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Proofgold Proof

pf
Let x0 of type (ι → ι → ι → ι → ι) → ((ι → ι) → ι) → ι → ι → ι → ι be given.
Let x1 of type ((((ι → ι → ι) → (ι → ι) → ι) → ι) → ι) → (ι → ι → (ι → ι) → ι) → ι be given.
Let x2 of type (ι → ((ι → ι → ι) → (ι → ι) → ι) → ((ι → ι) → ι) → (ι → ι) → ι → ι) → (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 be given.
Assume H0: ∀ x4 : (ι → ι) → ι . ∀ x5 : ι → ι . ∀ x6 : ((ι → ι) → ι → ι) → ι → ι → ι → ι . ∀ x7 : (((ι → ι) → ι) → (ι → ι) → ι) → ι → ι → ι → ι . x3 (λ x8 : (ι → (ι → ι) → ι) → ι → ι . x8 (λ x9 . λ x10 : ι → ι . x10 (x2 (λ x11 . λ x12 : (ι → ι → ι) → (ι → ι) → ι . λ x13 : (ι → ι) → ι . λ x14 : ι → ι . λ x15 . Inj1 0) (λ x11 . λ x12 : ι → ι . 0))) (x7 (λ x9 : (ι → ι) → ι . λ x10 : ι → ι . Inj1 (Inj0 0)) (x7 (λ x9 : (ι → ι) → ι . λ x10 : ι → ι . x1 (λ x11 : ((ι → ι → ι) → (ι → ι) → ι) → ι . 0) (λ x11 x12 . λ x13 : ι → ι . 0)) (x1 (λ x9 : ((ι → ι → ι) → (ι → ι) → ι) → ι . 0) (λ x9 x10 . λ x11 : ι → ι . 0)) (x3 (λ x9 : (ι → (ι → ι) → ι) → ι → ι . 0) (λ x9 : (ι → ι) → ι → ι . λ x10 . λ x11 : ι → ι . λ x12 . 0)) (x2 (λ x9 . λ x10 : (ι → ι → ι) → (ι → ι) → ι . λ x11 : (ι → ι) → ι . λ x12 : ι → ι . λ x13 . 0) (λ x9 . λ x10 : ι → ι . 0))) (x7 (λ x9 : (ι → ι) → ι . λ x10 : ι → ι . 0) (x0 (λ x9 x10 x11 x12 . 0) (λ x9 : ι → ι . 0) 0 0 0) (x1 (λ x9 : ((ι → ι → ι) → (ι → ι) → ι) → ι . 0) (λ x9 x10 . λ x11 : ι → ι . 0)) (x8 (λ x9 . λ x10 : ι → ι . 0) 0)) 0)) (λ x8 : (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι . λ x11 . x11) = x7 (λ x8 : (ι → ι) → ι . λ x9 : ι → ι . x3 (λ x10 : (ι → (ι → ι) → ι) → ι → ι . x9 (setsum (x2 (λ x11 . λ x12 : (ι → ι → ι) → (ι → ι) → ι . λ x13 : (ι → ι) → ι . λ x14 : ι → ι . λ x15 . 0) (λ x11 . λ x12 : ι → ι . 0)) (setsum 0 0))) (λ x10 : (ι → ι) → ι → ι . λ x11 . λ x12 : ι → ι . λ x13 . Inj1 0)) (setsum 0 (x3 (λ x8 : (ι → (ι → ι) → ι) → ι → ι . x7 (λ x9 : (ι → ι) → ι . λ x10 : ι → ι . setsum 0 0) (Inj1 0) (x3 (λ x9 : (ι → (ι → ι) → ι) → ι → ι . 0) (λ x9 : (ι → ι) → ι → ι . λ x10 . λ x11 : ι → ι . λ x12 . 0)) (x8 (λ x9 . λ x10 : ι → ι . 0) 0)) (λ x8 : (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι . λ x11 . 0))) (Inj0 (x5 (x4 (λ x8 . x5 0)))) (x1 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι) → ι . x0 (λ x9 x10 x11 x12 . x1 (λ x13 : ((ι → ι → ι) → (ι → ι) → ι) → ι . x1 (λ x14 : ((ι → ι → ι) → (ι → ι) → ι) → ι . 0) (λ x14 x15 . λ x16 : ι → ι . 0)) (λ x13 x14 . λ x15 : ι → ι . setsum 0 0)) (λ x9 : ι → ι . x0 (λ x10 x11 x12 x13 . x2 (λ x14 . λ x15 : (ι → ι → ι) → (ι → ι) → ι . λ x16 : (ι → ι) → ι . λ x17 : ι → ι . λ x18 . 0) (λ x14 . λ x15 : ι → ι . 0)) (λ x10 : ι → ι . x2 (λ x11 . λ x12 : (ι → ι → ι) → (ι → ι) → ι . λ x13 : (ι → ι) → ι . λ x14 : ι → ι . λ x15 . 0) (λ x11 . λ x12 : ι → ι . 0)) (Inj0 0) (x9 0) 0) (x3 (λ x9 : (ι → (ι → ι) → ι) → ι → ι . Inj1 0) (λ x9 : (ι → ι) → ι → ι . λ x10 . λ x11 : ι → ι . λ x12 . setsum 0 0)) (x1 (λ x9 : ((ι → ι → ι) → (ι → ι) → ι) → ι . x6 (λ x10 : ι → ι . λ x11 . 0) 0 0 0) (λ x9 x10 . λ x11 : ι → ι . setsum 0 0)) (x0 (λ x9 x10 x11 x12 . 0) (λ x9 : ι → ι . 0) (x1 (λ x9 : ((ι → ι → ι) → (ι → ι) → ι) → ι . 0) (λ x9 x10 . λ x11 : ι → ι . 0)) (x3 (λ x9 : (ι → (ι → ι) → ι) → ι → ι . 0) (λ x9 : (ι → ι) → ι → ι . λ x10 . λ x11 : ι → ι . λ x12 . 0)) (x1 (λ x9 : ((ι → ι → ι) → (ι → ι) → ι) → ι . 0) (λ x9 x10 . λ x11 : ι → ι . 0)))) (λ x8 x9 . λ x10 : ι → ι . 0)).
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