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)))))))),
3,
λ x10 x11 . x11 = x3 leaving 2 subgoals.
The subproof is completed by applying unknownprop_676c401f8098489ea0734657c4430df5000c9b26ac4bcf84e71557d406b8053d.
Apply If_i_0 with
3 = 0,
x0,
If_i (3 = 1) x1 (If_i (3 = 2) x2 (If_i (3 = 3) x3 (If_i (3 = 4) x4 (If_i (3 = 5) x5 (If_i (3 = 6) x6 (If_i (3 = 7) x7 (If_i (3 = 8) x8 x9))))))),
λ x10 x11 . x11 = x3 leaving 2 subgoals.
The subproof is completed by applying neq_3_0.
Apply If_i_0 with
3 = 1,
x1,
If_i (3 = 2) x2 (If_i (3 = 3) x3 (If_i (3 = 4) x4 (If_i (3 = 5) x5 (If_i (3 = 6) x6 (If_i (3 = 7) x7 (If_i (3 = 8) x8 x9)))))),
λ x10 x11 . x11 = x3 leaving 2 subgoals.
The subproof is completed by applying neq_3_1.
Apply If_i_0 with
3 = 2,
x2,
If_i (3 = 3) x3 (If_i (3 = 4) x4 (If_i (3 = 5) x5 (If_i (3 = 6) x6 (If_i (3 = 7) x7 (If_i (3 = 8) x8 x9))))),
λ x10 x11 . x11 = x3 leaving 2 subgoals.
The subproof is completed by applying neq_3_2.
Apply If_i_1 with
3 = 3,
x3,
If_i (3 = 4) x4 (If_i (3 = 5) x5 (If_i (3 = 6) x6 (If_i (3 = 7) x7 (If_i (3 = 8) x8 x9)))).
Let x10 of type ι → ι → ο be given.
Assume H0: x10 3 3.
The subproof is completed by applying H0.