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 SNoLt_tra with
add_SNo x0 (add_SNo x1 x2),
add_SNo x1 (add_SNo x3 x5),
add_SNo x3 x4 leaving 5 subgoals.
Apply SNo_add_SNo with
x0,
add_SNo x1 x2 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply SNo_add_SNo with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply SNo_add_SNo with
x1,
add_SNo x3 x5 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply SNo_add_SNo with
x3,
x5 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
Apply SNo_add_SNo with
x3,
x4 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Apply add_SNo_com_3_0_1 with
x0,
x1,
x2,
λ x6 x7 . SNoLt x7 (add_SNo x1 (add_SNo x3 x5)) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply add_SNo_Lt2 with
x1,
add_SNo x0 x2,
add_SNo x3 x5 leaving 4 subgoals.
The subproof is completed by applying H1.
Apply SNo_add_SNo with
x0,
x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply SNo_add_SNo with
x3,
x5 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
Apply add_SNo_rotate_3_1 with
x3,
x5,
x1,
λ x6 x7 . SNoLt x6 (add_SNo x3 x4) leaving 4 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
The subproof is completed by applying H1.
Apply add_SNo_Lt2 with
x3,
add_SNo x5 x1,
x4 leaving 4 subgoals.
The subproof is completed by applying H3.
Apply SNo_add_SNo with
x5,
x1 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
Apply add_SNo_com with
x5,
x1,
λ x6 x7 . SNoLt x7 x4 leaving 3 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H1.
The subproof is completed by applying H7.