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Proofgold Proof
pf
Apply unknownprop_f04ee3f2db7dade92ee7be101bff40fb89f733bb0c0340f09b616cd72154ad93 with
λ x0 .
and
(
and
(
and
(
80242..
(
f4dc0..
x0
)
)
(
∀ x1 .
prim1
x1
(
23e07..
x0
)
⟶
099f3..
(
f4dc0..
x0
)
(
f4dc0..
x1
)
)
)
(
∀ x1 .
prim1
x1
(
5246e..
x0
)
⟶
099f3..
(
f4dc0..
x1
)
(
f4dc0..
x0
)
)
)
(
02b90..
(
94f9e..
(
5246e..
x0
)
(
λ x1 .
f4dc0..
x1
)
)
(
94f9e..
(
23e07..
x0
)
(
λ x1 .
f4dc0..
x1
)
)
)
.
Let x0 of type
ι
be given.
Assume H0:
80242..
x0
.
Assume H1:
∀ x1 .
prim1
x1
(
56ded..
(
e4431..
x0
)
)
⟶
and
(
and
(
and
(
80242..
(
f4dc0..
x1
)
)
(
∀ x2 .
prim1
x2
(
23e07..
x1
)
⟶
099f3..
(
f4dc0..
x1
)
(
f4dc0..
x2
)
)
)
(
∀ x2 .
prim1
x2
(
5246e..
x1
)
⟶
099f3..
(
f4dc0..
x2
)
(
f4dc0..
x1
)
)
)
(
02b90..
(
94f9e..
(
5246e..
x1
)
(
λ x2 .
f4dc0..
x2
)
)
(
94f9e..
(
23e07..
x1
)
(
λ x2 .
f4dc0..
x2
)
)
)
.
Claim L2:
...
...
Claim L3:
...
...
Claim L4:
...
...
Claim L5:
...
...
Apply andI with
and
(
and
(
80242..
(
f4dc0..
x0
)
)
(
∀ x1 .
prim1
x1
(
23e07..
x0
)
⟶
099f3..
(
f4dc0..
x0
)
(
f4dc0..
x1
)
)
)
(
∀ x1 .
prim1
x1
(
5246e..
x0
)
⟶
099f3..
(
f4dc0..
x1
)
(
f4dc0..
x0
)
)
,
02b90..
(
94f9e..
(
5246e..
x0
)
(
λ x1 .
f4dc0..
x1
)
)
(
94f9e..
(
23e07..
x0
)
(
λ x1 .
f4dc0..
x1
)
)
leaving 2 subgoals.
Apply unknownprop_cb0c4ac573f21d515ac1c4f8e66909b50826695f34b524950ade1bb9cca6aa7b with
x0
,
λ x1 x2 .
and
(
and
(
80242..
x2
)
(
∀ x3 .
prim1
x3
(
23e07..
x0
)
⟶
099f3..
x2
(
f4dc0..
x3
)
)
)
(
∀ x3 .
prim1
x3
(
5246e..
x0
)
⟶
099f3..
(
f4dc0..
x3
)
x2
)
leaving 2 subgoals.
The subproof is completed by applying H0.
Apply and3I with
80242..
(
02a50..
(
94f9e..
(
5246e..
x0
)
f4dc0..
)
(
94f9e..
(
23e07..
x0
)
f4dc0..
)
)
,
∀ x1 .
prim1
x1
(
23e07..
x0
)
⟶
099f3..
(
02a50..
(
94f9e..
(
5246e..
x0
)
f4dc0..
)
(
94f9e..
(
23e07..
x0
)
f4dc0..
)
)
(
f4dc0..
x1
)
,
∀ x1 .
prim1
x1
(
5246e..
x0
)
⟶
099f3..
(
f4dc0..
x1
)
(
02a50..
(
94f9e..
(
5246e..
x0
)
f4dc0..
)
(
94f9e..
(
23e07..
x0
)
f4dc0..
)
)
leaving 3 subgoals.
The subproof is completed by applying L5.
Let x1 of type
ι
be given.
Assume H6:
prim1
x1
(
23e07..
x0
)
.
Apply unknownprop_eac7a6683c5ba7aa6ad74545e4a4f065634321e5b116001d3b7a7c41d91b1bf6 with
x0
,
x1
,
099f3..
(
02a50..
(
94f9e..
(
5246e..
x0
)
(
λ x2 .
f4dc0..
x2
)
)
(
94f9e..
(
23e07..
x0
)
(
λ x2 .
f4dc0..
x2
)
)
)
(
f4dc0..
x1
)
leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H6.
Assume H7:
80242..
x1
.
Assume H8:
prim1
(
e4431..
x1
)
(
e4431..
x0
)
.
Assume H9:
099f3..
x1
x0
.
Claim L10:
prim1
(
f4dc0..
x1
)
(
94f9e..
(
23e07..
x0
)
(
λ x2 .
f4dc0..
...
)
)
...
Apply unknownprop_aac490d730560b209b0a25a516afc93b5f7d88e788264e1e144da0f803647c0b with
94f9e..
(
5246e..
x0
)
(
λ x2 .
f4dc0..
x2
)
,
94f9e..
(
23e07..
x0
)
(
λ x2 .
f4dc0..
x2
)
,
f4dc0..
x1
leaving 2 subgoals.
The subproof is completed by applying L4.
The subproof is completed by applying L10.
...
...
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