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PUKNThmYK9PWhTaEhNUEqATpYH9PxmZhmVN
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fd47a../c4340.. bday: 11884 doc published by PrGVS..
Known 52346..pair_0_0 : setsum 0 0 = 0
Known fc3ab..Inj0_0 : Inj0 0 = 0
Known 8d83e..Inj1I1 : ∀ x0 . In 0 (Inj1 x0)
Theorem 2c11d.. : ∀ x0 : ((ι → ι → ι → ι) → ((ι → ι → ι) → ι → ι → ι) → ι) → ι → ο . ∀ x1 : (ι → ι → ι) → ι → ι → ο . ∀ x2 : (ι → ι) → ((ι → (ι → ι) → ι) → ι → ι) → ο . ∀ x3 : (ι → ι → ι) → ι → ο . (∀ x4 : ι → ι → ι → ι → ι . ∀ x5 x6 . ∀ x7 : ι → ι . x0 (λ x8 : ι → ι → ι → ι . λ x9 : (ι → ι → ι) → ι → ι → ι . setsum (Inj1 (Inj1 (Inj0 0))) x6) (Inj0 (x7 (x7 (Inj1 0)))) ⟶ x3 (λ x8 x9 . 0) (setsum 0 (setsum x5 (Inj1 (x4 0 0 0 0))))) ⟶ (∀ x4 x5 x6 . ∀ x7 : (ι → ι) → (ι → ι → ι) → ι . In (Inj0 x5) x4 ⟶ x3 (λ x8 x9 . 0) (setsum (setsum (Inj1 x4) x5) (setsum (x7 (λ x8 . x6) (λ x8 x9 . 0)) (x7 (λ x8 . 0) (λ x8 x9 . 0)))) ⟶ x1 (λ x8 x9 . setsum (Inj1 (x7 (λ x10 . Inj0 0) (λ x10 x11 . 0))) (Inj1 0)) 0 0) ⟶ (∀ x4 . ∀ x5 : ((ι → ι → ι) → ι → ι) → ι . ∀ x6 x7 . In (Inj1 (Inj0 0)) (Inj0 (Inj0 x7)) ⟶ x0 (λ x8 : ι → ι → ι → ι . λ x9 : (ι → ι → ι) → ι → ι → ι . 0) (setsum (x5 (λ x8 : ι → ι → ι . λ x9 . setsum (setsum 0 0) (x8 0 0))) x7) ⟶ x2 (λ x8 . Inj1 (Inj0 (setsum (setsum 0 0) x8))) (λ x8 : ι → (ι → ι) → ι . λ x9 . 0)) ⟶ (∀ x4 . ∀ x5 : ι → (ι → ι) → ι . ∀ x6 : ι → ι → ι . ∀ x7 . x2 (λ x8 . 0) (λ x8 : ι → (ι → ι) → ι . λ x9 . Inj0 (Inj1 x7)) ⟶ False) ⟶ (∀ x4 : ι → ι → (ι → ι) → ι . ∀ x5 : ι → ι . ∀ x6 . ∀ x7 : ι → ι . x1 (λ x8 x9 . setsum (setsum (Inj0 (Inj1 0)) x8) x6) (Inj1 (Inj1 0)) (x5 (setsum (setsum (setsum 0 0) (setsum 0 0)) (Inj0 (setsum 0 0))))) ⟶ (∀ x4 : ι → ι . ∀ x5 x6 x7 . In x7 (setsum x5 (x4 (setsum (Inj0 0) (setsum 0 0)))) ⟶ x1 (λ x8 x9 . setsum x7 (setsum (setsum x8 (setsum 0 0)) (setsum (setsum 0 0) 0))) (Inj1 (setsum (Inj0 (Inj1 0)) (setsum (setsum 0 0) (Inj1 0)))) (Inj1 0) ⟶ x3 (λ x8 x9 . Inj0 (Inj1 x8)) (x4 0)) ⟶ (∀ x4 : ι → ι → ι . ∀ x5 x6 x7 . In (Inj0 (setsum (setsum (setsum 0 0) (setsum 0 0)) x5)) (Inj1 x6) ⟶ x0 (λ x8 : ι → ι → ι → ι . λ x9 : (ι → ι → ι) → ι → ι → ι . 0) x5) ⟶ (∀ x4 : ι → ι . ∀ x5 x6 x7 . x0 (λ x8 : ι → ι → ι → ι . λ x9 : (ι → ι → ι) → ι → ι → ι . 0) 0 ⟶ False) ⟶ False (proof)
Known 8106d..notI : ∀ x0 : ο . (x0 ⟶ False) ⟶ not x0
Known TrueITrueI : True
Theorem d06ba.. : not (∀ x0 : (((((ι → ι) → ι) → ι → ι → ι) → ι) → (ι → ι → ι) → ι) → ι → ο . ∀ x1 : (ι → ι) → ((((ι → ι) → ι) → ι) → ι) → ι → ο . ∀ x2 : (ι → ι → ι) → ((((ι → ι) → ι) → (ι → ι) → ι → ι) → ι → ι → ι) → ο . ∀ x3 : (ι → ι → ι) → ι → ι → ι → ο . (∀ x4 : ι → (ι → ι → ι) → ι → ι → ι . ∀ x5 x6 x7 . In (setsum (x4 (Inj1 0) (λ x8 x9 . setsum (setsum 0 0) (setsum 0 0)) x5 0) (setsum (x4 (Inj0 0) (λ x8 x9 . 0) (setsum 0 0) 0) (setsum (Inj1 0) 0))) (Inj1 (setsum x6 0)) ⟶ x0 (λ x8 : (((ι → ι) → ι) → ι → ι → ι) → ι . λ x9 : ι → ι → ι . setsum 0 (setsum (x9 (Inj1 0) (x9 0 0)) x7)) (x4 (setsum (x4 (setsum 0 0) (λ x8 x9 . setsum 0 0) x5 (setsum 0 0)) x7) (λ x8 x9 . x9) (setsum (Inj1 (x4 0 (λ x8 x9 . 0) 0 0)) (Inj1 (Inj1 0))) 0) ⟶ x3 (λ x8 x9 . x9) (setsum x5 0) (Inj1 0) x6) ⟶ (∀ x4 . ∀ x5 : ι → (ι → ι) → ι . ∀ x6 : (((ι → ι) → ι) → (ι → ι) → ι → ι) → (ι → ι) → ι . ∀ x7 . In (Inj1 0) (setsum 0 0) ⟶ x3 (λ x8 x9 . 0) (setsum 0 (setsum 0 (Inj1 0))) 0 (Inj1 0) ⟶ x1 (λ x8 . setsum (setsum (Inj0 (x5 0 (λ x9 . 0))) x8) 0) (λ x8 : ((ι → ι) → ι) → ι . Inj1 0) (Inj1 (setsum (Inj1 (setsum 0 0)) (x5 0 (λ x8 . 0))))) ⟶ (∀ x4 : ι → ι . ∀ x5 : (ι → (ι → ι) → ι → ι) → ι → ι . ∀ x6 : (ι → ι) → ι → ι . ∀ x7 : (ι → ι) → ι . In (Inj1 (x4 (setsum (x6 (λ x8 . 0) 0) (setsum 0 0)))) (setsum (x4 (setsum (Inj0 0) 0)) (x4 0)) ⟶ x2 (λ x8 x9 . x7 (λ x10 . 0)) (λ x8 : ((ι → ι) → ι) → (ι → ι) → ι → ι . λ x9 x10 . setsum (Inj0 0) 0)) ⟶ (∀ x4 : (((ι → ι) → ι) → (ι → ι) → ι → ι) → ι → ι → ι . ∀ x5 : (((ι → ι) → ι → ι) → (ι → ι) → ι → ι) → (ι → ι → ι) → (ι → ι) → ι . ∀ x6 : ι → ((ι → ι) → ι) → ι → ι → ι . ∀ x7 . In (setsum 0 (setsum (setsum (Inj0 0) 0) (x5 (λ x8 : (ι → ι) → ι → ι . λ x9 : ι → ι . λ x10 . Inj0 0) (λ x8 x9 . 0) (λ x8 . x5 (λ x9 : (ι → ι) → ι → ι . λ x10 : ι → ι . λ x11 . 0) (λ x9 x10 . 0) (λ x9 . 0))))) (Inj1 (x5 (λ x8 : (ι → ι) → ι → ι . λ x9 : ι → ι . λ x10 . setsum x10 0) (λ x8 x9 . x8) (λ x8 . Inj1 (Inj1 0)))) ⟶ x2 (λ x8 x9 . 0) (λ x8 : ((ι → ι) → ι) → (ι → ι) → ι → ι . λ x9 x10 . Inj1 (Inj0 0)) ⟶ x1 (λ x8 . Inj1 0) (λ x8 : ((ι → ι) → ι) → ι . setsum (setsum (setsum (setsum 0 0) (Inj1 0)) (setsum 0 x7)) 0) (setsum (setsum (x4 (λ x8 : (ι → ι) → ι . λ x9 : ι → ι . λ x10 . 0) (x6 0 (λ x8 : ι → ι . 0) 0 0) 0) (x4 (λ x8 : (ι → ι) → ι . λ x9 : ι → ι . λ x10 . 0) (setsum 0 0) (setsum 0 0))) (setsum 0 (x6 x7 (λ x8 : ι → ι . setsum 0 0) (setsum 0 0) (Inj1 0))))) ⟶ (∀ x4 . ∀ x5 : ι → ι → (ι → ι) → ι → ι . ∀ x6 x7 . x3 (λ x8 x9 . x8) 0 (Inj1 0) (setsum 0 0) ⟶ x1 (λ x8 . setsum (Inj0 x8) (setsum 0 (setsum 0 (setsum 0 0)))) (λ x8 : ((ι → ι) → ι) → ι . 