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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.
Assume H0: SNo x0.
Assume H1: SNo x1.
Assume H2: SNo x2.
Assume H3: SNoLt x2 (add_SNo x0 (minus_SNo x1)).
Apply add_SNo_Lt1_cancel with add_SNo x2 x1, minus_SNo x1, x0 leaving 4 subgoals.
Apply SNo_add_SNo with x2, x1 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply SNo_minus_SNo with x1.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
Apply add_SNo_assoc with x2, x1, minus_SNo x1, λ x3 x4 . SNoLt x3 (add_SNo x0 (minus_SNo x1)) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply SNo_minus_SNo with x1.
The subproof is completed by applying H1.
Apply add_SNo_minus_SNo_rinv with x1, λ x3 x4 . SNoLt (add_SNo x2 x4) (add_SNo x0 (minus_SNo x1)) leaving 2 subgoals.
The subproof is completed by applying H1.
Apply add_SNo_0R with x2, λ x3 x4 . SNoLt x4 (add_SNo x0 (minus_SNo x1)) leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.