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.
Apply SNoLeE with
x2,
x0,
SNoLe (add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3)) (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3)) leaving 5 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H0.
The subproof is completed by applying H4.
Apply SNoLeE with
x3,
x1,
SNoLe (add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3)) (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3)) leaving 5 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H1.
The subproof is completed by applying H5.
Apply SNoLtLe with
add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3),
add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3).
Apply mul_SNo_Lt with
x0,
x1,
x2,
x3 leaving 6 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
Assume H7: x3 = x1.
Apply H7 with
λ x4 x5 . SNoLe (add_SNo (mul_SNo x2 x1) (mul_SNo x0 x5)) (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x5)).
Apply add_SNo_com with
mul_SNo x2 x1,
mul_SNo x0 x1,
λ x4 x5 . SNoLe x5 (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x1)) leaving 3 subgoals.
Apply SNo_mul_SNo with
x2,
x1 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply SNo_mul_SNo with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.