3
.
Claim L5:
...
...
Claim L6:
...
...
Claim L7:
...
...
Claim L8:
...
...
Claim L9:
...
...
Claim L10:
...
...
Apply unknownprop_6419f8fc1ff70ca950cf4e79a0f0cff9b9f9b876ea55c76aec20f2ed9adc5971 with
λ x2 .
λ x3 :
ι → ο
.
and
(
ordinal
x2
)
(
prim1
(
09072..
x2
x3
)
x0
)
,
λ x2 .
λ x3 :
ι → ο
.
and
(
ordinal
x2
)
(
prim1
(
09072..
x2
x3
)
x1
)
,
0ac37..
(
a842e..
x0
(
λ x2 .
4ae4a..
(
e4431..
x2
)
)
)
(
a842e..
x1
(
λ x2 .
4ae4a..
(
e4431..
x2
)
)
)
,
and
(
and
(
and
(
and
(
80242..
(
02a50..
x0
...
)
)
...
)
...
)
...
)
...
leaving 5 subgoals.
...
...
...
...
...
■