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Proofgold Proof

pf
Claim L0: ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ ∀ x2 x3 : ι → ι → ι . (∀ x4 . x4 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x5 . SNo x5 ⟶ x2 x4 x5 = x3 x4 x5) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ x2 x0 x4 = x3 x0 x4) ⟶ (λ x4 x5 . λ x6 : ι → ι → ι . SNoCut (binunion {x6 x7 x5|x7 ∈ SNoL x4} {x6 x4 x7|x7 ∈ SNoL x5}) (binunion {x6 x7 x5|x7 ∈ SNoR x4} {x6 x4 x7|x7 ∈ SNoR x5})) x0 x1 x2 = (λ x4 x5 . λ x6 : ι → ι → ι . SNoCut (binunion {x6 x7 x5|x7 ∈ SNoL x4} {x6 x4 x7|x7 ∈ SNoL x5}) (binunion {x6 x7 x5|x7 ∈ SNoR x4} {x6 x4 x7|x7 ∈ SNoR x5})) x0 x1 x3
Let x0 of type ι be given.
Assume H0: SNo x0.
Let x1 of type ι be given.
Assume H1: SNo x1.
Let x2 of type ι → ι → ι be given.
Let x3 of type ι → ι → ι be given.
Assume H2: ∀ x4 . x4 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x5 . SNo x5 ⟶ x2 x4 x5 = x3 x4 x5.
Assume H3: ∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ x2 x0 x4 = x3 x0 x4.
Claim L4: ...
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Claim L5: ...
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Claim L6: ...
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Claim L7: ...
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Apply L4 with λ x4 x5 . SNoCut (binunion x5 {x2 x0 x6|x6 ∈ SNoL x1}) (binunion {x2 x6 x1|x6 ∈ SNoR x0} {x2 x0 x6|x6 ∈ SNoR x1}) = SNoCut (binunion {x3 x6 x1|x6 ∈ SNoL x0} {x3 x0 x6|x6 ∈ SNoL x1}) (binunion {x3 x6 x1|x6 ∈ SNoR x0} {x3 x0 x6|x6 ∈ SNoR x1}).
Apply L5 with λ x4 x5 . SNoCut (binunion {x3 x6 x1|x6 ∈ SNoL x0} x5) (binunion {x2 x6 x1|x6 ∈ SNoR x0} {x2 x0 x6|x6 ∈ SNoR x1}) = SNoCut (binunion {x3 x6 x1|x6 ∈ SNoL x0} {x3 x0 x6|x6 ∈ SNoL x1}) (binunion {x3 x6 x1|x6 ∈ SNoR x0} {x3 x0 ...|x6 ∈ SNoR x1}).
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Apply SNo_rec2_eq with λ x0 x1 . λ x2 : ι → ι → ι . SNoCut (binunion {x2 x3 x1|x3 ∈ SNoL x0} {x2 x0 x3|x3 ∈ SNoL x1}) (binunion {x2 x3 x1|x3 ∈ SNoR x0} {x2 x0 x3|x3 ∈ SNoR x1}).
The subproof is completed by applying L0.
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