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.
Let x6 of type ι be given.
Let x7 of type ι be given.
Let x8 of type ι be given.
Let x9 of type ι be given.
Apply idl_negcycle_4 with
x0,
x1,
x2,
add_SNo x3 x4,
x5,
x6,
add_SNo x7 x4,
add_SNo x9 (minus_SNo x3) leaving 13 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply SNo_add_SNo with
x3,
x4 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
Apply SNo_add_SNo with
x7,
x4 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H4.
Apply SNo_add_SNo with
x9,
minus_SNo x3 leaving 2 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying L13.
Apply add_SNo_com with
x9,
minus_SNo x3,
λ x10 x11 . SNoLt (add_SNo x5 (add_SNo x6 (add_SNo (add_SNo x7 x4) x11))) 0 leaving 3 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying L13.
Apply add_SNo_assoc with
x7,
x4,
add_SNo (minus_SNo x3) x9,
λ x10 x11 . SNoLt (add_SNo x5 (add_SNo x6 x10)) 0 leaving 4 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H4.
Apply SNo_add_SNo with
minus_SNo x3,
x9 leaving 2 subgoals.
The subproof is completed by applying L13.
The subproof is completed by applying H9.
Apply add_SNo_assoc with
x4,
minus_SNo x3,
x9,
λ x10 x11 . SNoLt (add_SNo x5 (add_SNo x6 (add_SNo x7 x11))) 0 leaving 4 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying L13.
The subproof is completed by applying H9.
Apply SNoLeLt_tra with
add_SNo x5 (add_SNo x6 (add_SNo x7 (add_SNo (add_SNo x4 (minus_SNo x3)) x9))),
add_SNo x5 (add_SNo x6 (add_SNo x7 (add_SNo x8 x9))),
0 leaving 5 subgoals.
Apply L22 with
add_SNo x4 (minus_SNo x3).
The subproof is completed by applying L15.
The subproof is completed by applying L25.
The subproof is completed by applying SNo_0.
Apply add_SNo_Le2 with
x5,
add_SNo x6 (add_SNo x7 (add_SNo (add_SNo x4 (minus_SNo x3)) x9)),
add_SNo x6 (add_SNo x7 (add_SNo x8 x9)) leaving 4 subgoals.
The subproof is completed by applying H5.
Apply L21 with
add_SNo x4 (minus_SNo x3).
The subproof is completed by applying L15.
The subproof is completed by applying L24.
Apply add_SNo_Le2 with
x6,
add_SNo x7 (add_SNo (add_SNo x4 (minus_SNo x3)) x9),
add_SNo x7 (add_SNo x8 x9) leaving 4 subgoals.
The subproof is completed by applying H6.
Apply L20 with
add_SNo x4 (minus_SNo x3).
The subproof is completed by applying L15.
The subproof is completed by applying L23.
Apply add_SNo_Le2 with
x7,
add_SNo (add_SNo x4 (minus_SNo x3)) x9,
add_SNo x8 x9 leaving 4 subgoals.
The subproof is completed by applying H7.
Apply SNo_add_SNo with
add_SNo x4 (minus_SNo x3),
x9 leaving 2 subgoals.
The subproof is completed by applying L15.
The subproof is completed by applying H9.
The subproof is completed by applying L16.
Apply add_SNo_Le1 with
add_SNo x4 (minus_SNo x3),
x9,
x8 leaving 4 subgoals.
The subproof is completed by applying L15.
The subproof is completed by applying H9.
The subproof is completed by applying H8.
The subproof is completed by applying H30.
The subproof is completed by applying H26.
The subproof is completed by applying H27.
The subproof is completed by applying H28.
Apply add_SNo_com_3b_1_2 with
x3,
x4,
minus_SNo x2,
λ x10 x11 . SNoLe x11 (add_SNo x7 x4) leaving 4 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying L12.
Apply add_SNo_Le1 with
add_SNo x3 (minus_SNo x2),
x4,
x7 leaving 4 subgoals.
Apply SNo_add_SNo with
x3,
... leaving 2 subgoals.