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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: SNo x0.
Assume H1: SNo x1.
Claim L2: ...
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Claim L3: ...
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Claim L4: ...
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Apply SNoLtLe_or with x0, 0, SNoLe (abs_SNo (add_SNo x0 x1)) (add_SNo (abs_SNo x0) (abs_SNo x1)) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Assume H5: SNoLt x0 0.
Apply neg_abs_SNo with x0, λ x2 x3 . SNoLe (abs_SNo (add_SNo x0 x1)) (add_SNo x3 (abs_SNo x1)) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H5.
Apply SNoLtLe_or with x1, 0, SNoLe (abs_SNo (add_SNo x0 x1)) (add_SNo (minus_SNo x0) (abs_SNo x1)) leaving 4 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying SNo_0.
Assume H6: SNoLt x1 0.
Apply neg_abs_SNo with x1, λ x2 x3 . SNoLe (abs_SNo (add_SNo x0 x1)) (add_SNo (minus_SNo x0) x3) leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H6.
Claim L7: SNoLt (add_SNo x0 x1) 0
Apply add_SNo_0L with 0, λ x2 x3 . SNoLt (add_SNo x0 x1) x2 leaving 2 subgoals.
The subproof is completed by applying SNo_0.
Apply add_SNo_Lt3 with x0, x1, 0, 0 leaving 6 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying SNo_0.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
Apply neg_abs_SNo with add_SNo x0 x1, λ x2 x3 . SNoLe x3 (add_SNo (minus_SNo x0) (minus_SNo x1)) leaving 3 subgoals.
The subproof is completed by applying L2.
The subproof is completed by applying L7.
Apply minus_add_SNo_distr with x0, x1, λ x2 x3 . SNoLe x3 (add_SNo (minus_SNo x0) (minus_SNo x1)) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying SNoLe_ref with add_SNo (minus_SNo x0) (minus_SNo x1).
Assume H6: SNoLe 0 x1.
Apply nonneg_abs_SNo with x1, λ x2 x3 . SNoLe (abs_SNo (add_SNo x0 x1)) (add_SNo (minus_SNo x0) x3) leaving 2 subgoals.
The subproof is completed by applying H6.
Apply xm with SNoLe 0 (add_SNo x0 x1), SNoLe (abs_SNo (add_SNo x0 x1)) (add_SNo (minus_SNo x0) x1) leaving 2 subgoals.
Assume H7: SNoLe 0 (add_SNo x0 x1).
Apply nonneg_abs_SNo with add_SNo x0 x1, λ x2 x3 . SNoLe x3 (add_SNo (minus_SNo x0) x1) leaving 2 subgoals.
The subproof is completed by applying H7.
Apply add_SNo_Le1 with x0, x1, minus_SNo x0 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying L3.
Apply SNoLtLe with x0, minus_SNo x0.
Apply SNoLt_tra with x0, 0, minus_SNo x0 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
The subproof is completed by applying L3.
The subproof is completed by applying H5.
Apply minus_SNo_Lt_contra2 with x0, 0 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Apply minus_SNo_0 with λ x2 x3 . SNoLt x0 x3.
The subproof is completed by applying H5.
Assume H7: not (SNoLe 0 (add_SNo x0 x1)).
Apply not_nonneg_abs_SNo with add_SNo x0 x1, λ x2 x3 . SNoLe x3 (add_SNo (minus_SNo x0) x1) leaving 2 subgoals.
The subproof is completed by applying H7.
Apply minus_add_SNo_distr with x0, x1, λ x2 x3 . SNoLe x3 (add_SNo (minus_SNo x0) x1) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply add_SNo_Le2 with minus_SNo x0, minus_SNo x1, x1 leaving 4 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying L4.
The subproof is completed by applying H1.
Claim L8: SNoLe (minus_SNo x1) 0
Apply minus_SNo_0 with λ x2 x3 . SNoLe (minus_SNo x1) x2.
Apply minus_SNo_Le_contra with 0, x1 leaving 3 subgoals.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H1.
The subproof is completed by applying H6.
Apply SNoLe_tra with minus_SNo x1, 0, x1 leaving 5 subgoals.
The subproof is completed by applying L4.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H1.
The subproof is completed by applying L8.
The subproof is completed by applying H6.
Assume H5: SNoLe 0 x0.
Apply nonneg_abs_SNo with x0, λ x2 x3 . SNoLe (abs_SNo (add_SNo x0 x1)) (add_SNo x3 (abs_SNo x1)) leaving 2 subgoals.
The subproof is completed by applying H5.
Apply SNoLtLe_or with x1, 0, SNoLe (abs_SNo (add_SNo x0 x1)) (add_SNo x0 (abs_SNo x1)) leaving 4 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying SNo_0.
Assume H6: SNoLt x1 0.
Apply neg_abs_SNo with x1, λ x2 x3 . ... leaving 3 subgoals.
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