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)))))))),
5,
λ x10 x11 . x11 = x5 leaving 2 subgoals.
The subproof is completed by applying unknownprop_f92773479be56466b39b85209c949bc89027283dc3c6bdb4a9c58e34c6e67297.
Apply If_i_0 with
5 = 0,
x0,
If_i (5 = 1) x1 (If_i (5 = 2) x2 (If_i (5 = 3) x3 (If_i (5 = 4) x4 (If_i (5 = 5) x5 (If_i (5 = 6) x6 (If_i (5 = 7) x7 (If_i (5 = 8) x8 x9))))))),
λ x10 x11 . x11 = x5 leaving 2 subgoals.
The subproof is completed by applying neq_5_0.
Apply If_i_0 with
5 = 1,
x1,
If_i (5 = 2) x2 (If_i (5 = 3) x3 (If_i (5 = 4) x4 (If_i (5 = 5) x5 (If_i (5 = 6) x6 (If_i (5 = 7) x7 (If_i (5 = 8) x8 x9)))))),
λ x10 x11 . x11 = x5 leaving 2 subgoals.
The subproof is completed by applying neq_5_1.
Apply If_i_0 with
5 = 2,
x2,
If_i (5 = 3) x3 (If_i (5 = 4) x4 (If_i (5 = 5) x5 (If_i (5 = 6) x6 (If_i (5 = 7) x7 (If_i (5 = 8) x8 x9))))),
λ x10 x11 . x11 = x5 leaving 2 subgoals.
The subproof is completed by applying neq_5_2.
Apply If_i_0 with
5 = 3,
x3,
If_i (5 = 4) x4 (If_i (5 = 5) x5 (If_i (5 = 6) x6 (If_i (5 = 7) x7 (If_i (5 = 8) x8 x9)))),
λ x10 x11 . x11 = x5 leaving 2 subgoals.
The subproof is completed by applying neq_5_3.
Apply If_i_0 with
5 = 4,
x4,
If_i (5 = 5) x5 (If_i (5 = 6) x6 (If_i (5 = 7) x7 (If_i (5 = 8) x8 x9))),
λ x10 x11 . x11 = x5 leaving 2 subgoals.
The subproof is completed by applying neq_5_4.
Apply If_i_1 with
5 = 5,
x5,
If_i (5 = 6) x6 (If_i (5 = 7) x7 (If_i (5 = 8) x8 x9)).
Let x10 of type ι → ι → ο be given.
Assume H0: x10 5 5.
The subproof is completed by applying H0.