0) 0) ⟶ (∀ x4 : ι → (ι → ι → ι) → (ι → ι) → ι → ι . ∀ x5 : ι → ι . ∀ x6 : ι → ι → (ι → ι) → ι → ι . ∀ x7 : (((ι → ι) → ι → ι) → (ι → ι) → ι → ι) → ι . x1 (λ x8 . x7 (λ x9 : (ι → ι) → ι → ι . λ x10 : ι → ι . λ x11 . setsum (Inj1 0) 0)) (λ x8 : ((ι → ι) → ι) → ι . Inj1 0) (x5 0) ⟶ x1 (λ x8 . setsum (x5 0) (Inj0 0)) (λ x8 : ((ι → ι) → ι) → ι . Inj0 0) (setsum (setsum 0 0) (Inj0 (setsum 0 (setsum 0 0))))) ⟶ (∀ x4 : ((ι → ι) → ι) → ι . ∀ x5 : (ι → (ι → ι) → ι) → ι . ∀ x6 : ((ι → ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ι . ∀ x7 : ι → ι . x0 (λ x8 : (((ι → ι) → ι) → ι → ι → ι) → ι . λ x9 : ι → ι → ι . x9 0 0) (Inj1 (setsum 0 (x6 (λ x8 : ι → ι → ι . λ x9 . Inj0 0) (λ x8 : ι → ι . λ x9 . x9))))) ⟶ (∀ x4 : ι → ι . ∀ x5 : (ι → ι) → ι → ι . ∀ x6 : ι → ((ι → ι) → ι → ι) → (ι → ι) → ι . ∀ x7 : ι → ((ι → ι) → ι → ι) → ι . In (Inj1 (setsum (x7 (x4 0) (λ x8 : ι → ι . λ x9 . setsum 0 0)) 0)) (Inj0 0) ⟶ x0 (λ x8 : (((ι → ι) → ι) → ι → ι → ι) → ι . λ x9 : ι → ι → ι . setsum 0 (x7 (x6 (setsum 0 0) (λ x10 : ι → ι . λ x11 . Inj1 0) (λ x10 . setsum 0 0)) (λ x10 : ι → ι . λ x11 . setsum (x10 0) 0))) (Inj1 (setsum (x4 (Inj1 0)) (Inj0 0))) ⟶ x3 (λ x8 x9 . setsum (setsum (x6 (setsum 0 0) (λ x10 : ι → ι . λ x11 . setsum 0 0) (λ x10 . x10)) (Inj1 0)) (Inj0 0)) 0 0 (x7 0 (λ x8 : ι → ι . λ x9 . x9))) ⟶ False) (proof)
Theorem 2c194.. : not (∀ x0 : ((((ι → ι → ι) → ι → ι → ι) → ι) → ι) → (ι → ι → ι → ι) → ο . ∀ x1 : (((ι → (ι → ι) → ι) → ι → ι → ι) → ι → ι) → (ι → ι) → ι → ((ι → ι) → ι → ι) → ι → ο . ∀ x2 : (ι → ((ι → ι → ι) → (ι → ι) → ι) → ι) → ι → ι → (ι → ι) → ι → ο . ∀ x3 : (ι → ι) → ((((ι → ι) → ι → ι) → ι → ι) → ((ι → ι) → ι) → ι) → ο . (∀ x4 . ∀ x5 : ι → ι . ∀ x6 . ∀ x7 : ι → ι . In (Inj1 (setsum (setsum (Inj0 0) (x5 0)) (x5 0))) (Inj1 (Inj0 (x5 (x7 0)))) ⟶ x3 (λ x8 . setsum (setsum (Inj0 (Inj1 0)) (Inj1 (setsum 0 0))) x6) (λ x8 : ((ι → ι) → ι → ι) → ι → ι . λ x9 : (ι → ι) → ι . setsum 0 (x8 (λ x10 : ι → ι . λ x11 . setsum (setsum 0 0) 0) (x7 (x7 0))))) ⟶ (∀ x4 x5 . ∀ x6 : (ι → ι → ι → ι) → (ι → ι) → (ι → ι) → ι → ι . ∀ x7 : (ι → ι) → (ι → ι → ι) → ι . x3 (λ x8 . Inj0 (setsum (setsum (Inj0 0) x8) (x6 (λ x9 x10 x11 . x11) (λ x9 . 0) (λ x9 . 0) (Inj0 0)))) (λ x8 : ((ι → ι) → ι → ι) → ι → ι . λ x9 : (ι → ι) → ι . x8 (λ x10 : ι → ι . λ x11 . x10 (Inj1 0)) (Inj0 0)) ⟶ x1 (λ x8 : (ι → (ι → ι) → ι) → ι → ι → ι . λ x9 . x9) (λ x8 . setsum (setsum 0 0) 0) x4 (λ x8 : ι → ι . λ x9 . 0) 0) ⟶ (∀ x4 : ((ι → ι → ι) → ι → ι → ι) → ι . ∀ x5 : ι → ι → (ι → ι) → ι . ∀ x6 : (ι → ι) → (ι → ι) → ι → ι → ι . ∀ x7 . x3 (λ x8 . setsum (Inj1 0) (Inj1 (x5 0 (setsum 0 0) (λ x9 . x6 (λ x10 . 0) (λ x10 . 0) 0 0)))) (λ x8 : ((ι → ι) → ι → ι) → ι → ι . λ x9 : (ι → ι) → ι . Inj1 (x9 (λ x10 . x8 (λ x11 : ι → ι . λ x12 . Inj1 0) (x8 (λ x11 : ι → ι . λ x12 . 0) 0)))) ⟶ x2 (λ x8 . λ x9 : (ι → ι → ι) → (ι → ι) → ι . 0) 0 (Inj0 (setsum (Inj1 (x5 0 0 (λ x8 . 0))) (x4 (λ x8 : ι → ι → ι . λ x9 x10 . Inj0 0)))) (λ x8 . 0) 0) ⟶ (∀ x4 : ι → (ι → ι) → ι → ι → ι . ∀ x5 : ((ι → ι) → (ι → ι) → ι → ι) → ι → ι → ι → ι . ∀ x6 : ι → (ι → ι → ι) → ι → ι → ι . ∀ x7 . In x7 (Inj0 (x5 (λ x8 x9 : ι → ι . λ x10 . Inj1 (setsum 0 0)) (Inj0 0) (x6 0 (λ x8 x9 . setsum 0 0) (x4 0 (λ x8 . 0) 0 0) 0) x7)) ⟶ x2 (λ x8 . λ x9 : (ι → ι → ι) → (ι → ι) → ι . setsum (setsum x8 (x9 (λ x10 x11 . Inj1 0) (λ x10 . setsum 0 0))) (setsum x8 (Inj0 (x9 (λ x10 x11 . 0) (λ x10 . 0))))) 0 0 (setsum (x5 (λ x8 x9 : ι → ι . λ x10 . Inj0 (x9 0)) (x6 (Inj1 0) (λ x8 x9 . setsum 0 0) (setsum 0 0) (Inj1 0)) (Inj1 (setsum 0 0)) (setsum (Inj1 0) 0))) 0 ⟶ x0 (λ x8 : ((ι → ι → ι) → ι → ι → ι) → ι . setsum 0 (Inj0 (setsum (Inj0 0) (setsum 0 0)))) (λ x8 x9 x10 . x7)) ⟶ (∀ x4 : ι → ι . ∀ x5 : (((ι → ι) → ι → ι) → ι) → ι . ∀ x6 : (((ι → ι) → ι → ι) → ι) → ι → ι . ∀ x7 : ι → ι . In (Inj0 (setsum (Inj1 (x7 0)) 0)) (setsum (setsum 0 (setsum (Inj0 0) (x5 (λ x8 : (ι → ι) → ι → ι . 0)))) (Inj1 (Inj1 0))) ⟶ x0 (λ x8 : ((ι → ι → ι) → ι → ι → ι) → ι . Inj1 (Inj0 (setsum (setsum 0 0) (setsum 0 0)))) (λ x8 x9 x10 . x8) ⟶ x1 (λ x8 : (ι → (ι → ι) → ι) → ι → ι → ι . x8 (λ x9 . λ x10 : ι → ι . 0) (x7 0)) (λ x8 . 0) 0 (λ x8 : ι → ι . λ x9 . x7 (Inj0 (Inj1 (x6 (λ x10 : (ι → ι) → ι → ι . 0) 0)))) (setsum (Inj1 0) (setsum (Inj1 (Inj0 0)) (setsum (x7 0) (Inj0 0))))) ⟶ (∀ x4 x5 x6 . ∀ x7 : ι → ι → ι → ι . In x5 (setsum x6 (setsum 0 0)) ⟶ x1 (λ x8 : (ι → (ι → ι) → ι) → ι → ι → ι . λ x9 . x9) (λ x8 . Inj1 (Inj1 x8)) 0 (λ x8 : ι → ι . λ x9 . Inj0 (x8 x9)) x5 ⟶ x3 (λ x8 . Inj1 (setsum x6 (setsum 0 x8))) (λ x8 : ((ι → ι) → ι → ι) → ι → ι . λ x9 : (ι → ι) → ι . Inj0 (setsum (Inj1 (x9 (λ x10 . 0))) (x7 (x7 0 0 0) (Inj1 0) (Inj1 0))))) ⟶ (∀ x4 x5 x6 . ∀ x7 : ι → ι . x0 (λ x8 : ((ι → ι → ι) → ι → ι → ι) → ι . setsum x6 (Inj1 (setsum (Inj1 0) (setsum 0 0)))) (λ x8 x9 x10 . Inj1 (Inj0 0)) ⟶ x0 (λ x8 : ((ι → ι → ι) → ι → ι → ι) → ι . 0) (λ x8 x9 x10 . Inj1 (Inj1 (Inj1 (Inj1 0))))) ⟶ (∀ x4 : ι → ι → (ι → ι) → ι . ∀ x5 : (((ι → ι) → ι → ι) → ι) → ι . ∀ x6 : ι → ι → (ι → ι) → ι → ι . ∀ x7 . x0 (λ x8 : ((ι → ι → ι) → ι → ι → ι) → ι . x6 0 0 (λ x9 . 0) 0) (λ x8 x9 x10 . 0) ⟶ x0 (λ x8 : ((ι → ι → ι) → ι → ι → ι) → ι . x6 (x8 (λ x9 : ι → ι → ι . λ x10 x11 . x9 (x8 (λ x12 : ι → ι → ι . λ x13 x14 . 0)) (setsum 0 0))) (setsum 0 (setsum (Inj0 0) 0)) (λ x9 . setsum 0 (setsum (x6 0 0 (λ x10 . 0) 0) 0)) (setsum (Inj1 (setsum 0 0)) (x8 (λ x9 : ι → ι → ι . λ x10 x11 . 0)))) (λ x8 x9 x10 . x10)) ⟶ False) (proof)
Known FalseEFalseE : False ⟶ ∀ x0 : ο . x0
