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.
Apply add_SNo_com with
x0,
x1,
λ x6 x7 . SNoLt x7 (add_SNo x2 (add_SNo x3 x4)) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply add_SNo_rotate_3_1 with
x3,
x4,
x2,
λ x6 x7 . SNoLt (add_SNo x1 x0) x6 leaving 4 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H2.
Apply add_SNo_0R with
x0,
λ x6 x7 . SNoLt (add_SNo x1 x6) (add_SNo x3 (add_SNo x4 x2)) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply add_SNo_Lt_subprop3a with
x1,
x0,
0,
x3,
add_SNo x4 x2,
x5 leaving 8 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H3.
Apply SNo_add_SNo with
x4,
x2 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H2.
The subproof is completed by applying H5.
Apply add_SNo_0R with
x1,
λ x6 x7 . SNoLt x7 (add_SNo x3 x5) leaving 2 subgoals.
The subproof is completed by applying H1.
Apply add_SNo_com with
x3,
x5,
λ x6 x7 . SNoLt x1 x7 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
The subproof is completed by applying H7.
Apply add_SNo_com with
x4,
x2,
λ x6 x7 . SNoLt (add_SNo x0 x5) x7 leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H2.
The subproof is completed by applying H6.