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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
.
Let x2 of type
ο
be given.
Assume H1:
80242..
(
02a50..
x0
x1
)
⟶
prim1
(
e4431..
(
02a50..
x0
x1
)
)
(
4ae4a..
(
0ac37..
(
a842e..
x0
(
λ x3 .
4ae4a..
(
e4431..
x3
)
)
)
(
a842e..
x1
(
λ x3 .
4ae4a..
(
e4431..
x3
)
)
)
)
)
⟶
(
∀ x3 .
prim1
x3
x0
⟶
099f3..
x3
(
02a50..
x0
x1
)
)
⟶
(
∀ x3 .
prim1
x3
x1
⟶
099f3..
(
02a50..
x0
x1
)
x3
)
⟶
(
∀ x3 .
80242..
x3
⟶
(
∀ x4 .
prim1
x4
x0
⟶
099f3..
x4
x3
)
⟶
(
∀ x4 .
prim1
x4
x1
⟶
099f3..
x3
x4
)
⟶
and
(
Subq
(
e4431..
(
02a50..
x0
x1
)
)
(
e4431..
x3
)
)
(
SNoEq_
(
e4431..
(
02a50..
x0
x1
)
)
(
02a50..
x0
x1
)
x3
)
)
⟶
x2
.
Apply unknownprop_e277188ae242e07bd6727f267e38747aecd739d129890076e65b92339f7beb98 with
x0
,
x1
,
x2
leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H2:
and
(
and
(
and
(
80242..
(
02a50..
x0
x1
)
)
(
prim1
(
e4431..
(
02a50..
x0
x1
)
)
(
4ae4a..
(
0ac37..
(
a842e..
x0
(
λ x3 .
4ae4a..
(
e4431..
x3
)
)
)
(
a842e..
x1
(
λ x3 .
4ae4a..
(
e4431..
x3
)
)
)
)
)
)
)
(
∀ x3 .
prim1
x3
x0
⟶
099f3..
x3
(
02a50..
x0
x1
)
)
)
(
∀ x3 .
prim1
x3
x1
⟶
099f3..
(
02a50..
x0
x1
)
x3
)
.
Apply H2 with
(
∀ x3 .
80242..
x3
⟶
(
∀ x4 .
prim1
x4
x0
⟶
099f3..
x4
x3
)
⟶
(
∀ x4 .
prim1
x4
x1
⟶
099f3..
x3
x4
)
⟶
and
(
Subq
(
e4431..
(
02a50..
x0
x1
)
)
(
e4431..
x3
)
)
(
SNoEq_
(
e4431..
(
02a50..
x0
x1
)
)
(
02a50..
x0
x1
)
x3
)
)
⟶
x2
.
Assume H3:
and
(
and
(
80242..
(
02a50..
x0
x1
)
)
(
prim1
(
e4431..
(
02a50..
x0
x1
)
)
(
4ae4a..
(
0ac37..
(
a842e..
x0
(
λ x3 .
4ae4a..
(
e4431..
x3
)
)
)
(
a842e..
x1
(
λ x3 .
4ae4a..
(
e4431..
x3
)
)
)
)
)
)
)
(
∀ x3 .
prim1
x3
x0
⟶
099f3..
x3
(
02a50..
x0
x1
)
)
.
Apply H3 with
(
∀ x3 .
prim1
x3
x1
⟶
099f3..
(
02a50..
x0
x1
)
x3
)
⟶
(
∀ x3 .
80242..
x3
⟶
(
∀ x4 .
prim1
x4
x0
⟶
099f3..
x4
x3
)
⟶
(
∀ x4 .
prim1
x4
x1
⟶
099f3..
x3
x4
)
⟶
and
(
Subq
(
e4431..
(
02a50..
x0
x1
)
)
(
e4431..
x3
)
)
(
SNoEq_
(
e4431..
(
02a50..
x0
x1
)
)
(
02a50..
x0
x1
)
x3
)
)
⟶
x2
.
Assume H4:
and
(
80242..
(
02a50..
x0
x1
)
)
(
prim1
(
e4431..
(
02a50..
...
...
)
)
...
)
.
...
■