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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: ...
...
Claim L3: ∀ x2 . SNo x2 ⟶ SNoLev x2 ∈ SNoLev (add_SNo x0 x1) ⟶ SNoLt x2 (add_SNo x0 x1) ⟶ or (∃ x3 . and (x3 ∈ SNoL x0) (SNoLe x2 (add_SNo x3 x1))) (∃ x3 . and (x3 ∈ SNoL x1) (SNoLe x2 (add_SNo x0 x3)))
Apply SNoLev_ind with λ x2 . SNoLev x2 ∈ SNoLev (add_SNo x0 x1) ⟶ SNoLt x2 (add_SNo x0 x1) ⟶ or (∃ x3 . and (x3 ∈ SNoL x0) (SNoLe x2 (add_SNo x3 x1))) (∃ x3 . and (x3 ∈ SNoL x1) (SNoLe x2 (add_SNo x0 x3))).
Let x2 of type ι be given.
Assume H3: SNo x2.
Assume H4: ∀ x3 . x3 ∈ SNoS_ (SNoLev x2) ⟶ SNoLev x3 ∈ SNoLev (add_SNo x0 x1) ⟶ SNoLt x3 (add_SNo x0 x1) ⟶ or (∃ x4 . and (x4 ∈ SNoL x0) (SNoLe x3 (add_SNo x4 x1))) (∃ x4 . and (x4 ∈ SNoL x1) (SNoLe x3 (add_SNo x0 x4))).
Assume H5: SNoLev x2 ∈ SNoLev (add_SNo x0 x1).
Assume H6: SNoLt x2 (add_SNo x0 x1).
Apply dneg with or (∃ x3 . and (x3 ∈ SNoL x0) (SNoLe x2 (add_SNo x3 x1))) (∃ x3 . and (x3 ∈ SNoL x1) (SNoLe x2 (add_SNo x0 x3))).
Assume H7: not (or (∃ x3 . and (x3 ∈ SNoL x0) (SNoLe x2 (add_SNo x3 x1))) (∃ x3 . and (x3 ∈ SNoL x1) (SNoLe x2 (add_SNo x0 x3)))).
Apply SNoLt_irref with x2.
Apply SNoLtLe_tra with x2, add_SNo x0 x1, x2 leaving 5 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying L2.
The subproof is completed by applying H3.
The subproof is completed by applying H6.
Apply add_SNo_eq with x0, x1, λ x3 x4 . SNoLe x4 x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply SNo_eta with x2, λ x3 x4 . SNoLe (SNoCut (binunion {...|x5 ∈ SNoL x0} ...) ...) ... leaving 2 subgoals.
...
...
Let x2 of type ι be given.
Assume H4: x2 ∈ SNoL (add_SNo x0 x1).
Apply SNoL_E with add_SNo x0 x1, x2, or (∃ x3 . and (x3 ∈ SNoL x0) (SNoLe x2 (add_SNo x3 x1))) (∃ x3 . and (x3 ∈ SNoL x1) (SNoLe x2 (add_SNo x0 x3))) leaving 3 subgoals.
The subproof is completed by applying L2.
The subproof is completed by applying H4.
Assume H5: SNo x2.
Assume H6: SNoLev x2 ∈ SNoLev (add_SNo x0 x1).
Assume H7: SNoLt x2 (add_SNo x0 x1).
Apply L3 with x2 leaving 3 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
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