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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 x2) (add_SNo x3 x5).
Assume H7: SNoLt (add_SNo x1 x5) x4.
Apply SNoLt_tra with add_SNo x0 (add_SNo x1 x2), add_SNo x1 (add_SNo x3 x5), add_SNo x3 x4 leaving 5 subgoals.
Apply SNo_add_SNo with x0, add_SNo x1 x2 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply SNo_add_SNo with x1, x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply SNo_add_SNo with x1, add_SNo x3 x5 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply SNo_add_SNo with x3, x5 leaving 2 subgoals.
The subproof is completed by applying H3.
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.
Apply add_SNo_com_3_0_1 with x0, x1, x2, λ x6 x7 . SNoLt x7 (add_SNo x1 (add_SNo x3 x5)) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply add_SNo_Lt2 with x1, add_SNo x0 x2, add_SNo x3 x5 leaving 4 subgoals.
The subproof is completed by applying H1.
Apply SNo_add_SNo with x0, x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply SNo_add_SNo with x3, x5 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
Apply add_SNo_rotate_3_1 with x3, x5, x1, λ x6 x7 . SNoLt x6 (add_SNo x3 x4) leaving 4 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
The subproof is completed by applying H1.
Apply add_SNo_Lt2 with x3, add_SNo x5 x1, x4 leaving 4 subgoals.
The subproof is completed by applying H3.
Apply SNo_add_SNo with x5, x1 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
Apply add_SNo_com with x5, x1, λ x6 x7 . SNoLt x7 x4 leaving 3 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H1.
The subproof is completed by applying H7.