Theorem 95285.. : not (∀ x0 : ((ι → ι → ι → ι) → ι → ι → ι) → ι → ο . ∀ x1 : ((ι → ι) → ι) → (ι → ι) → ι → ο . ∀ x2 : (ι → ι) → ι → ο . ∀ x3 : (ι → ι → ι → ι → ι → ι) → ((ι → ι) → (ι → ι → ι) → ι) → ο . (∀ x4 : ι → ι → ι . ∀ x5 . ∀ x6 : (((ι → ι) → ι → ι) → ι) → ι . ∀ x7 : (ι → (ι → ι) → ι) → (ι → ι) → ι → ι . In (setsum 0 (x7 (λ x8 . λ x9 : ι → ι . x7 (λ x10 . λ x11 : ι → ι . 0) (λ x10 . setsum 0 0) 0) (λ x8 . 0) (Inj1 (x6 (λ x8 : (ι → ι) → ι → ι . 0))))) (setsum (x7 (λ x8 . λ x9 : ι → ι . x7 (λ x10 . λ x11 : ι → ι . 0) (λ x10 . Inj1 0) (x6 (λ x10 : (ι → ι) → ι → ι . 0))) (λ x8 . Inj1 (setsum 0 0)) 0) (setsum (x6 (λ x8 : (ι → ι) → ι → ι . 0)) (setsum (setsum 0 0) (Inj1 0)))) ⟶ x3 (λ x8 x9 x10 x11 x12 . Inj0 0) (λ x8 : ι → ι . λ x9 : ι → ι → ι . x8 (x6 (λ x10 : (ι → ι) → ι → ι . 0)))) ⟶ (∀ x4 : ι → (ι → ι → ι) → ι → ι → ι . ∀ x5 : (ι → ι → ι → ι) → ι → ι → ι . ∀ x6 . ∀ x7 : (ι → ι) → ι . In (Inj1 (setsum (x5 (λ x8 x9 x10 . Inj0 0) (setsum 0 0) (setsum 0 0)) (setsum x6 (x7 (λ x8 . 0))))) x6 ⟶ x3 (λ x8 x9 x10 x11 x12 . Inj1 (setsum (Inj1 0) x12)) (λ x8 : ι → ι . λ x9 : ι → ι → ι . Inj1 (Inj1 (Inj0 (Inj1 0)))) ⟶ x3 (λ x8 x9 x10 x11 x12 . Inj1 (setsum 0 (setsum (Inj0 0) 0))) (λ x8 : ι → ι . λ x9 : ι → ι → ι . Inj0 (Inj0 (x9 (x7 (λ x10 . 0)) (x7 (λ x10 . 0)))))) ⟶ (∀ x4 . ∀ x5 : (ι → ι) → ι . ∀ x6 x7 . In (Inj0 x7) (setsum 0 0) ⟶ x2 (λ x8 . 0) x6 ⟶ x2 (λ x8 . 0) (Inj1 0)) ⟶ (∀ x4 x5 x6 x7 . x2 (λ x8 . x6) x4 ⟶ In x5 (Inj1 0)) ⟶ (∀ x4 : (ι → ι) → ι → ι . ∀ x5 : ((ι → ι → ι) → ι) → ι . ∀ x6 . ∀ x7 : ι → ι . In (x5 (λ x8 : ι → ι → ι . 0)) (x5 (λ x8 : ι → ι → ι . Inj0 (x8 (Inj1 0) (setsum 0 0)))) ⟶ x1 (λ x8 : ι → ι . Inj0 0) (λ x8 . 0) (x5 (λ x8 : ι → ι → ι . 0))) ⟶ (∀ x4 x5 . ∀ x6 : (ι → ι) → ι → ι . ∀ x7 . x1 (λ x8 : ι → ι . x6 (λ x9 . x7) (Inj1 (Inj0 (Inj1 0)))) (λ x8 . setsum (x6 (λ x9 . setsum x9 0) (Inj1 (Inj1 0))) 0) (setsum 0 (Inj1 0)) ⟶ x0 (λ x8 : ι → ι → ι → ι . λ x9 x10 . setsum x7 (setsum x10 (x8 (setsum 0 0) 0 (setsum 0 0)))) 0) ⟶ (∀ x4 x5 . ∀ x6 : ι → ι . ∀ x7 . x0 (λ x8 : ι → ι → ι → ι . λ x9 x10 . 0) 0 ⟶ x0 (λ x8 : ι → ι → ι → ι . λ x9 x10 . 0) (Inj1 x7)) ⟶ (∀ x4 x5 x6 x7 . x0 (λ x8 : ι → ι → ι → ι . λ x9 x10 . x10) (Inj0 0) ⟶ x3 (λ x8 x9 x10 x11 x12 . Inj1 0) (λ x8 : ι → ι . λ x9 : ι → ι → ι . 0)) ⟶ False) (proof)
Theorem acf72.. : not (∀ x0 : (((ι → ι) → (ι → ι → ι) → (ι → ι) → ι → ι) → ((ι → ι) → ι) → ((ι → ι) → ι → ι) → ι) → ι → ι → ((ι → ι) → ι → ι) → ο . ∀ x1 : (ι → ι → ι) → ((ι → ι) → ι) → ο . ∀ x2 : (ι → ι) → (ι → ι → ι) → ο . ∀ x3 : (((ι → ι) → ι) → ι → ι) → ((ι → ι → ι) → ι) → (((ι → ι) → ι) → ι) → ι → ο . (∀ x4 . ∀ x5 : ι → ι . ∀ x6 : (ι → (ι → ι) → ι → ι) → ι . ∀ x7 : (ι → ι) → ((ι → ι) → ι → ι) → ι . In (Inj1 (x5 0)) (Inj0 (Inj1 (Inj1 (setsum 0 0)))) ⟶ x2 (λ x8 . x7 (λ x9 . setsum (setsum (setsum 0 0) 0) (x6 (λ x10 . λ x11 : ι → ι . λ x12 . Inj1 0))) (λ x9 : ι → ι . λ x10 . setsum 0 (x7 (λ x11 . Inj0 0) (λ x11 : ι → ι . λ x12 . 0)))) (λ x8 x9 . 0) ⟶ x3 (λ x8 : (ι → ι) → ι . λ x9 . setsum (x8 (λ x10 . setsum (setsum 0 0) (Inj0 0))) 0) (λ x8 : ι → ι → ι . Inj0 0) (λ x8 : (ι → ι) → ι . 0) (Inj1 0)) ⟶ (∀ x4 : ι → ι . ∀ x5 x6 . ∀ x7 : ι → ι . x3 (λ x8 : (ι → ι) → ι . λ x9 . setsum (x7 (setsum x9 (Inj1 0))) (Inj0 (setsum 0 (Inj1 0)))) (λ x8 : ι → ι → ι . 0) (λ x8 : (ι → ι) → ι . setsum x6 (setsum (x8 (λ x9 . Inj0 0)) (x7 0))) (setsum (setsum (x7 0) x5) (Inj1 (setsum (Inj1 0) (setsum 0 0)))) ⟶ In (Inj0 0) (Inj1 0)) ⟶ (∀ x4 x5 x6 x7 . In (Inj1 x7) (Inj0 (Inj0 x5)) ⟶ x3 (λ x8 : (ι → ι) → ι . λ x9 . Inj0 (setsum (setsum (Inj0 0) 0) (setsum 0 (x8 (λ x10 . 0))))) (λ x8 : ι → ι → ι . 0) (λ x8 : (ι → ι) → ι . x6) (Inj0 0) ⟶ x2 (λ x8 . Inj0 x6) (λ x8 x9 . setsum (setsum 0 0) x6)) ⟶ (∀ x4 : (((ι → ι) → ι → ι) → ι) → ι → ι → ι . ∀ x5 x6 . ∀ x7 : ι → ι . x2 (λ x8 . 0) (λ x8 x9 . 0) ⟶ In (Inj0 x6) (setsum x6 (Inj1 (Inj0 (Inj0 0))))) ⟶ (∀ x4 x5 x6 . ∀ x7 : ι → ι . In (Inj0 0) (Inj1 (setsum x6 0)) ⟶ x2 (λ x8 . Inj0 0) (λ x8 x9 . setsum (x7 (setsum 0 0)) 0) ⟶ x1 (λ x8 x9 . 0) (λ x8 : ι → ι . x6)) ⟶ (∀ x4 x5 x6 x7 . x1 (λ x8 x9 . setsum (setsum 0 (setsum (Inj1 0) x7)) (Inj1 (setsum x6 x6))) (λ x8 : ι → ι . 0) ⟶ x1 (λ x8 x9 . x6) (λ x8 : ι → ι . x7)) ⟶ (∀ x4 : ι → ι → (ι → ι) → ι → ι . ∀ x5 : (((ι → ι) → ι) → (ι → ι) → ι) → (ι → ι → ι) → ι . ∀ x6 . ∀ x7 : ι → ι → ι . x2 (λ x8 . 0) (λ x8 x9 . Inj1 0) ⟶ x0 (λ x8 : (ι → ι) → (ι → ι → ι) → (ι → ι) → ι → ι . λ x9 : (ι → ι) → ι . λ x10 : (ι → ι) → ι → ι . 0) (x7 (setsum (Inj0 (setsum 0 0)) (setsum (setsum 0 0) (Inj0 0))) (Inj1 (setsum 0 (setsum 0 0)))) (setsum 0 (setsum (x4 (setsum 0 0) 0 (λ x8 . setsum 0 0) (setsum 0 0)) (setsum (x7 0 0) (Inj1 0)))) (λ x8 : ι → ι . λ x9 . setsum x9 (Inj0 x9))) ⟶ (∀ x4 x5 x6 . ∀ x7 : ι → ι . x0 (λ x8 : (ι → ι) → (ι → ι → ι) → (ι → ι) → ι → ι . λ x9 : (ι → ι) → ι . λ x10 : (ι → ι) → ι → ι . 0) x6 x5 (λ x8 : ι → ι . λ x9 . setsum (Inj1 (setsum (Inj1 0) x9)) (Inj0 x9)) ⟶ x2 (λ x8 . Inj0 (x7 0)) (λ x8 x9 . x6)) ⟶ False) (proof)
