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Proofgold Proof

pf
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.
Assume H0: SNo x0.
Assume H1: SNo x1.
Assume H2: SNo x2.
Assume H3: SNo x3.
Assume H4: SNo x4.
Assume H5: SNo x5.
Assume H6: SNoLt (add_SNo x0 x5) (add_SNo x2 x4).
Assume H7: SNoLt x1 (add_SNo x5 x3).
Apply add_SNo_com with x0, x1, λ x6 x7 . SNoLt x7 (add_SNo x2 (add_SNo x3 x4)) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply add_SNo_rotate_3_1 with x3, x4, x2, λ x6 x7 . SNoLt (add_SNo x1 x0) x6 leaving 4 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H2.
Apply add_SNo_0R with x0, λ x6 x7 . SNoLt (add_SNo x1 x6) (add_SNo x3 (add_SNo x4 x2)) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply add_SNo_Lt_subprop3a with x1, x0, 0, x3, add_SNo x4 x2, x5 leaving 8 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H3.
Apply SNo_add_SNo with x4, x2 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H2.
The subproof is completed by applying H5.
Apply add_SNo_0R with x1, λ x6 x7 . SNoLt x7 (add_SNo x3 x5) leaving 2 subgoals.
The subproof is completed by applying H1.
Apply add_SNo_com with x3, x5, λ x6 x7 . SNoLt x1 x7 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
The subproof is completed by applying H7.
Apply add_SNo_com with x4, x2, λ x6 x7 . SNoLt (add_SNo x0 x5) x7 leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H2.
The subproof is completed by applying H6.