Let x0 of type ι be given.
Apply unknownprop_a0e1670aba181da94860f563c67514747cbc863abd8e41c828885e11fbde7e1e with
λ x1 x2 . mul_SNo x1 x0 = add_SNo x0 (add_SNo x0 (add_SNo x0 x0)).
Apply mul_SNo_distrR with
2,
2,
x0,
λ x1 x2 . x2 = add_SNo x0 (add_SNo x0 (add_SNo x0 x0)) leaving 4 subgoals.
The subproof is completed by applying SNo_2.
The subproof is completed by applying SNo_2.
The subproof is completed by applying H0.
Apply add_SNo_1_1_2 with
λ x1 x2 . add_SNo (mul_SNo x1 x0) (mul_SNo x1 x0) = add_SNo x0 (add_SNo x0 (add_SNo x0 x0)).
Apply mul_SNo_distrR with
1,
1,
x0,
λ x1 x2 . add_SNo x2 x2 = add_SNo x0 (add_SNo x0 (add_SNo x0 x0)) leaving 4 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 mul_SNo_oneL with
x0,
λ x1 x2 . add_SNo (add_SNo x2 x2) (add_SNo x2 x2) = add_SNo x0 (add_SNo x0 (add_SNo x0 x0)) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x1 of type ι → ι → ο be given.
Apply add_SNo_assoc with
x0,
x0,
add_SNo x0 x0,
λ x2 x3 . x1 x3 x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H0.
Apply SNo_add_SNo with
x0,
x0 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H0.