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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.
Let x6 of type ι be given.
Let x7 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: SNo x6.
Assume H7: SNo x7.
Assume H8: SNoLt (add_SNo x0 x5) (add_SNo x6 x7).
Assume H9: SNoLt (add_SNo x1 x7) x4.
Assume H10: SNoLt (add_SNo x6 x2) (add_SNo x3 x5).
Claim L11: SNoLt (add_SNo x0 x2) (add_SNo x7 x3)
Apply add_SNo_Lt_subprop2 with x0, x2, x7, x3, x5, x6 leaving 8 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
The subproof is completed by applying H7.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
Apply add_SNo_com with x7, x6, λ x8 x9 . SNoLt (add_SNo x0 x5) x9 leaving 3 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H6.
The subproof is completed by applying H8.
Apply add_SNo_com with x2, x6, λ x8 x9 . SNoLt x9 (add_SNo x3 x5) leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H6.
The subproof is completed by applying H10.
Claim L12: SNo (add_SNo x0 x2)
Apply SNo_add_SNo with x0, x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Claim L13: SNo (add_SNo x7 x3)
Apply SNo_add_SNo with x7, x3 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H3.
Apply SNoLt_tra with add_SNo x0 (add_SNo x1 x2), add_SNo x7 (add_SNo x3 x1), 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 x7, add_SNo x3 x1 leaving 2 subgoals.
The subproof is completed by applying H7.
Apply SNo_add_SNo with x3, x1 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H1.
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 with x1, x2, λ x8 x9 . SNoLt (add_SNo x0 x9) (add_SNo x7 (add_SNo x3 x1)) leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply add_SNo_assoc with x0, x2, x1, λ x8 x9 . SNoLt x9 (add_SNo x7 (add_SNo x3 x1)) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply add_SNo_assoc with x7, x3, x1, λ x8 x9 . SNoLt (add_SNo (add_SNo x0 x2) x1) x9 leaving 4 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H3.
The subproof is completed by applying H1.
Apply add_SNo_Lt1 with add_SNo x0 x2, x1, add_SNo x7 x3 leaving 4 subgoals.
The subproof is completed by applying L12.
The subproof is completed by applying H1.
The subproof is completed by applying L13.
The subproof is completed by applying L11.
Apply add_SNo_rotate_3_1 with x3, x1, x7, λ x8 x9 . SNoLt x8 (add_SNo x3 x4) leaving 4 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H1.
The subproof is completed by applying H7.
Apply add_SNo_Lt2 with x3, add_SNo x1 x7, x4 leaving 4 subgoals.
The subproof is completed by applying H3.
Apply SNo_add_SNo with x1, x7 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H7.
The subproof is completed by applying H4.
The subproof is completed by applying H9.