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 add_SNo_Lt1_cancel with
add_SNo x0 x1,
add_SNo x4 x5,
add_SNo x2 x3 leaving 4 subgoals.
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 SNo_add_SNo with
x4,
x5 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Apply SNo_add_SNo with
x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply add_SNo_com_4_inner_mid with
x0,
x1,
x4,
x5,
λ x6 x7 . SNoLt x7 (add_SNo (add_SNo x2 x3) (add_SNo x4 x5)) leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Apply add_SNo_assoc with
x2,
x3,
add_SNo x4 x5,
λ x6 x7 . x6 = add_SNo (add_SNo x2 x5) (add_SNo x3 x4) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply SNo_add_SNo with
x4,
x5 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Apply add_SNo_assoc with
x2,
x5,
add_SNo x3 x4,
λ x6 x7 . add_SNo x2 (add_SNo x3 (add_SNo x4 x5)) = x6 leaving 4 subgoals.
The subproof is completed by applying H2.
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.
Claim L8: ∀ x8 : ι → ο . x8 y7 ⟶ x8 y6
Let x8 of type ι → ο be given.
set y9 to be λ x9 . x8
Apply add_SNo_rotate_3_1 with
x5,
y6,
y7,
λ x10 x11 . y9 (add_SNo x4 x10) (add_SNo x4 x11) leaving 4 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 H8.
Let x8 of type ι → ι → ο be given.
Apply L8 with
λ x9 . x8 x9 y7 ⟶ x8 y7 x9.
Assume H9: x8 y7 y7.
The subproof is completed by applying H9.
Apply L8 with
λ x6 x7 . SNoLt (add_SNo (add_SNo x0 x4) (add_SNo x1 x5)) x7.
Apply add_SNo_Lt3 with
add_SNo x0 x4,
add_SNo x1 x5,
add_SNo x2 x5,
add_SNo x3 x4 leaving 6 subgoals.
Apply SNo_add_SNo with
x0,
x4 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H4.
Apply SNo_add_SNo with
x1,
x5 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H5.
Apply SNo_add_SNo with
x2,
x5 leaving 2 subgoals.
The subproof is completed by applying H2.
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.
The subproof is completed by applying H6.
The subproof is completed by applying H7.