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Proofgold Proof

pf
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.
Apply beta with 7, λ x7 . If_i (x7 = 0) x0 (If_i (x7 = 1) x1 (If_i (x7 = 2) x2 (If_i (x7 = 3) x3 (If_i (x7 = 4) x4 (If_i (x7 = 5) x5 x6))))), 5, λ x7 x8 . x8 = x5 leaving 2 subgoals.
The subproof is completed by applying In_5_7.
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 x6)))), λ x7 x8 . x8 = 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 x6))), λ x7 x8 . x8 = 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 x6)), λ x7 x8 . x8 = 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 x6), λ x7 x8 . x8 = 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 x6, λ x7 x8 . x8 = x5 leaving 2 subgoals.
The subproof is completed by applying neq_5_4.
Apply If_i_1 with 5 = 5, x5, x6.
Let x7 of type ιιο be given.
Assume H0: x7 5 5.
The subproof is completed by applying H0.