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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_38ee93c68aca60318bbeaa48caeaf22e4a66dbcceb33f1720f11297f26ed11bf with
x0
,
x1
,
explicit_Group_inverse
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x2 .
bij
x0
x0
(
λ x3 .
f482f..
x2
x3
)
)
)
(
λ x2 x3 .
0fc90..
x0
(
λ x4 .
f482f..
x3
(
f482f..
x2
x4
)
)
)
x1
=
0fc90..
x0
(
λ x2 .
inv
x0
(
λ x3 .
f482f..
x1
x3
)
x2
)
leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H1:
and
(
prim1
(
0fc90..
x0
(
inv
x0
(
f482f..
x1
)
)
)
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x2 .
bij
x0
x0
(
f482f..
x2
)
)
)
)
(
0fc90..
x0
(
λ x2 .
f482f..
(
0fc90..
x0
(
inv
x0
(
f482f..
x1
)
)
)
(
f482f..
x1
x2
)
)
=
0fc90..
x0
(
λ x2 .
x2
)
)
.
Assume H2:
0fc90..
x0
(
λ x2 .
f482f..
x1
(
f482f..
(
0fc90..
x0
(
inv
x0
(
f482f..
x1
)
)
)
x2
)
)
=
0fc90..
x0
(
λ x2 .
x2
)
.
Apply H1 with
explicit_Group_inverse
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x2 .
bij
x0
x0
(
λ x3 .
f482f..
x2
x3
)
)
)
(
λ x2 x3 .
0fc90..
x0
(
λ x4 .
f482f..
x3
(
f482f..
x2
x4
)
)
)
x1
=
0fc90..
x0
(
λ x2 .
inv
x0
(
λ x3 .
f482f..
x1
x3
)
x2
)
.
Assume H3:
prim1
(
0fc90..
x0
(
inv
x0
(
f482f..
x1
)
)
)
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x2 .
bij
x0
x0
(
f482f..
x2
)
)
)
.
Assume H4:
0fc90..
x0
(
λ x2 .
f482f..
(
0fc90..
x0
(
inv
x0
(
f482f..
x1
)
)
)
(
f482f..
x1
x2
)
)
=
0fc90..
x0
(
λ x2 .
x2
)
.
Apply explicit_Group_lcancel with
1216a..
(
b5c9f..
x0
x0
)
(
λ x2 .
bij
x0
x0
(
λ x3 .
f482f..
x2
x3
)
)
,
λ x2 x3 .
0fc90..
x0
(
λ x4 .
f482f..
x3
(
f482f..
x2
x4
)
)
,
x1
,
explicit_Group_inverse
(
1216a..
(
b5c9f..
x0
x0
)
(
λ x2 .
bij
x0
x0
(
λ x3 .
f482f..
x2
x3
)
)
)
(
λ x2 x3 .
0fc90..
x0
(
λ x4 .
f482f..
x3
(
f482f..
x2
x4
)
)
)
x1
,
0fc90..
x0
(
λ x2 .
inv
x0
(
λ x3 .
f482f..
x1
x3
)
x2
)
leaving 5 subgoals.
The subproof is completed by applying unknownprop_bc372826b7f0da52f3a8a3ea86862b41fc91267615627787d3fb4a389b8c72dc with
x0
.
The subproof is completed by applying H0.
Apply explicit_Group_inverse_in with
1216a..
(
b5c9f..
x0
x0
)
(
λ x2 .
bij
x0
x0
(
λ x3 .
f482f..
x2
x3
)
)
,
λ x2 x3 .
0fc90..
x0
(
λ x4 .
f482f..
x3
(
f482f..
x2
x4
)
)
,
x1
leaving 2 subgoals.
The subproof is completed by applying unknownprop_bc372826b7f0da52f3a8a3ea86862b41fc91267615627787d3fb4a389b8c72dc with
x0
.
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