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Proofgold Proof

pf
Let x0 of type (ιι) → (ιι) → (ιι) → ι be given.
Let x1 of type (((((ιι) → ιι) → ι) → ι(ιι) → ι) → ιιιι) → (ιCT2 ι) → ι be given.
Let x2 of type (ιι) → ιι(ιι) → ιιι be given.
Let x3 of type ((ιι) → ι) → ιιι be given.
Assume H0: ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι . ∀ x7 : (ι → (ι → ι) → ι) → ι . x3 (λ x8 : ι → ι . 0) (x2 (λ x8 . x5 (x2 (λ x9 . setsum 0 0) (x5 0 0) (Inj1 0) (λ x9 . x1 (λ x10 : (((ι → ι)ι → ι) → ι)ι → (ι → ι) → ι . λ x11 x12 x13 . 0) (λ x10 . λ x11 : ι → ι → ι . 0)) (x5 0 0) 0) (x6 0)) (x2 (λ x8 . x2 (λ x9 . x8) (setsum 0 0) (x7 (λ x9 . λ x10 : ι → ι . 0)) (λ x9 . 0) x8 (x5 0 0)) (x3 (λ x8 : ι → ι . x2 (λ x9 . 0) 0 0 (λ x9 . 0) 0 0) (x3 (λ x8 : ι → ι . 0) 0 0) (x7 (λ x8 . λ x9 : ι → ι . 0))) (x7 (λ x8 . λ x9 : ι → ι . Inj1 0)) (λ x8 . x5 0 (setsum 0 0)) (x3 (λ x8 : ι → ι . 0) 0 (setsum 0 0)) (x3 (λ x8 : ι → ι . Inj1 0) (x2 (λ x8 . 0) 0 0 (λ x8 . 0) 0 0) (setsum 0 0))) (x6 0) (λ x8 . setsum (Inj1 (x3 (λ x9 : ι → ι . 0) 0 0)) (x1 (λ x9 : (((ι → ι)ι → ι) → ι)ι → (ι → ι) → ι . λ x10 x11 x12 . 0) (λ x9 . λ x10 : ι → ι → ι . 0))) (x1 (λ x8 : (((ι → ι)ι → ι) → ι)ι → (ι → ι) → ι . λ x9 x10 x11 . Inj0 (x0 (λ x12 . 0) (λ x12 . 0) (λ x12 . 0))) (λ x8 . λ x9 : ι → ι → ι . 0)) (setsum (x5 (x6 0) (x0 (λ x8 . 0) (λ x8 . 0) (λ x8 . 0))) 0)) (x3 (λ x8 : ι → ι . 0) (x1 (λ x8 : (((ι → ι)ι → ι) → ι)ι → (ι → ι) → ι . λ x9 x10 x11 . 0) (λ x8 . λ x9 : ι → ι → ι . setsum 0 0)) (x5 (x7 (λ x8 . λ x9 : ι → ι . 0)) 0)) = x2 (λ x8 . x8) (Inj0 x4) (Inj0 x4) (λ x8 . x5 0 (x7 (λ x9 . λ x10 : ι → ι . x3 (λ x11 : ι → ι . x11 0) 0 (x1 (λ x11 : (((ι → ι)ι → ι) → ι)ι → (ι → ι) → ι . λ x12 x13 x14 . 0) (λ x11 . λ x12 : ι → ι → ι . 0))))) (Inj0 (x1 (λ x8 : (((ι → ι)ι → ι) → ι)ι → (ι → ι) → ι . λ x9 x10 x11 . 0) (λ x8 . λ x9 : ι → ι → ι . Inj1 0))) (setsum 0 (x7 (λ x8 . λ x9 : ι → ι . x9 0))).
Assume H1: ∀ x4 : (ι → ι) → ι . ∀ x5 . ∀ x6 : ((ι → ι → ι)ι → ι → ι) → ι . ∀ x7 : (ι → ι)((ι → ι) → ι)(ι → ι)ι → ι . x3 (λ x8 : ι → ι . x0 (λ x9 . Inj0 (x1 (λ x10 : (((ι → ι)ι → ι) → ι)ι → (ι → ι) → ι . λ x11 x12 x13 . setsum 0 0) (λ x10 . λ x11 : ι → ι → ι . 0))) (λ x9 . Inj0 (x0 (λ x10 . setsum 0 0) (λ x10 . 0) (λ x10 . x7 (λ x11 . 0) (λ x11 : ι → ι . 0) (λ x11 . 0) 0))) (λ x9 . Inj0 (x8 0))) ... ... = ....
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