Theorem 748af.. : not (∀ x0 : (ι → ι) → ι → ο . ∀ x1 : (ι → (ι → ι) → ι → (ι → ι) → ι → ι) → ((ι → (ι → ι) → ι) → ι) → ο . ∀ x2 : (ι → ι) → ι → ι → ι → ο . ∀ x3 : (((ι → ι → ι → ι) → (ι → ι) → ι → ι) → ι → (ι → ι → ι) → ι → ι) → ι → ο . (∀ x4 . ∀ x5 : ι → ι . ∀ x6 : (ι → ι) → ι . ∀ x7 : ι → ι . In (Inj1 (setsum (Inj0 (setsum 0 0)) x4)) (Inj1 (x7 (setsum 0 (setsum 0 0)))) ⟶ x1 (λ x8 . λ x9 : ι → ι . λ x10 . λ x11 : ι → ι . λ x12 . x12) (λ x8 : ι → (ι → ι) → ι . x5 0) ⟶ x3 (λ x8 : (ι → ι → ι → ι) → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι → ι . λ x11 . Inj1 (Inj1 (setsum (setsum 0 0) (Inj0 0)))) (x5 (setsum (Inj1 (Inj1 0)) 0))) ⟶ (∀ x4 : ι → (ι → ι → ι) → ι → ι . ∀ x5 x6 x7 . x3 (λ x8 : (ι → ι → ι → ι) → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι → ι . λ x11 . setsum (x8 (λ x12 x13 x14 . Inj1 (Inj1 0)) (λ x12 . Inj0 0) (x10 (x10 0 0) 0)) (Inj1 0)) x7 ⟶ x3 (λ x8 : (ι → ι → ι → ι) → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι → ι . λ x11 . 0) x7) ⟶ (∀ x4 . ∀ x5 : ι → ((ι → ι) → ι) → ι . ∀ x6 : ι → ι . ∀ x7 : ((ι → ι) → ι) → ι → ι . x2 (λ x8 . x7 (λ x9 : ι → ι . 0) (Inj0 (setsum 0 0))) 0 (Inj1 0) (Inj0 (setsum (x5 (Inj1 0) (λ x8 : ι → ι . x7 (λ x9 : ι → ι . 0) 0)) (x6 x4))) ⟶ x2 (λ x8 . Inj0 0) (setsum x4 (Inj0 (x5 0 (λ x8 : ι → ι . setsum 0 0)))) x4 0) ⟶ (∀ x4 x5 x6 x7 . x2 (λ x8 . 0) (Inj0 (Inj1 x4)) (Inj0 (setsum (Inj0 0) 0)) (setsum (setsum x4 0) 0) ⟶ In x7 (Inj1 0)) ⟶ (∀ x4 . ∀ x5 : ι → ι → (ι → ι) → ι → ι . ∀ x6 . ∀ x7 : (ι → ι → ι) → ι → ι → ι . x3 (λ x8 : (ι → ι → ι → ι) → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι → ι . λ x11 . setsum 0 (Inj0 (Inj1 (x10 0 0)))) (Inj1 0) ⟶ x1 (λ x8 . λ x9 : ι → ι . λ x10 . λ x11 : ι → ι . λ x12 . x12) (λ x8 : ι → (ι → ι) → ι . x5 0 (setsum (Inj0 (setsum 0 0)) 0) (λ x9 . x6) (x8 (setsum (Inj1 0) 0) (λ x9 . 0)))) ⟶ (∀ x4 : ι → ι . ∀ x5 . ∀ x6 : (ι → ι) → ι → ι → ι . ∀ x7 . x1 (λ x8 . λ x9 : ι → ι . λ x10 . λ x11 : ι → ι . λ x12 . 0) (λ x8 : ι → (ι → ι) → ι . 0) ⟶ False) ⟶ (∀ x4 . ∀ x5 : ι → ι . ∀ x6 : (ι → ι → ι → ι) → ι . ∀ x7 . In (Inj1 (Inj0 (setsum 0 (x6 (λ x8 x9 x10 . 0))))) (Inj0 (setsum x4 0)) ⟶ x3 (λ x8 : (ι → ι → ι → ι) → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι → ι . λ x11 . 0) x7 ⟶ x0 (λ x8 . Inj0 (Inj0 (setsum (x5 0) (Inj1 0)))) (Inj1 0)) ⟶ (∀ x4 x5 . ∀ x6 : ((ι → ι) → ι → ι) → ι . ∀ x7 . In (Inj1 0) x4 ⟶ x0 (λ x8 . setsum 0 (Inj0 (x6 (λ x9 : ι → ι . λ x10 . 0)))) (setsum x7 x7) ⟶ x3 (λ x8 : (ι → ι → ι → ι) → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι → ι . λ x11 . 0) (Inj1 (Inj1 x5))) ⟶ False) (proof)
Theorem 56955.. : not (∀ x0 : (ι → (ι → (ι → ι) → ι → ι) → ι) → ι → (((ι → ι) → ι → ι) → ι) → ο . ∀ x1 : ((((ι → ι) → ι) → ι → (ι → ι) → ι) → ι → ι → ι) → ι → ο . ∀ x2 : (ι → (((ι → ι) → ι → ι) → ι → ι → ι) → ι) → ι → ο . ∀ x3 : (ι → ι) → ((ι → (ι → ι) → ι → ι) → ι → (ι → ι) → ι) → ο . (∀ x4 x5 . ∀ x6 : (((ι → ι) → ι) → ι) → ι . ∀ x7 : ι → ι → ι . In (Inj1 (Inj1 x4)) x4 ⟶ x0 (λ x8 . λ x9 : ι → (ι → ι) → ι → ι . 0) (setsum (x7 0 x4) (Inj0 (setsum (Inj0 0) 0))) (λ x8 : (ι → ι) → ι → ι . setsum (Inj0 (setsum x5 x5)) (Inj1 0)) ⟶ x3 (λ x8 . x6 (λ x9 : (ι → ι) → ι . setsum (Inj1 (setsum 0 0)) (x7 (Inj1 0) 0))) (λ x8 : ι → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι . 0)) ⟶ (∀ x4 . ∀ x5 : (((ι → ι) → ι) → ι → ι → ι) → ι . ∀ x6 x7 . x3 (λ x8 . 0) (λ x8 : ι → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι . setsum (Inj1 (setsum x7 (Inj1 0))) 0) ⟶ False) ⟶ (∀ x4 : ι → ι → ι . ∀ x5 : (((ι → ι) → ι) → ι) → ι . ∀ x6 x7 . x1 (λ x8 : ((ι → ι) → ι) → ι → (ι → ι) → ι . λ x9 x10 . 0) (setsum (setsum (x5 (λ x8 : (ι → ι) → ι . setsum 0 0)) (Inj0 (Inj1 0))) (x4 (Inj1 (setsum 0 0)) (Inj0 0))) ⟶ x2 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι → ι → ι . Inj0 x8) (Inj1 (setsum x7 (setsum 0 (setsum 0 0))))) ⟶ (∀ x4 . ∀ x5 : ι → (ι → ι → ι) → ι . ∀ x6 . ∀ x7 : ((ι → ι) → ι → ι → ι) → ι . x2 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι → ι → ι . Inj0 (setsum (Inj1 (setsum 0 0)) x6)) (Inj0 (Inj0 (Inj1 (Inj1 0)))) ⟶ x1 (λ x8 : ((ι → ι) → ι) → ι → (ι → ι) → ι . λ x9 x10 . 0) 0) ⟶ (∀ x4 . ∀ x5 : (ι → (ι → ι) → ι → ι) → ι . ∀ x6 x7 : ι → ι . In (setsum 0 (setsum (setsum (x5 (λ x8 . λ x9 : ι → ι . λ x10 . 0)) (setsum 0 0)) (Inj1 (Inj0 0)))) (Inj0 (setsum (Inj0 (Inj0 0)) 0)) ⟶ x3 (λ x8 . setsum 0 (Inj0 (Inj1 (Inj0 0)))) (λ x8 : ι → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι . 0) ⟶ x1 (λ x8 : ((ι → ι) → ι) → ι → (ι → ι) → ι . λ x9 x10 . setsum (Inj1 0) (Inj1 0)) (setsum 0 (setsum 0 0))) ⟶ (∀ x4 : (ι → ι → ι) → ι → (ι → ι) → ι → ι . ∀ x5 : ι → (ι → ι → ι) → ι . ∀ x6 . ∀ x7 : ι → ι . x1 (λ x8 : ((ι → ι) → ι) → ι → (ι → ι) → ι . λ x9 x10 . 0) (setsum (x7 (x5 0 (λ x8 x9 . 0))) (setsum x6 x6)) ⟶ In (Inj1 (setsum (Inj0 (setsum 0 0)) (setsum (Inj1 0) (setsum 0 0)))) (x4 (λ x8 x9 . setsum (setsum x6 0) (Inj0 (x7 0))) (setsum 0 0) (λ x8 . 0) (setsum (x7 (setsum 0 0)) (Inj1 (Inj0 0))))) ⟶ (∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι . ∀ x7 . x3 (λ x8 . x7) (λ x8 : ι → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι . setsum 0 (Inj0 (setsum (x8 0 (λ x11 . 0) 0) 0))) ⟶ x0 (λ x8 . λ x9 : ι → (ι → ι) → ι → ι . x8) (setsum (setsum (Inj0 0) x4) 0) (λ x8 : (ι → ι) → ι → ι . Inj1 (setsum 0 (x6 (x5 0 0))))) ⟶ (∀ x4 x5 . ∀ x6 : ((ι → ι) → ι → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . ∀ x7 : (((ι → ι) → ι → ι) → ι) → ι . x0 (λ x8 . λ x9 : ι → (ι → ι) → ι → ι . Inj0 0) (setsum 0 (x7 (λ x8 : (ι → ι) → ι → ι . setsum (x7 (λ x9 : (ι → ι) → ι → ι . 0)) (Inj1 0)))) (λ x8 : (ι → ι) → ι → ι . x5) ⟶ False) ⟶ False) (proof)
