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.
Apply beta with
9,
λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) x1 (If_i (x9 = 2) x2 (If_i (x9 = 3) x3 (If_i (x9 = 4) x4 (If_i (x9 = 5) x5 (If_i (x9 = 6) x6 (If_i (x9 = 7) x7 x8))))))),
2,
λ x9 x10 . x10 = x2 leaving 2 subgoals.
The subproof is completed by applying In_2_9.
Apply If_i_0 with
2 = 0,
x0,
If_i (2 = 1) x1 (If_i (2 = 2) x2 (If_i (2 = 3) x3 (If_i (2 = 4) x4 (If_i (2 = 5) x5 (If_i (2 = 6) x6 (If_i (2 = 7) x7 x8)))))),
λ x9 x10 . x10 = x2 leaving 2 subgoals.
The subproof is completed by applying neq_2_0.
Apply If_i_0 with
2 = 1,
x1,
If_i (2 = 2) x2 (If_i (2 = 3) x3 (If_i (2 = 4) x4 (If_i (2 = 5) x5 (If_i (2 = 6) x6 (If_i (2 = 7) x7 x8))))),
λ x9 x10 . x10 = x2 leaving 2 subgoals.
The subproof is completed by applying neq_2_1.
Apply If_i_1 with
2 = 2,
x2,
If_i (2 = 3) x3 (If_i (2 = 4) x4 (If_i (2 = 5) x5 (If_i (2 = 6) x6 (If_i (2 = 7) x7 x8)))).
Let x9 of type ι → ι → ο be given.
Assume H0: x9 2 2.
The subproof is completed by applying H0.