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Proofgold Proof

pf
Let x0 of type (ι → ι) → ι → ι be given.
Let x1 of type (ι → ι) → ι → ι be given.
Let x2 of type (((CT2 ι) → ι) → ι) → ι → ι → ι be given.
Let x3 of type (ι → ι) → ι → ι → ι be given.
Assume H0: ∀ x4 x5 . ∀ x6 : ι → ι . ∀ x7 : ι → ι → ι → ι . x3 (λ x8 . x7 (Inj1 0) x8 (x7 (x6 (setsum 0 0)) 0 0)) (Inj0 (Inj0 (setsum (x2 (λ x8 : ((ι → ι → ι) → ι) → ι . 0) 0 0) x4))) (x6 (x0 (λ x8 . 0) (x6 (setsum 0 0)))) = x6 (x7 0 x4 (Inj0 (Inj1 0))).
Assume H1: ∀ x4 : (ι → (ι → ι) → ι → ι) → ι → ι . ∀ x5 : (ι → (ι → ι) → ι → ι) → ι . ∀ x6 : ι → ι . ∀ x7 . x3 (λ x8 . x8) 0 (Inj0 0) = x5 (λ x8 . λ x9 : ι → ι . λ x10 . x8).
Apply FalseE with ... ⟶ (∀ x4 : ι → ι → (ι → ι) → ι . ∀ x5 : (ι → ι) → ι . ∀ x6 : ((ι → ι) → ι) → (ι → ι → ι) → ι → ι . ∀ x7 : (((ι → ι) → ι) → ι → ι → ι) → ((ι → ι) → ι) → ι → ι . x2 (λ x8 : ((ι → ι → ι) → ι) → ι . Inj1 (x6 (λ x9 : ι → ι . 0) (λ x9 x10 . x1 (λ x11 . x1 (λ x12 . 0) 0) x9) (x8 (λ x9 : ι → ι → ι . x9 0 0)))) (x1 (λ x8 . x1 (λ x9 . x7 (λ x10 : (ι → ι) → ι . λ x11 x12 . Inj1 0) (λ x10 : ι → ι . 0) 0) (x3 (λ x9 . setsum 0 0) (x7 (λ x9 : (ι → ι) → ι . λ x10 x11 . 0) (λ x9 : ι → ι . 0) 0) x8)) ...) ... = ...) ⟶ (∀ x4 : (ι → ι) → ι . ∀ x5 x6 x7 . x1 (λ x8 . x5) (x2 (λ x8 : ((ι → ι → ι) → ι) → ι . setsum 0 (x0 (λ x9 . x9) (x0 (λ x9 . 0) 0))) (Inj0 x7) (x1 (λ x8 . setsum (setsum 0 0) x6) (setsum (x1 (λ x8 . 0) 0) 0))) = Inj1 (setsum (setsum (setsum x5 (Inj0 0)) (x2 (λ x8 : ((ι → ι → ι) → ι) → ι . setsum 0 0) 0 0)) (setsum (x1 (λ x8 . setsum 0 0) x5) (setsum x6 x5)))) ⟶ (∀ x4 x5 x6 x7 . x1 (λ x8 . x0 (λ x9 . x8) (Inj1 (x2 (λ x9 : ((ι → ι → ι) → ι) → ι . Inj1 0) x5 (x3 (λ x9 . 0) 0 0)))) (x1 (λ x8 . x3 (λ x9 . x7) (setsum x6 x8) (setsum (Inj0 0) (Inj0 0))) (Inj0 0)) = Inj1 0) ⟶ (∀ x4 : ι → ι . ∀ x5 : ι → ((ι → ι) → ι → ι) → ι . ∀ x6 . ∀ x7 : ι → ι → ι → ι . x0 (λ x8 . 0) x6 = Inj1 (x3 (λ x8 . 0) (setsum 0 (Inj0 (Inj1 0))) (setsum (x1 (λ x8 . 0) (Inj1 0)) (setsum (x1 (λ x8 . 0) 0) (setsum 0 0))))) ⟶ (∀ x4 : ι → ι . ∀ x5 x6 x7 . x0 (λ x8 . setsum x8 0) (setsum (x0 (λ x8 . x0 (λ x9 . x8) (x3 (λ x9 . 0) 0 0)) 0) x6) = x6) ⟶ False.
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