Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Let x8 of type ι be given.
Let x9 of type ι be given.
Apply beta with
10,
λ x10 . If_i (x10 = 0) x0 (If_i (x10 = 1) x1 (If_i (x10 = 2) x2 (If_i (x10 = 3) x3 (If_i (x10 = 4) x4 (If_i (x10 = 5) x5 (If_i (x10 = 6) x6 (If_i (x10 = 7) x7 (If_i (x10 = 8) x8 x9)))))))),
8,
λ x10 x11 . x11 = x8 leaving 2 subgoals.
The subproof is completed by applying unknownprop_afbf689bc11c7ff9c9e9875ca76148f88a08e7b5261e935945ff12d04ab37d90.
Apply If_i_0 with
8 = 0,
x0,
If_i (8 = 1) x1 (If_i (8 = 2) x2 (If_i (8 = 3) x3 (If_i (8 = 4) x4 (If_i (8 = 5) x5 (If_i (8 = 6) x6 (If_i (8 = 7) x7 (If_i (8 = 8) x8 x9))))))),
λ x10 x11 . x11 = x8 leaving 2 subgoals.
The subproof is completed by applying neq_8_0.
Apply If_i_0 with
8 = 1,
x1,
If_i (8 = 2) x2 (If_i (8 = 3) x3 (If_i (8 = 4) x4 (If_i (8 = 5) x5 (If_i (8 = 6) x6 (If_i (8 = 7) x7 (If_i (8 = 8) x8 x9)))))),
λ x10 x11 . x11 = x8 leaving 2 subgoals.
The subproof is completed by applying neq_8_1.
Apply If_i_0 with
8 = 2,
x2,
If_i (8 = 3) x3 (If_i (8 = 4) x4 (If_i (8 = 5) x5 (If_i (8 = 6) x6 (If_i (8 = 7) x7 (If_i (8 = 8) x8 x9))))),
λ x10 x11 . x11 = x8 leaving 2 subgoals.
The subproof is completed by applying neq_8_2.
Apply If_i_0 with
8 = 3,
x3,
If_i (8 = 4) x4 (If_i (8 = 5) x5 (If_i (8 = 6) x6 (If_i (8 = 7) x7 (If_i (8 = 8) x8 x9)))),
λ x10 x11 . x11 = x8 leaving 2 subgoals.
The subproof is completed by applying neq_8_3.
Apply If_i_0 with
8 = 4,
x4,
If_i (8 = 5) x5 (If_i (8 = 6) x6 (If_i (8 = 7) x7 (If_i (8 = 8) x8 x9))),
λ x10 x11 . x11 = x8 leaving 2 subgoals.
The subproof is completed by applying neq_8_4.
Apply If_i_0 with
8 = 5,
x5,
If_i (8 = 6) x6 (If_i (8 = 7) x7 (If_i (8 = 8) x8 x9)),
λ x10 x11 . x11 = x8 leaving 2 subgoals.
The subproof is completed by applying neq_8_5.
Apply If_i_0 with
8 = 6,
x6,
If_i (8 = 7) x7 ...,
... leaving 2 subgoals.