Theorem 16fbe.. : not (∀ x0 : (ι → ι) → ι → ι → ο . ∀ x1 : ((ι → ι) → ι) → ((ι → ι → ι → ι) → ι) → (ι → ι → ι → ι) → ο . ∀ x2 : (ι → ι) → ((((ι → ι) → ι) → ι) → ((ι → ι) → ι → ι) → ι → ι → ι) → (((ι → ι) → ι) → ι → ι → ι) → ι → (ι → ι) → ι → ο . ∀ x3 : ((ι → (ι → ι) → ι) → (((ι → ι) → ι) → ι) → ι) → ((ι → ι → ι → ι) → ι) → (ι → ι → ι → ι) → ι → ο . (∀ x4 : ι → (ι → ι → ι) → ι . ∀ x5 . ∀ x6 : ι → ι → ι . ∀ x7 . x0 (λ x8 . 0) (setsum 0 (Inj1 (setsum x7 (setsum 0 0)))) (setsum (x4 (x6 (Inj0 0) (setsum 0 0)) (λ x8 x9 . x9)) (setsum 0 x7)) ⟶ x3 (λ x8 : ι → (ι → ι) → ι . λ x9 : ((ι → ι) → ι) → ι . 0) (λ x8 : ι → ι → ι → ι . 0) (λ x8 x9 x10 . setsum (Inj1 (Inj0 (Inj1 0))) (Inj0 0)) (setsum (Inj0 (setsum (Inj1 0) (Inj1 0))) 0)) ⟶ (∀ x4 : ((ι → ι → ι) → ι) → ι . ∀ x5 : ι → ι . ∀ x6 x7 . In x7 (Inj1 (Inj1 (Inj0 x6))) ⟶ x3 (λ x8 : ι → (ι → ι) → ι . λ x9 : ((ι → ι) → ι) → ι . 0) (λ x8 : ι → ι → ι → ι . setsum (Inj1 (Inj1 (setsum 0 0))) 0) (λ x8 x9 x10 . setsum 0 (Inj0 (Inj1 (Inj0 0)))) 0 ⟶ x3 (λ x8 : ι → (ι → ι) → ι . λ x9 : ((ι → ι) → ι) → ι . Inj1 (setsum (Inj1 (x8 0 (λ x10 . 0))) 0)) (λ x8 : ι → ι → ι → ι . setsum (Inj1 0) (setsum 0 0)) (λ x8 x9 x10 . x9) (x4 (λ x8 : ι → ι → ι . setsum x7 (setsum (Inj0 0) (x8 0 0))))) ⟶ (∀ x4 x5 x6 x7 . x2 (λ x8 . setsum (Inj1 0) (Inj1 0)) (λ x8 : ((ι → ι) → ι) → ι . λ x9 : (ι → ι) → ι → ι . λ x10 x11 . x9 (λ x12 . Inj0 (Inj0 (setsum 0 0))) 0) (λ x8 : (ι → ι) → ι . λ x9 x10 . Inj0 (Inj1 0)) (Inj1 (Inj0 (setsum 0 0))) (λ x8 . setsum (Inj1 0) (setsum 0 (setsum (Inj0 0) x5))) 0 ⟶ x2 (λ x8 . setsum 0 0) (λ x8 : ((ι → ι) → ι) → ι . λ x9 : (ι → ι) → ι → ι . λ x10 x11 . setsum (setsum x10 0) (setsum (Inj1 (Inj1 0)) (setsum (x8 (λ x12 : ι → ι . 0)) x10))) (λ x8 : (ι → ι) → ι . λ x9 x10 . Inj1 (x8 (λ x11 . setsum (setsum 0 0) (Inj0 0)))) (setsum (Inj1 (Inj0 x6)) (Inj1 (setsum 0 (Inj1 0)))) (λ x8 . Inj1 (Inj0 (setsum (Inj0 0) (Inj0 0)))) (setsum x7 (Inj1 0))) ⟶ (∀ x4 : ι → ι . ∀ x5 x6 x7 . In x7 (Inj0 0) ⟶ x2 (λ x8 . 0) (λ x8 : ((ι → ι) → ι) → ι . λ x9 : (ι → ι) → ι → ι . λ x10 x11 . x10) (λ x8 : (ι → ι) → ι . λ x9 x10 . 0) (Inj1 (setsum x7 (setsum (Inj0 0) 0))) (λ x8 . x8) (setsum (Inj1 (x4 (Inj0 0))) (Inj1 x7)) ⟶ x2 Inj0 (λ x8 : ((ι → ι) → ι) → ι . λ x9 : (ι → ι) → ι → ι . λ x10 x11 . x8 (λ x12 : ι → ι . 0)) (λ x8 : (ι → ι) → ι . λ x9 x10 . setsum x7 0) (setsum (setsum 0 0) 0) (λ x8 . Inj0 (Inj0 x5)) 0) ⟶ (∀ x4 : ι → ι . ∀ x5 . ∀ x6 : ι → ι . ∀ x7 : (ι → ι → ι → ι) → ι . In (setsum (Inj0 (Inj0 (Inj1 0))) (setsum 0 x5)) (x4 (setsum (setsum x5 (setsum 0 0)) (Inj1 x5))) ⟶ x2 (λ x8 . setsum (x7 (λ x9 x10 x11 . 0)) x5) (λ x8 : ((ι → ι) → ι) → ι . λ x9 : (ι → ι) → ι → ι . λ x10 x11 . 0) (λ x8 : (ι → ι) → ι . λ x9 x10 . x8 (λ x11 . setsum x10 0)) (setsum 0 (setsum (Inj1 0) (Inj0 x5))) (λ x8 . x7 (λ x9 x10 x11 . setsum (setsum (setsum 0 0) (Inj1 0)) (Inj0 (Inj1 0)))) (Inj1 (Inj1 0)) ⟶ x1 (λ x8 : ι → ι . 0) (λ x8 : ι → ι → ι → ι . Inj1 0) (λ x8 x9 x10 . x10)) ⟶ (∀ x4 : (ι → ι) → ι . ∀ x5 . ∀ x6 : (ι → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι . ∀ x7 : (ι → (ι → ι) → ι) → ι . x1 (λ x8 : ι → ι . Inj1 (setsum (x7 (λ x9 . λ x10 : ι → ι . 0)) (setsum 0 0))) (λ x8 : ι → ι → ι → ι . x6 (λ x9 x10 . x9) (λ x9 : ι → ι . λ x10 . x8 0 (x8 (setsum 0 0) (Inj1 0) (Inj0 0)) 0) (λ x9 . Inj1 (Inj0 0))) (λ x8 x9 x10 . x9) ⟶ x0 (λ x8 . setsum 0 (Inj1 (setsum 0 (x7 (λ x9 . λ x10 : ι → ι . 0))))) 0 (setsum (setsum (setsum (x6 (λ x8 x9 . 0) (λ x8 : ι → ι . λ x9 . 0) (λ x8 . 0)) 0) (setsum 0 (setsum 0 0))) (x6 (λ x8 x9 . x7 (λ x10 . λ x11 : ι → ι . Inj1 0)) (λ x8 : ι → ι . λ x9 . x8 (setsum 0 0)) (λ x8 . setsum 0 (x7 (λ x9 . λ x10 : ι → ι . 0)))))) ⟶ (∀ x4 : (ι → ι → ι → ι) → ι . ∀ x5 : ι → ι → ι . ∀ x6 : (((ι → ι) → ι → ι) → ι) → ι → ι → ι . ∀ x7 . x0 (λ x8 . 0) (setsum (Inj1 (x6 (λ x8 : (ι → ι) → ι → ι . 0) (Inj0 0) 0)) x7) (setsum (x4 (λ x8 x9 x10 . Inj0 (setsum 0 0))) 0) ⟶ x0 (λ x8 . setsum (setsum (Inj1 (Inj0 0)) (setsum (Inj0 0) 0)) (x5 0 (Inj0 (setsum 0 0)))) (Inj1 (Inj1 (x5 (x6 (λ x8 : (ι → ι) → ι → ι . 0) 0 0) (Inj0 0)))) (Inj1 (Inj1 (Inj1 0)))) ⟶ (∀ x4 x5 . ∀ x6 : ι → (ι → ι → ι) → ι . ∀ x7 : (((ι → ι) → ι) → ι) → ((ι → ι) → ι) → ι → ι → ι . x0 (λ x8 . x5) 0 (setsum 0 x5) ⟶ False) ⟶ False) (proof)
Theorem 5be0e.. : not (∀ x0 : ((((ι → ι → ι) → (ι → ι) → ι → ι) → ι) → (((ι → ι) → ι) → ι) → ι → ι → ι → ι) → ((ι → (ι → ι) → ι → ι) → ι → (ι → ι) → ι) → ι → ο . ∀ x1 : (((((ι → ι) → ι) → ι → ι) → ι → (ι → ι) → ι) → ι) → ((ι → (ι → ι) → ι → ι) → ι) → ο . ∀ x2 : ((ι → ι → (ι → ι) → ι → ι) → ι → ι) → ι → ο . ∀ x3 : ((ι → ι) → ι → (ι → ι → ι) → ι) → (ι → ι → ι → ι → ι) → ((ι → ι → ι) → ι → ι) → ο . (∀ x4 : ι → ι . ∀ x5 x6 . ∀ x7 : ι → (ι → ι) → (ι → ι) → ι . In (x7 (setsum 0 0) (λ x8 . 0) (setsum (setsum (setsum 0 0) (Inj0 0)))) (x7 (setsum (setsum (x4 0) (setsum 0 0)) (x4 (Inj0 0))) (λ x8 . setsum 0 (setsum (setsum 0 0) (setsum 0 0))) (λ x8 . 0)) ⟶ x0 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ι . λ x9 : ((ι → ι) → ι) → ι . λ x10 x11 x12 . setsum (x9 (λ x13 : ι → ι . x13 (x13 0))) (setsum 0 (x9 (λ x13 : ι → ι . 0)))) (λ x8 : ι → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι . x7 0 (λ x11 . setsum (setsum (setsum 0 0) (Inj0 0)) (Inj1 (setsum 0 0))) (λ x11 . setsum (x8 (setsum 0 0) (λ x12 . 0) (x8 0 (λ x12 . 0) 0)) (Inj0 0))) (Inj1 (Inj0 (setsum (Inj1 0) 0))) ⟶ x3 (λ x8 : ι → ι . λ x9 . λ x10 : ι → ι → ι . Inj1 (setsum (setsum (setsum 0 0) x9) 0)) (λ x8 x9 x10 x11 . x8) (λ x8 : ι → ι → ι . λ x9 . 