Let x0 of type ι be given.
Apply nat_ind with
λ x1 . SNoLe 1 (exp_SNo_nat x0 x1) leaving 2 subgoals.
Apply exp_SNo_nat_0 with
x0,
λ x1 x2 . SNoLe 1 x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNoLe_ref with 1.
Let x1 of type ι be given.
Apply exp_SNo_nat_S with
x0,
x1,
λ x2 x3 . SNoLe 1 x3 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply mul_SNo_oneL with
1,
λ x2 x3 . SNoLe x2 (mul_SNo x0 (exp_SNo_nat x0 x1)) leaving 2 subgoals.
The subproof is completed by applying SNo_1.
Apply nonneg_mul_SNo_Le2 with
1,
1,
x0,
exp_SNo_nat x0 x1 leaving 8 subgoals.
The subproof is completed by applying SNo_1.
The subproof is completed by applying SNo_1.
The subproof is completed by applying H0.
Apply SNo_exp_SNo_nat with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply SNoLtLe with
0,
1.
The subproof is completed by applying SNoLt_0_1.
Apply SNoLtLe with
0,
1.
The subproof is completed by applying SNoLt_0_1.
The subproof is completed by applying H1.
The subproof is completed by applying H3.