Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply add_SNo_Le1_cancel with
add_SNo x0 (minus_SNo x1),
x1,
x2 leaving 4 subgoals.
Apply SNo_add_SNo with
x0,
minus_SNo x1 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply SNo_minus_SNo with
x1.
The subproof is completed by applying H1.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply add_SNo_assoc with
x0,
minus_SNo x1,
x1,
λ x3 x4 . SNoLe x3 (add_SNo x2 x1) leaving 4 subgoals.
The subproof is completed by applying H0.
Apply SNo_minus_SNo with
x1.
The subproof is completed by applying H1.
The subproof is completed by applying H1.
Apply add_SNo_minus_SNo_linv with
x1,
λ x3 x4 . SNoLe (add_SNo x0 x4) (add_SNo x2 x1) leaving 2 subgoals.
The subproof is completed by applying H1.
Apply add_SNo_0R with
x0,
λ x3 x4 . SNoLe x4 (add_SNo x2 x1) leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H3.