0)) ⟶ (∀ x4 x5 . ∀ x6 : ι → ι . ∀ x7 : ι → (ι → ι) → ι . x3 (λ x8 : ι → ι . λ x9 . λ x10 : ι → ι → ι . Inj1 (Inj1 (setsum (setsum 0 0) (x10 0 0)))) (λ x8 x9 x10 x11 . setsum x9 (setsum (setsum x11 (Inj0 0)) x10)) (λ x8 : ι → ι → ι . λ x9 . x8 0 (Inj0 0)) ⟶ x2 (λ x8 : ι → ι → (ι → ι) → ι → ι . λ x9 . x7 x9 (λ x10 . Inj0 (Inj1 0))) 0) ⟶ (∀ x4 . ∀ x5 : (ι → ι) → ι . ∀ x6 : (ι → ι) → (ι → ι → ι) → ι . ∀ x7 : ι → (ι → ι → ι) → ι . x0 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ι . λ x9 : ((ι → ι) → ι) → ι . λ x10 x11 x12 . x12) (λ x8 : ι → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι . x9) (Inj1 (Inj0 0)) ⟶ x2 (λ x8 : ι → ι → (ι → ι) → ι → ι . λ x9 . 0) (Inj1 0)) ⟶ (∀ x4 x5 x6 x7 . x2 (λ x8 : ι → ι → (ι → ι) → ι → ι . λ x9 . x7) (setsum (setsum (setsum (Inj0 0) (Inj0 0)) 0) 0) ⟶ In (Inj0 0) x7) ⟶ (∀ x4 x5 . ∀ x6 : (((ι → ι) → ι) → (ι → ι) → ι) → ι → (ι → ι) → ι → ι . ∀ x7 . In (setsum 0 (setsum 0 0)) x5 ⟶ x2 (λ x8 : ι → ι → (ι → ι) → ι → ι . λ x9 . setsum (Inj1 0) (setsum 0 (x6 (λ x10 : (ι → ι) → ι . λ x11 : ι → ι . x11 0) 0 (λ x10 . x7) (Inj0 0)))) 0 ⟶ x1 (λ x8 : (((ι → ι) → ι) → ι → ι) → ι → (ι → ι) → ι . x6 (λ x9 : (ι → ι) → ι . λ x10 : ι → ι . setsum (x9 (λ x11 . Inj0 0)) 0) x5 (λ x9 . setsum x7 0) x5) (λ x8 : ι → (ι → ι) → ι → ι . Inj1 0)) ⟶ (∀ x4 . ∀ x5 : (ι → ι → ι) → ι → ι . ∀ x6 : (ι → ι → ι) → ((ι → ι) → ι) → ι → ι . ∀ x7 : ι → (ι → ι → ι) → ι . x1 (λ x8 : (((ι → ι) → ι) → ι → ι) → ι → (ι → ι) → ι . Inj0 0) (λ x8 : ι → (ι → ι) → ι → ι . 0) ⟶ x1 (λ x8 : (((ι → ι) → ι) → ι → ι) → ι → (ι → ι) → ι . 0) (λ x8 : ι → (ι → ι) → ι → ι . Inj1 (x6 (λ x9 x10 . Inj1 (setsum 0 0)) (λ x9 : ι → ι . x8 0 (λ x10 . setsum 0 0) 0) (Inj0 0)))) ⟶ (∀ x4 : (((ι → ι) → ι) → ι → ι → ι) → ι . ∀ x5 : (((ι → ι) → ι) → ι) → ι → (ι → ι) → ι → ι . ∀ x6 . ∀ x7 : ((ι → ι) → (ι → ι) → ι → ι) → (ι → ι → ι) → ι → ι → ι . x2 (λ x8 : ι → ι → (ι → ι) → ι → ι . λ x9 . Inj0 0) 0 ⟶ x0 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ι . λ x9 : ((ι → ι) → ι) → ι . λ x10 x11 x12 . setsum x10 x11) (λ x8 : ι → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι . setsum (x8 (x7 (λ x11 x12 : ι → ι . λ x13 . setsum 0 0) (λ x11 x12 . setsum 0 0) (setsum 0 0) (x8 0 (λ x11 . 0) 0)) (λ x11 . Inj1 0) (Inj1 (setsum 0 0))) 0) (x4 (λ x8 : (ι → ι) → ι . λ x9 x10 . 0))) ⟶ (∀ x4 x5 . ∀ x6 : ι → (ι → ι) → ι . ∀ x7 : (ι → ι → ι) → ι → ι . x0 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ι . λ x9 : ((ι → ι) → ι) → ι . λ x10 x11 x12 . 0) (λ x8 : ι → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι . setsum (Inj1 (Inj0 x9)) x9) (Inj0 (Inj1 (Inj0 x4))) ⟶ x0 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ι . λ x9 : ((ι → ι) → ι) → ι . λ x10 x11 x12 . x11) (λ x8 : ι → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι . Inj1 (x8 (Inj0 0) (λ x11 . Inj0 (x10 0)) (Inj0 (Inj1 0)))) (Inj1 (x7 (λ x8 x9 . setsum (setsum 0 0) x8) 0))) ⟶ False) (proof)
Theorem 3311e.. : not (∀ x0 : (ι → ι) → ι → ο . ∀ x1 x2 : ((ι → ι) → ι) → ι → ι → ο . ∀ x3 : (ι → ((ι → ι) → ι) → ι) → ι → (((ι → ι) → ι) → ι → ι) → ο . (∀ x4 x5 x6 x7 . In (setsum (setsum (setsum (Inj1 0) 0) (Inj0 (Inj0 0))) (setsum (setsum 0 0) (Inj0 (setsum 0 0)))) (setsum (setsum (setsum (setsum 0 0) x4) x7) 0) ⟶ x1 (λ x8 : ι → ι . setsum (setsum (Inj0 x5) (Inj0 0)) 0) (Inj0 (setsum (setsum (setsum 0 0) (setsum 0 0)) (Inj0 (Inj0 0)))) (setsum (Inj0 (setsum 0 (Inj1 0))) 0) ⟶ x3 (λ x8 . λ x9 : (ι → ι) → ι . x9 (λ x10 . x8)) x5 (λ x8 : (ι → ι) → ι . λ x9 . Inj0 0)) ⟶ (∀ x4 : ι → ι → ι . ∀ x5 . ∀ x6 : ι → ι → ι . ∀ x7 : ι → (ι → ι → ι) → ι . x3 (λ x8 . λ x9 : (ι → ι) → ι . setsum (x9 (λ x10 . setsum (setsum 0 0) x8)) 0) 0 (λ x8 : (ι → ι) → ι . λ x9 . setsum (x8 (λ x10 . x8 (λ x11 . setsum 0 0))) 0) ⟶ False) ⟶ (∀ x4 . ∀ x5 : ι → (ι → ι) → ι . ∀ x6 : ι → ι . ∀ x7 : (ι → ι) → (ι → ι → ι) → ι → ι . x0 (λ x8 . 0) (x6 (Inj0 (setsum (setsum 0 0) (setsum 0 0)))) ⟶ x2 (λ x8 : ι → ι . setsum (setsum (x8 0) 0) (x7 (λ x9 . 0) (λ x9 x10 . setsum (Inj0 0) 0) (x6 (Inj0 0)))) 0 (x7 (λ x8 . Inj1 (x5 0 (λ x9 . 0))) (λ x8 x9 . x8) 0)) ⟶ (∀ x4 . ∀ x5 : ι → ((ι → ι) → ι → ι) → (ι → ι) → ι . ∀ x6 x7 . x2 (λ x8 : ι → ι . x6) (setsum (setsum x4 x7) (setsum 0 0)) (Inj0 0) ⟶ x0 (λ x8 . Inj0 (x5 0 (λ x9 : ι → ι . λ x10 . Inj1 (Inj0 0)) (λ x9 . 0))) (setsum x6 (Inj0 (Inj0 (Inj1 0))))) ⟶ (∀ x4 . ∀ x5 : ι → ι . ∀ x6 : (ι → ι) → ι → ι . ∀ x7 . x0 (λ x8 . 0) (Inj1 x4) ⟶ x1 (λ x8 : ι → ι . setsum 0 (Inj0 (Inj1 (Inj1 0)))) x4 (setsum 0 0)) ⟶ (∀ x4 : ι → ι . ∀ x5 . ∀ x6 : ι → (ι → ι) → ι . ∀ x7 : (((ι → ι) → ι) → ι) → ι → ι → ι → ι . x1 (λ x8 : ι → ι . setsum 0 x5) (setsum (Inj1 0) (setsum (Inj0 (setsum 0 0)) 0)) (Inj1 (x7 (λ x8 : (ι → ι) → ι . x6 (Inj0 0) (λ x9 . Inj1 0)) (Inj1 (Inj0 0)) (Inj0 (setsum 0 0)) 0)) ⟶ In (Inj0 (Inj0 (x7 (λ x8 : (ι → ι) → ι . x8 (λ x9 . 0)) x5 (Inj1 0) (Inj0 0)))) (x4 x5)) ⟶ (∀ x4 x5 . ∀ x6 : ι → ι → ι . ∀ x7 . In (setsum x7 (setsum 0 0)) (Inj0 x7) ⟶ x0 (λ x8 . x8) (Inj1 0) ⟶ x0 (λ x8 . x6 0 (setsum 0 (Inj0 0))) (setsum (setsum (Inj1 x5) 0) (setsum (x6 0 x4) (setsum (Inj1 0) (x6 0 0))))) ⟶ (∀ x4 x5 . ∀ x6 : (ι → ι) → ι . ∀ x7 : (((ι → ι) → ι) → (ι → ι) → ι → ι) → (ι → ι → ι) → (ι → ι) → ι → ι . x0 (λ x8 . x6 (λ x9 . Inj1 0)) (Inj1 (setsum 0 (Inj1 (setsum 0 0)))) ⟶ False) ⟶ False) (proof)
