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Proofgold Proof

pf
Let x0 of type (ι → (((ι → ι) → ι) → ι) → ι → ι → ι) → ι → ι → ι → ι → ι → ι be given.
Let x1 of type ((ι → ι) → ι) → ι → ι be given.
Let x2 of type (ι → ι → ι) → ι → ι be given.
Let x3 of type ((ι → ι) → ι → ((ι → ι) → ι → ι) → (ι → ι) → ι) → (ι → ((ι → ι) → ι) → ι) → (ι → ι) → ι be given.
Assume H0: ∀ x4 : ι → ι . ∀ x5 : (ι → ι) → ι . ∀ x6 . ∀ x7 : ((ι → ι) → (ι → ι) → ι) → (ι → ι → ι) → ι . x3 (λ x8 : ι → ι . λ x9 . λ x10 : (ι → ι) → ι → ι . λ x11 : ι → ι . 0) (λ x8 . λ x9 : (ι → ι) → ι . x0 (λ x10 . λ x11 : ((ι → ι) → ι) → ι . λ x12 x13 . x2 (λ x14 x15 . x14) 0) (setsum (x9 (λ x10 . 0)) (x2 (λ x10 x11 . 0) (setsum 0 0))) (setsum (Inj1 x6) x8) (setsum (setsum (x0 (λ x10 . λ x11 : ((ι → ι) → ι) → ι . λ x12 x13 . 0) 0 0 0 0 0) (Inj0 0)) 0) (x9 (λ x10 . x2 (λ x11 x12 . Inj0 0) 0)) 0) (λ x8 . x6) = x0 (λ x8 . λ x9 : ((ι → ι) → ι) → ι . λ x10 x11 . setsum (x2 (λ x12 x13 . setsum (setsum 0 0) (x0 (λ x14 . λ x15 : ((ι → ι) → ι) → ι . λ x16 x17 . 0) 0 0 0 0 0)) (Inj1 x10)) (Inj0 (Inj1 (x2 (λ x12 x13 . 0) 0)))) (setsum (Inj1 0) (x1 (λ x8 : ι → ι . setsum x6 (setsum 0 0)) 0)) (setsum (x0 (λ x8 . λ x9 : ((ι → ι) → ι) → ι . λ x10 x11 . 0) 0 (Inj1 (x1 (λ x8 : ι → ι . 0) 0)) (setsum (x4 0) 0) (x7 (λ x8 x9 : ι → ι . Inj1 0) (λ x8 x9 . Inj0 0)) (x0 (λ x8 . λ x9 : ((ι → ι) → ι) → ι . λ x10 x11 . Inj0 0) (x4 0) (x2 (λ x8 x9 . 0) 0) (Inj1 0) (x7 (λ x8 x9 : ι → ι . 0) (λ x8 x9 . 0)) x6)) (x5 (λ x8 . Inj1 0))) (x5 (λ x8 . x0 (λ x9 . λ x10 : ((ι → ι) → ι) → ι . λ x11 x12 . x2 (λ x13 x14 . Inj1 0) (x2 (λ x13 x14 . 0) 0)) (setsum (x0 (λ x9 . λ x10 : ((ι → ι) → ι) → ι . λ x11 x12 . 0) 0 0 0 0 0) x6) (setsum (x3 (λ x9 : ι → ι . λ x10 . λ x11 : (ι → ι) → ι → ι . λ x12 : ι → ι . 0) (λ x9 . λ x10 : (ι → ι) → ι . 0) (λ x9 . 0)) (x0 (λ x9 . λ x10 : ((ι → ι) → ι) → ι . λ x11 x12 . 0) 0 0 0 0 0)) x6 0 0)) (setsum (x7 (λ x8 x9 : ι → ι . Inj1 (setsum 0 0)) (λ x8 x9 . setsum (x2 (λ x10 x11 . 0) 0) 0)) (x0 (λ x8 . λ x9 : ((ι → ι) → ι) → ι . λ x10 x11 . 0) (x1 (λ x8 : ι → ι . x0 (λ x9 . λ x10 : ((ι → ι) → ι) → ι . λ x11 x12 . 0) 0 0 0 0 0) 0) (x2 (λ x8 x9 . Inj1 0) (setsum 0 0)) 0 0 0)) (x2 (λ x8 x9 . x8) (x4 (setsum 0 (x2 (λ x8 x9 . 0) 0)))).
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