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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
ordinal
x0
.
Apply ordinal_ind with
λ x1 .
bc82c..
(
4ae4a..
x0
)
x1
=
4ae4a..
(
bc82c..
x0
x1
)
.
Let x1 of type
ι
be given.
Assume H1:
ordinal
x1
.
Assume H2:
∀ x2 .
prim1
x2
x1
⟶
bc82c..
(
4ae4a..
x0
)
x2
=
4ae4a..
(
bc82c..
x0
x2
)
.
Claim L3:
...
...
Claim L4:
...
...
Claim L5:
...
...
Claim L6:
...
...
Claim L7:
...
...
Claim L8:
...
...
Apply unknownprop_e277188ae242e07bd6727f267e38747aecd739d129890076e65b92339f7beb98 with
0ac37..
(
94f9e..
(
56ded..
(
4ae4a..
x0
)
)
(
λ x2 .
bc82c..
x2
x1
)
)
(
94f9e..
(
56ded..
x1
)
(
λ x2 .
bc82c..
(
4ae4a..
x0
)
x2
)
)
,
4a7ef..
,
bc82c..
(
4ae4a..
x0
)
x1
=
4ae4a..
(
bc82c..
x0
x1
)
leaving 2 subgoals.
Apply unknownprop_360dad628eb310c4b99b99306a537b749071911afda713bd180f99c61063736f with
4ae4a..
x0
,
x1
leaving 2 subgoals.
The subproof is completed by applying L6.
The subproof is completed by applying H1.
Assume H9:
and
(
and
(
and
(
80242..
(
02a50..
(
0ac37..
(
94f9e..
(
56ded..
(
4ae4a..
x0
)
)
(
λ x2 .
bc82c..
x2
x1
)
)
(
94f9e..
(
56ded..
x1
)
(
bc82c..
(
4ae4a..
x0
)
)
)
)
4a7ef..
)
)
(
prim1
(
e4431..
(
02a50..
(
0ac37..
(
94f9e..
(
56ded..
(
4ae4a..
x0
)
)
(
λ x2 .
bc82c..
x2
x1
)
)
(
94f9e..
(
56ded..
x1
)
(
bc82c..
(
4ae4a..
x0
)
)
)
)
4a7ef..
)
)
(
4ae4a..
(
0ac37..
(
a842e..
(
0ac37..
(
94f9e..
(
56ded..
(
4ae4a..
x0
)
)
(
λ x2 .
bc82c..
x2
x1
)
)
(
94f9e..
(
56ded..
x1
)
(
bc82c..
(
4ae4a..
x0
)
)
)
)
(
λ x2 .
4ae4a..
(
e4431..
x2
)
)
)
(
a842e..
4a7ef..
(
λ x2 .
4ae4a..
(
e4431..
x2
)
)
)
)
)
)
)
(
∀ x2 .
prim1
x2
(
0ac37..
(
94f9e..
(
56ded..
(
4ae4a..
x0
)
)
(
λ x3 .
bc82c..
x3
x1
)
)
(
94f9e..
(
56ded..
x1
)
(
bc82c..
(
4ae4a..
x0
)
)
)
)
⟶
099f3..
x2
(
02a50..
(
0ac37..
(
94f9e..
(
56ded..
(
4ae4a..
x0
)
)
(
λ x3 .
bc82c..
x3
x1
)
)
(
94f9e..
(
56ded..
x1
)
(
bc82c..
(
4ae4a..
x0
)
)
)
)
4a7ef..
)
)
)
(
∀ x2 .
prim1
x2
4a7ef..
⟶
099f3..
(
02a50..
(
0ac37..
(
94f9e..
(
56ded..
(
4ae4a..
x0
)
)
(
λ x3 .
bc82c..
x3
x1
)
)
(
94f9e..
(
56ded..
x1
)
(
bc82c..
(
4ae4a..
x0
)
)
)
)
4a7ef..
)
x2
)
.
Apply H9 with
(
∀ x2 .
...
⟶
...
⟶
...
⟶
and
(
Subq
(
e4431..
(
02a50..
(
0ac37..
(
94f9e..
(
56ded..
(
4ae4a..
x0
)
)
(
λ x3 .
bc82c..
x3
x1
)
)
(
94f9e..
(
56ded..
x1
)
(
bc82c..
(
4ae4a..
x0
)
)
)
)
4a7ef..
)
)
(
e4431..
x2
)
)
(
SNoEq_
(
e4431..
(
02a50..
(
0ac37..
(
94f9e..
(
56ded..
(
4ae4a..
x0
)
)
(
λ x3 .
bc82c..
x3
x1
)
)
(
94f9e..
(
56ded..
x1
)
(
bc82c..
(
4ae4a..
x0
)
)
)
)
4a7ef..
)
)
(
02a50..
(
0ac37..
(
94f9e..
(
56ded..
(
4ae4a..
x0
)
)
(
λ x3 .
bc82c..
x3
x1
)
)
(
94f9e..
(
56ded..
x1
)
(
bc82c..
...
)
)
)
...
)
...
)
)
⟶
bc82c..
(
4ae4a..
x0
)
x1
=
4ae4a..
(
bc82c..
x0
x1
)
.
...
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