Theorem 63b62.. : not (∀ x0 : (ι → (ι → (ι → ι) → ι) → ι) → ((ι → ι) → ι) → ο . ∀ x1 : ((ι → ((ι → ι) → ι) → ι) → ((ι → ι → ι) → ι) → ι → ι) → ι → ι → ((ι → ι) → ι → ι) → (ι → ι) → ο . ∀ x2 : (ι → (((ι → ι) → ι → ι) → ι) → ι) → ι → ι → ι → (ι → ι) → ι → ο . ∀ x3 : (ι → (ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι) → (((ι → ι → ι) → ι) → ι) → (ι → ι) → ο . (∀ x4 : ι → (ι → ι) → (ι → ι) → ι → ι . ∀ x5 x6 . ∀ x7 : (ι → ι) → ι → ι → ι . In (Inj0 x6) (setsum 0 (setsum 0 0)) ⟶ x2 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι . Inj1 x6) (setsum (Inj0 (Inj1 (setsum 0 0))) (Inj0 (setsum (setsum 0 0) (setsum 0 0)))) (x7 (λ x8 . setsum 0 0) 0 0) (setsum (x7 (setsum x6) (Inj1 (Inj1 0)) 0) x5) (λ x8 . 0) (x7 (λ x8 . Inj0 0) (x4 (setsum (Inj1 0) (Inj0 0)) (λ x8 . x6) (λ x8 . 0) 0) (Inj0 (setsum x5 (Inj1 0)))) ⟶ x3 (λ x8 . λ x9 : ι → ι . λ x10 : (ι → ι) → ι → ι . λ x11 : ι → ι . 0) (λ x8 : (ι → ι → ι) → ι . x6) (λ x8 . x5)) ⟶ (∀ x4 : ((ι → ι) → (ι → ι) → ι) → ι → ι → ι → ι . ∀ x5 x6 . ∀ x7 : ι → ι → (ι → ι) → ι → ι . x3 (λ x8 . λ x9 : ι → ι . λ x10 : (ι → ι) → ι → ι . λ x11 : ι → ι . x11 0) (λ x8 : (ι → ι → ι) → ι . 0) (λ x8 . 0) ⟶ x2 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι . Inj0 (x7 0 (Inj0 0) (λ x10 . 0) (Inj0 (setsum 0 0)))) x6 (x4 (λ x8 x9 : ι → ι . 0) (x4 (λ x8 x9 : ι → ι . 0) (Inj0 (Inj0 0)) 0 0) (setsum (setsum 0 x6) (setsum (Inj0 0) (Inj1 0))) x6) x6 (λ x8 . setsum (Inj1 (setsum (x7 0 0 (λ x9 . 0) 0) (setsum 0 0))) (setsum (setsum 0 (setsum 0 0)) (setsum x8 (Inj0 0)))) (Inj1 (x7 (setsum (setsum 0 0) 0) (setsum (setsum 0 0) (setsum 0 0)) (λ x8 . Inj0 (Inj1 0)) 0))) ⟶ (∀ x4 : (ι → ι) → ι . ∀ x5 x6 x7 . In x7 (setsum (Inj0 (x4 (λ x8 . 0))) 0) ⟶ x2 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι . 0) (x4 (λ x8 . x5)) (Inj1 (Inj0 (setsum (Inj1 0) 0))) (Inj0 0) (λ x8 . Inj1 0) (Inj0 (Inj0 0))) ⟶ (∀ x4 : (ι → ι) → ((ι → ι) → ι) → ι → ι . ∀ x5 x6 x7 . x2 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι . 0) x6 0 x7 (λ x8 . setsum 0 (Inj0 (Inj1 (Inj1 0)))) (x4 (λ x8 . 0) (λ x8 : ι → ι . Inj1 x6) (setsum 0 (setsum (setsum 0 0) (setsum 0 0)))) ⟶ x3 (λ x8 . λ x9 : ι → ι . λ x10 : (ι → ι) → ι → ι . λ x11 : ι → ι . 0) (λ x8 : (ι → ι → ι) → ι . 0) (λ x8 . setsum x5 0)) ⟶ (∀ x4 : ι → ι → ι → ι → ι . ∀ x5 . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ((ι → ι) → ι) → ι . x3 (λ x8 . λ x9 : ι → ι . λ x10 : (ι → ι) → ι → ι . λ x11 : ι → ι . setsum (Inj0 (Inj1 (x11 0))) (x9 0)) (λ x8 : (ι → ι → ι) → ι . 0) Inj0 ⟶ x1 (λ x8 : ι → ((ι → ι) → ι) → ι . λ x9 : (ι → ι → ι) → ι . λ x10 . setsum (x7 (x9 (λ x11 x12 . setsum 0 0)) (λ x11 : ι → ι . x10)) 0) (Inj1 0) x5 (λ x8 : ι → ι . λ x9 . 0) (λ x8 . 0)) ⟶ (∀ x4 . ∀ x5 : ι → ι . ∀ x6 x7 . x1 (λ x8 : ι → ((ι → ι) → ι) → ι . λ x9 : (ι → ι → ι) → ι . λ x10 . setsum (x9 (λ x11 x12 . 0)) (x8 0 (λ x11 : ι → ι . Inj0 (x11 0)))) (Inj0 (setsum (Inj1 0) (setsum x4 (setsum 0 0)))) (setsum x7 (Inj1 (setsum (setsum 0 0) 0))) (λ x8 : ι → ι . λ x9 . x7) (λ x8 . 0) ⟶ x1 (λ x8 : ι → ((ι → ι) → ι) → ι . λ x9 : (ι → ι → ι) → ι . λ x10 . x10) (Inj0 x7) 0 (λ x8 : ι → ι . λ x9 . 0) (λ x8 . x8)) ⟶ (∀ x4 x5 . ∀ x6 : ι → ι → ι . ∀ x7 . x0 (λ x8 . λ x9 : ι → (ι → ι) → ι . Inj0 (Inj1 (Inj1 (setsum 0 0)))) (λ x8 : ι → ι . 0) ⟶ x0 (λ x8 . λ x9 : ι → (ι → ι) → ι . setsum (Inj1 (Inj1 (setsum 0 0))) (setsum (x9 0 (λ x10 . setsum 0 0)) x7)) (λ x8 : ι → ι . setsum 0 (x6 (x6 0 0) x5))) ⟶ (∀ x4 : ι → ι → ι . ∀ x5 : ι → ι . ∀ x6 . ∀ x7 : ι → ι . In x6 (x5 (setsum (x4 0 (Inj1 0)) (x4 x6 (Inj0 0)))) ⟶ x0 (λ x8 . λ x9 : ι → (ι → ι) → ι . setsum (x9 0 (λ x10 . 0)) (x9 (Inj1 (x9 0 (λ x10 . 0))) (λ x10 . x8))) (λ x8 : ι → ι . x8 0) ⟶ x1 (λ x8 : ι → ((ι → ι) → ι) → ι . λ x9 : (ι → ι → ι) → ι . λ x10 . setsum (setsum (Inj0 (Inj1 0)) (Inj1 (setsum 0 0))) (x8 (setsum (x9 (λ x11 x12 . 0)) (setsum 0 0)) (λ x11 : ι → ι . x8 (setsum 0 0) (λ x12 : ι → ι . x10)))) (x4 (x4 (x4 (setsum 0 0) (Inj1 0)) (x7 0)) (setsum (setsum (x4 0 0) (setsum 0 0)) 0)) (x5 (x4 (setsum 0 (x5 0)) 0)) (λ x8 : ι → ι . λ x9 . Inj0 (setsum (Inj1 (Inj0 0)) 0)) (λ x8 . Inj1 (setsum (x7 (setsum 0 0)) (Inj1 x6)))) ⟶ False) (proof)
Theorem bfeb0.. : not (∀ x0 : (ι → ι) → ι → (ι → ι) → ο . ∀ x1 : (((((ι → ι) → ι) → (ι → ι) → ι) → ι → (ι → ι) → ι) → ι → ι) → ((ι → ι) → ι → (ι → ι) → ι) → ο . ∀ x2 : ((ι → ι) → ι → (ι → ι) → ι) → ι → ο . ∀ x3 : ((ι → (ι → ι) → ι) → ι → ι) → ι → ο . (∀ x4 : ι → ι . ∀ x5 : (((ι → ι) → ι) → ι → ι → ι) → ((ι → ι) → ι → ι) → ι . ∀ x6 : (ι → (ι → ι) → ι) → ι → ι . ∀ x7 . In (Inj1 0) x7 ⟶ x3 (λ x8 : ι → (ι → ι) → ι . λ x9 . Inj1 0) (Inj0 0)) ⟶ (∀ x4 : ι → ι → ι → ι . ∀ x5 : ι → (ι → ι → ι) → ι . ∀ x6 : (ι → (ι → ι) → ι → ι) → ι → (ι → ι) → ι → ι . ∀ x7 . In (setsum (x4 (setsum x7 (x6 (λ x8 . λ x9 : ι → ι . λ x10 . 0) 0 (λ x8 . 0) 0)) (Inj1 (Inj1 0)) 0) (Inj1 0)) (Inj0 x7) ⟶ x3 (λ x8 : ι → (ι → ι) → ι . λ x9 . Inj1 (setsum 0 (Inj1 x7))) (x6 (λ x8 . λ x9 : ι → ι . λ x10 . setsum (setsum x10 0) (Inj1 (setsum 0 0))) (Inj0 (x4 0 0 (setsum 0 0))) (λ x8 . Inj0 0) (Inj1 0)) ⟶ x3 (λ x8 : ι → (ι → ι) → ι . λ x9 . setsum (Inj0 (x8 x9 (λ x10 . Inj0 0))) (Inj1 x9)) (setsum (setsum (x5 (setsum 0 0) (λ x8 x9 . setsum 0 0)) (setsum (setsum 0 0) (x6 (λ x8 . λ x9 : ι → ι . λ x10 . 0) 0 (λ x8 . 0) 0))) x7)) ⟶ (∀ x4 x5 . ∀ x6 : (ι → ι → ι) → ι . ∀ x7 : ι → ι . x2 (λ x8 : ι → ι . λ x9 . λ x10 : ι → ι . setsum (setsum (setsum 0 (setsum 0 0)) (x10 0)) 0) 0) ⟶ (∀ x4 : ι → ι → ι . ∀ x5 . ∀ x6 : (ι → ι → ι → ι) → ι . ∀ x7 . x2 (λ x8 : ι → ι . λ x9 . λ x10 : ι → ι . x10 (setsum (setsum (setsum 0 0) 0) (x10 (Inj0 0)))) (setsum 0 (setsum (x4 (Inj0 0) x5) (Inj1 0))) ⟶ x2 (λ x8 : ι → ι . λ x9 . λ x10 : ι → ι . x8 (setsum x7 (Inj0 x7))) (Inj0 (setsum (setsum (setsum 0 0) (setsum 0 0)) (Inj0 (setsum 0 0))))) ⟶ (∀ x4 : ι → ι . ∀ x5 . ∀ x6 : (ι → ι) → (ι → ι → ι) → ι . ∀ x7 . x1 (λ x8 : (((ι → ι) → ι) → (ι → ι) → ι) → ι → (ι → ι) → ι . λ x9 . 