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Proofgold Proof
pf
Let x0 of type
ι
be given.
Let x1 of type
ι
be given.
Assume H0:
prim1
x1
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x2 .
bij
x0
x0
(
λ x3 .
f482f..
x2
x3
)
)
)
.
Apply unknownprop_e4362c04e65a765de9cf61045b78be0adc0f9e51a17754420e1088df0891ff67 with
b5c9f..
x0
x0
,
λ x2 .
bij
x0
x0
(
λ x3 .
f482f..
x2
x3
)
,
x1
,
and
(
and
(
prim1
(
0fc90..
x0
(
λ x2 .
inv
x0
(
λ x3 .
f482f..
x1
x3
)
x2
)
)
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x2 .
bij
x0
x0
(
λ x3 .
f482f..
x2
x3
)
)
)
)
(
0fc90..
x0
(
λ x2 .
f482f..
(
0fc90..
x0
(
λ x3 .
inv
x0
(
λ x4 .
f482f..
x1
x4
)
x3
)
)
(
f482f..
x1
x2
)
)
=
0fc90..
x0
(
λ x2 .
x2
)
)
)
(
0fc90..
x0
(
λ x2 .
f482f..
x1
(
f482f..
(
0fc90..
x0
(
λ x3 .
inv
x0
(
λ x4 .
f482f..
x1
x4
)
x3
)
)
x2
)
)
=
0fc90..
x0
(
λ x2 .
x2
)
)
leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H1:
prim1
x1
(
b5c9f..
x0
x0
)
.
Assume H2:
bij
x0
x0
(
λ x2 .
f482f..
x1
x2
)
.
Claim L3:
...
...
Claim L4:
...
...
Claim L5:
...
...
Claim L6:
...
...
Claim L7:
...
...
Claim L8:
...
...
Claim L9:
...
...
Apply and3I with
prim1
(
0fc90..
x0
(
λ x2 .
inv
x0
(
λ x3 .
f482f..
x1
x3
)
x2
)
)
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x2 .
bij
x0
x0
(
λ x3 .
f482f..
x2
x3
)
)
)
,
0fc90..
x0
(
λ x2 .
f482f..
(
0fc90..
x0
(
λ x3 .
inv
x0
(
λ x4 .
f482f..
x1
x4
)
x3
)
)
(
f482f..
x1
x2
)
)
=
0fc90..
x0
(
λ x2 .
x2
)
,
0fc90..
x0
(
λ x2 .
f482f..
x1
(
f482f..
(
0fc90..
x0
(
λ x3 .
inv
x0
(
λ x4 .
f482f..
x1
x4
)
x3
)
)
x2
)
)
=
0fc90..
x0
(
λ x2 .
x2
)
leaving 3 subgoals.
Apply unknownprop_1dada0fb38ff7f9b45b564ad11d6295d93250427446875218f17ee62431454a6 with
b5c9f..
x0
x0
,
λ x2 .
bij
x0
x0
(
λ x3 .
f482f..
x2
x3
)
,
0fc90..
x0
(
λ x2 .
inv
x0
(
λ x3 .
f482f..
x1
x3
)
x2
)
leaving 2 subgoals.
Apply unknownprop_78f4273a7b4d13f02d77d194b65be481121674fd021f4ffb88d69be4bcd0ab71 with
x0
,
λ x2 .
x0
,
λ x2 .
inv
x0
(
λ x3 .
f482f..
x1
x3
)
x2
.
The subproof is completed by applying L7.
Apply and3I with
∀ x2 .
prim1
x2
x0
⟶
prim1
(
f482f..
(
0fc90..
x0
(
λ x3 .
inv
x0
(
λ x4 .
f482f..
x1
x4
)
x3
)
)
x2
)
x0
,
∀ x2 .
...
⟶
∀ x3 .
prim1
...
...
⟶
f482f..
(
0fc90..
x0
(
λ x4 .
inv
x0
(
λ x5 .
f482f..
x1
x5
)
x4
)
)
x2
=
f482f..
(
0fc90..
x0
(
λ x4 .
inv
x0
(
λ x5 .
f482f..
x1
x5
)
x4
)
)
x3
⟶
x2
=
x3
,
...
leaving 3 subgoals.
...
...
...
...
...
■