Let x0 of type ι be given.
Let x1 of type ι be given.
Apply SNoLeE with
0,
x0,
SNoLe 0 (mul_SNo x0 x1) leaving 5 subgoals.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply SNoLeE with
0,
x1,
SNoLe 0 (mul_SNo x0 x1) leaving 5 subgoals.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Apply SNoLtLe with
0,
mul_SNo x0 x1.
Apply mul_SNo_pos_pos with
x0,
x1 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Assume H5: 0 = x1.
Apply H5 with
λ x2 x3 . SNoLe 0 (mul_SNo x0 x2).
Apply mul_SNo_zeroR with
x0,
λ x2 x3 . SNoLe 0 x3 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNoLe_ref with 0.
Assume H4: 0 = x0.
Apply H4 with
λ x2 x3 . SNoLe 0 (mul_SNo x2 x1).
Apply mul_SNo_zeroL with
x1,
λ x2 x3 . SNoLe 0 x3 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying SNoLe_ref with 0.