Let x0 of type ι be given.
Let x1 of type ι be given.
Apply SNo_minus_SNo with
x0.
The subproof is completed by applying H0.
Apply SNo_minus_SNo with
x1.
The subproof is completed by applying H1.
Apply SNo_add_SNo with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply add_SNo_minus_SNo_rinv with
add_SNo x0 x1,
λ x2 x3 . x3 = add_SNo (add_SNo x0 x1) (add_SNo (minus_SNo x0) (minus_SNo x1)) leaving 2 subgoals.
The subproof is completed by applying L4.
Apply add_SNo_assoc with
add_SNo x0 x1,
minus_SNo x0,
minus_SNo x1,
λ x2 x3 . 0 = x3 leaving 4 subgoals.
The subproof is completed by applying L4.
The subproof is completed by applying L2.
The subproof is completed by applying L3.
Apply add_SNo_com with
x0,
x1,
λ x2 x3 . 0 = add_SNo (add_SNo x3 (minus_SNo x0)) (minus_SNo x1) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply add_SNo_assoc with
x1,
x0,
minus_SNo x0,
λ x2 x3 . 0 = add_SNo x2 (minus_SNo x1) leaving 4 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
The subproof is completed by applying L2.
Apply add_SNo_minus_SNo_rinv with
x0,
λ x2 x3 . 0 = add_SNo (add_SNo x1 x3) (minus_SNo x1) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply add_SNo_0R with
x1,
λ x2 x3 . 0 = add_SNo x3 (minus_SNo x1) leaving 2 subgoals.
The subproof is completed by applying H1.
Apply add_SNo_minus_SNo_rinv with
x1,
λ x2 x3 . 0 = x3 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x2 of type ι → ι → ο be given.
Assume H5: x2 0 0.
The subproof is completed by applying H5.
Apply add_SNo_cancel_L with
add_SNo x0 x1,
minus_SNo (add_SNo x0 x1),
add_SNo (minus_SNo x0) (minus_SNo x1) leaving 4 subgoals.
The subproof is completed by applying L4.
Apply SNo_minus_SNo with
add_SNo x0 x1.
The subproof is completed by applying L4.
Apply SNo_add_SNo with
minus_SNo x0,
minus_SNo x1 leaving 2 subgoals.
The subproof is completed by applying L2.
The subproof is completed by applying L3.
The subproof is completed by applying L5.