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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: SNo x0.
Assume H1: SNoLe 1 x0.
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.
Assume H2: nat_p x1.
Assume H3: SNoLe 1 (exp_SNo_nat x0 x1).
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.