0) (λ x8 : ι → ι . λ x9 . λ x10 : ι → ι . 0)) ⟶ (∀ x4 x5 . ∀ x6 : ι → ι → ι . ∀ x7 : (ι → ι) → (ι → ι) → ι → ι → ι . In x5 (Inj0 (Inj0 (x7 (λ x8 . x7 (λ x9 . 0) (λ x9 . 0) 0 0) (λ x8 . 0) (Inj0 0) (Inj0 0)))) ⟶ x1 (λ x8 : (((ι → ι) → ι) → (ι → ι) → ι) → ι → (ι → ι) → ι . λ x9 . x6 (setsum (Inj0 (x6 0 0)) (setsum 0 0)) (Inj0 (x6 (Inj0 0) 0))) (λ x8 : ι → ι . λ x9 . λ x10 : ι → ι . setsum 0 x9) ⟶ x3 (λ x8 : ι → (ι → ι) → ι . λ x9 . x7 (λ x10 . x10) (λ x10 . x8 (Inj0 (x8 0 (λ x11 . 0))) (λ x11 . setsum x11 (setsum 0 0))) (setsum 0 (setsum (x7 (λ x10 . 0) (λ x10 . 0) 0 0) (Inj0 0))) (setsum x9 0)) (Inj1 (x7 (λ x8 . 0) (λ x8 . setsum (setsum 0 0) (Inj0 0)) (x7 (λ x8 . Inj0 0) (λ x8 . Inj0 0) (Inj1 0) (setsum 0 0)) (x6 (x6 0 0) 0)))) ⟶ (∀ x4 : (ι → ι) → ι . ∀ x5 . ∀ x6 : ι → ι → ι . ∀ x7 : ι → ((ι → ι) → ι → ι) → ι . In (Inj0 0) (setsum (Inj0 (setsum (Inj1 0) (x6 0 0))) 0) ⟶ x0 (setsum x5) (Inj0 (x7 (setsum (Inj0 0) (x6 0 0)) (λ x8 : ι → ι . λ x9 . setsum (Inj0 0) 0))) (λ x8 . x6 x5 (setsum (Inj0 x5) (x6 x8 (setsum 0 0)))) ⟶ x0 (λ x8 . 0) (x4 (λ x8 . x5)) (λ x8 . Inj0 (setsum x5 (x6 (setsum 0 0) (setsum 0 0))))) ⟶ (∀ x4 . ∀ x5 : (((ι → ι) → ι) → ι) → ι → (ι → ι) → ι . ∀ x6 x7 : ι → ι . x0 (λ x8 . Inj1 (Inj0 (x5 (λ x9 : (ι → ι) → ι . Inj0 0) (x6 0) (λ x9 . setsum 0 0)))) (setsum 0 (Inj1 0)) (λ x8 . 0) ⟶ x0 (λ x8 . x5 (λ x9 : (ι → ι) → ι . 0) x8 (λ x9 . 0)) (Inj1 (Inj1 (x7 0))) (λ x8 . setsum (setsum (Inj0 (x7 0)) (Inj0 0)) (Inj1 (Inj0 (x7 0))))) ⟶ False) (proof)
Theorem ebfdd.. : not (∀ x0 : (ι → ι → ι) → ((ι → ι) → ι) → ι → ((ι → ι) → ι) → ι → ο . ∀ x1 : (ι → ((ι → ι) → (ι → ι) → ι → ι) → ι → ι → ι → ι) → ι → (ι → (ι → ι) → ι) → ο . ∀ x2 : (ι → ι) → ((ι → ι) → ι) → (((ι → ι) → ι) → (ι → ι) → ι) → ι → ι → ο . ∀ x3 : ((ι → ι → (ι → ι) → ι) → ι → ((ι → ι) → ι → ι) → ι → ι → ι) → ((ι → ι → ι → ι) → ι) → ι → ο . (∀ x4 : ι → ι . ∀ x5 : ((ι → ι) → (ι → ι) → ι) → ι . ∀ x6 x7 . x3 (λ x8 : ι → ι → (ι → ι) → ι . λ x9 . λ x10 : (ι → ι) → ι → ι . λ x11 x12 . Inj1 (Inj0 x11)) (λ x8 : ι → ι → ι → ι . setsum x6 (Inj1 (Inj0 (setsum 0 0)))) (Inj0 (setsum (setsum (x4 0) 0) 0))) ⟶ (∀ x4 : ι → (ι → ι) → ι . ∀ x5 . ∀ x6 : (((ι → ι) → ι) → (ι → ι) → ι) → ι . ∀ x7 . In (setsum 0 x5) (setsum (Inj0 0) (Inj1 0)) ⟶ x3 (λ x8 : ι → ι → (ι → ι) → ι . λ x9 . λ x10 : (ι → ι) → ι → ι . λ x11 x12 . setsum (Inj0 x11) (Inj0 (Inj0 (x10 (λ x13 . 0) 0)))) (λ x8 : ι → ι → ι → ι . 0) 0 ⟶ x3 (λ x8 : ι → ι → (ι → ι) → ι . λ x9 . λ x10 : (ι → ι) → ι → ι . λ x11 x12 . setsum 0 (Inj1 (setsum (Inj1 0) 0))) (λ x8 : ι → ι → ι → ι . x5) (setsum x7 (Inj0 (Inj0 0)))) ⟶ (∀ x4 x5 x6 . ∀ x7 : ι → ι . x3 (λ x8 : ι → ι → (ι → ι) → ι . λ x9 . λ x10 : (ι → ι) → ι → ι . λ x11 x12 . x11) (λ x8 : ι → ι → ι → ι . x5) (Inj0 x5) ⟶ x2 (λ x8 . x7 x5) (λ x8 : ι → ι . 0) (λ x8 : (ι → ι) → ι . λ x9 : ι → ι . setsum (x8 (λ x10 . Inj1 (setsum 0 0))) (setsum (setsum (x8 (λ x10 . 0)) (x7 0)) (Inj0 (Inj0 0)))) (Inj1 (Inj1 (setsum x5 (Inj1 0)))) (setsum (Inj1 (setsum (setsum 0 0) x5)) (setsum (Inj1 x6) (Inj1 (Inj1 0))))) ⟶ (∀ x4 : (((ι → ι) → ι → ι) → (ι → ι) → ι) → (ι → ι) → (ι → ι) → ι → ι . ∀ x5 x6 x7 . In (Inj0 x5) (Inj1 (Inj1 (Inj0 0))) ⟶ x2 (λ x8 . x6) (λ x8 : ι → ι . Inj0 (Inj0 0)) (λ x8 : (ι → ι) → ι . λ x9 : ι → ι . 0) (Inj1 (setsum 0 (Inj0 0))) x5 ⟶ x2 (λ x8 . 0) (λ x8 : ι → ι . x5) (λ x8 : (ι → ι) → ι . λ x9 : ι → ι . Inj1 0) (Inj1 x5) x6) ⟶ (∀ x4 . ∀ x5 : ι → ι . ∀ x6 : (ι → ι → ι → ι) → (ι → ι) → (ι → ι) → ι . ∀ x7 . In (setsum (Inj1 (setsum (setsum 0 0) (setsum 0 0))) (setsum (Inj0 (x5 0)) 0)) (Inj1 (setsum (setsum (Inj1 0) (setsum 0 0)) (Inj0 (x6 (λ x8 x9 x10 . 0) (λ x8 . 0) (λ x8 . 0))))) ⟶ x0 (λ x8 x9 . setsum (setsum 0 0) (x6 (λ x10 x11 x12 . Inj0 (Inj1 0)) (λ x10 . 0) (λ x10 . x8))) (λ x8 : ι → ι . Inj0 (setsum (x6 (λ x9 x10 x11 . Inj0 0) (λ x9 . 0) (λ x9 . Inj0 0)) 0)) (Inj0 (setsum (setsum (setsum 0 0) 0) (Inj0 (setsum 0 0)))) (λ x8 : ι → ι . Inj0 (setsum 0 0)) (setsum x7 (Inj0 (Inj0 0))) ⟶ x1 (λ x8 . λ x9 : (ι → ι) → (ι → ι) → ι → ι . λ x10 x11 x12 . x11) (Inj1 (setsum (Inj0 0) 0)) (λ x8 . λ x9 : ι → ι . 0)) ⟶ (∀ x4 . ∀ x5 : ι → ι . ∀ x6 x7 . In (Inj0 0) (Inj1 (setsum 0 (Inj0 x6))) ⟶ x1 (λ x8 . λ x9 : (ι → ι) → (ι → ι) → ι → ι . λ x10 x11 x12 . 0) 0 (λ x8 . λ x9 : ι → ι . x9 (setsum 0 (Inj1 x7))) ⟶ x1 (λ x8 . λ x9 : (ι → ι) → (ι → ι) → ι → ι . λ x10 x11 x12 . setsum (setsum 0 (Inj0 (Inj0 0))) (Inj1 (x9 (λ x13 . setsum 0 0) (λ x13 . x13) x11))) x6 (λ x8 . λ x9 : ι → ι . setsum (setsum 0 0) 0)) ⟶ (∀ x4 . ∀ x5 : ι → ι . ∀ x6 x7 . In (x5 0) (setsum (Inj0 (setsum 0 (x5 0))) (setsum x7 (setsum (setsum 0 0) 0))) ⟶ x0 (λ x8 x9 . 0) (λ x8 : ι → ι . x8 (setsum (x8 (x5 0)) (Inj0 0))) (Inj1 x6) (λ x8 : ι → ι . Inj1 (setsum (setsum (x8 0) x6) (Inj1 (setsum 0 0)))) 0) ⟶ (∀ x4 . ∀ x5 : (ι → ι) → ι . ∀ x6 . ∀ x7 : ι → ι . x0 (λ x8 x9 . x8) (λ x8 : ι → ι . setsum 0 (setsum 0 (x7 (x5 (λ x9 . 0))))) 0 (λ x8 : ι → ι . Inj1 (Inj0 0)) (x7 0) ⟶ x0 (λ x8 x9 . 0) (λ x8 : ι → ι . x8 0) (setsum (setsum 0 (Inj1 (Inj1 0))) 0) (λ x8 : ι → ι . 0) (Inj0 (Inj1 0))) ⟶ False) (proof)

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