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5e9ba../49dc4.. bday: 11884 doc published by PrGVS..
Known 8106d..notI : ∀ x0 : ο . (x0 ⟶ False) ⟶ not x0
Known FalseEFalseE : False ⟶ ∀ x0 : ο . x0
Theorem 7a4d9.. : 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) → ο . ∀ x3 : ((((ι → ι) → ι) → ι → ι) → ι) → ι → ο . (∀ x4 : ι → ι . ∀ x5 x6 . ∀ x7 : ι → ι . x0 (λ x8 . λ x9 : ι → ι . 0) (λ x8 : ι → (ι → ι) → ι . setsum 0 x5) ⟶ x3 (λ x8 : ((ι → ι) → ι) → ι → ι . x8 (λ x9 : ι → ι . Inj1 0) x6) (setsum (Inj0 0) (setsum (Inj1 (x4 0)) (Inj1 (Inj0 0))))) ⟶ (∀ x4 : (((ι → ι) → ι) → (ι → ι) → ι → ι) → ι . ∀ x5 : ι → (ι → ι) → ι → ι . ∀ x6 : ι → ((ι → ι) → ι) → ι → ι . ∀ x7 . x3 (λ x8 : ((ι → ι) → ι) → ι → ι . x8 (λ x9 : ι → ι . Inj0 x7) 0) (Inj1 x7) ⟶ False) ⟶ (∀ x4 . ∀ x5 : (ι → ι) → ((ι → ι) → ι) → ι . ∀ x6 : (ι → ι → ι) → ι . ∀ x7 . x2 (λ x8 : ι → ι . λ x9 . setsum (x8 (setsum 0 (setsum 0 0))) x7) (λ x8 . λ x9 : (ι → ι) → ι → ι . x6 (λ x10 x11 . setsum (setsum x11 (setsum 0 0)) x11)) ⟶ x2 (λ x8 : ι → ι . λ x9 . 0) (λ x8 . λ x9 : (ι → ι) → ι → ι . Inj1 (Inj1 0))) ⟶ (∀ x4 : (ι → ι) → ι . ∀ x5 x6 x7 . x2 (λ x8 : ι → ι . Inj1) (λ x8 . λ x9 : (ι → ι) → ι → ι . 0) ⟶ False) ⟶ (∀ x4 : ι → ι → ι → ι → ι . ∀ x5 x6 . ∀ x7 : ι → ι . In (Inj1 0) (setsum (Inj0 (x4 (setsum 0 0) (Inj1 0) (Inj1 0) (setsum 0 0))) 0) ⟶ x1 (λ x8 . λ x9 : ι → ι . λ x10 x11 . setsum (Inj1 (setsum 0 (Inj1 0))) (x9 (Inj1 0))) (λ x8 : ι → (ι → ι) → ι → ι . λ x9 x10 x11 . x11) (λ x8 : (ι → ι) → ι . 0) ⟶ x1 (λ x8 . λ x9 : ι → ι . λ x10 x11 . 0) (λ x8 : ι → (ι → ι) → ι → ι . λ x9 x10 x11 . setsum (setsum 0 (Inj0 0)) (setsum (Inj0 0) 0)) (λ x8 : (ι → ι) → ι . setsum (Inj0 (x7 (setsum 0 0))) 0)) ⟶ (∀ x4 : ι → ι . ∀ x5 x6 x7 . x1 (λ x8 . λ x9 : ι → ι . λ x10 x11 . setsum (Inj0 (setsum (setsum 0 0) (Inj0 0))) (setsum (setsum (setsum 0 0) 0) (Inj1 0))) (λ x8 : ι → (ι → ι) → ι → ι . λ x9 x10 x11 . 0) (λ x8 : (ι → ι) → ι . setsum 0 (Inj0 x6)) ⟶ In (Inj1 (Inj1 0)) x7) ⟶ (∀ x4 x5 : ι → ι → ι . ∀ x6 x7 . x0 (λ x8 . λ x9 : ι → ι . 0) (λ x8 : ι → (ι → ι) → ι . x7) ⟶ x0 (λ x8 . λ x9 : ι → ι . x6) (λ x8 : ι → (ι → ι) → ι . 0)) ⟶ (∀ x4 : ι → ι → ι . ∀ x5 . ∀ x6 x7 : ι → ι . x0 (λ x8 . λ x9 : ι → ι . Inj0 (setsum 0 (Inj1 0))) (λ x8 : ι → (ι → ι) → ι . 0) ⟶ In (setsum 0 0) (Inj0 0)) ⟶ False) (proof)
Known TrueITrueI : True
Theorem 9033d.. : not (∀ x0 : ((ι → ι → (ι → ι) → ι) → ι) → ι → ι → ο . ∀ x1 : (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) → ο . (∀ x4 : ι → ι → ι → ι → ι . ∀ x5 x6 . ∀ x7 : ι → ι . x3 (λ x8 . λ x9 : ι → ι . x9 (Inj1 (Inj0 x6))) (λ x8 : (ι → ι) → ι . setsum (Inj1 (setsum x6 0)) (Inj0 0))) ⟶ (∀ x4 : ι → ι . ∀ x5 : (((ι → ι) → ι → ι) → ι) → ι . ∀ x6 : (((ι → ι) → ι) → ι → ι) → ι . ∀ x7 . In (setsum (Inj0 (setsum (Inj0 0) 0)) (Inj1 0)) (Inj0 x7) ⟶ x3 (λ x8 . λ x9 : ι → ι . x8) (λ x8 : (ι → ι) → ι . Inj0 (setsum (setsum (setsum 0 0) 0) (setsum (setsum 0 0) 0))) ⟶ x3 (λ x8 . λ x9 : ι → ι . 0) (λ x8 : (ι → ι) → ι . 0)) ⟶ (∀ x4 x5 . ∀ x6 : ι → ι → ι . ∀ x7 : (ι → ι) → ι → ι . x1 (λ x8 . x5) (setsum x5 (x6 (Inj1 0) (Inj1 (Inj0 0)))) ⟶ x2 (λ x8 : ((ι → ι) → ι) → ((ι → ι) → ι → ι) → ι . x6 (x8 (λ x9 : ι → ι . 0) (λ x9 : ι → ι . λ x10 . 0)) 0) (x6 0 x4) (λ x8 x9 . Inj1 (x7 (λ x10 . Inj1 x9) 0))) ⟶ (∀ x4 : ι → ι → (ι → ι) → ι → ι . ∀ x5 x6 x7 . x2 (λ x8 : ((ι → ι) → ι) → ((ι → ι) → ι → ι) → ι . Inj1 0) 0 (λ x8 x9 . setsum (Inj0 0) x7) ⟶ In (Inj0 (Inj0 (setsum x6 (Inj0 0)))) (Inj0 (setsum (Inj0 (setsum 0 0)) (setsum (x4 0 0 (λ x8 . 0) 0) x7)))) ⟶ (∀ x4 x5 . ∀ x6 : ((ι → ι) → (ι → ι) → ι → ι) → (ι → ι → ι) → ι . ∀ x7 . In (Inj1 x5) (Inj1 (setsum (Inj1 (x6 (λ x8 x9 : ι → ι . λ x10 . 0) (λ x8 x9 . 0))) (Inj1 (setsum 0 0)))) ⟶ x1 (λ x8 . x6 (λ x9 x10 : ι → ι . λ x11 . Inj0 0) (λ x9 x10 . setsum (Inj0 0) 0)) (setsum 0 (setsum (setsum 0 (setsum 0 0)) (setsum 0 0))) ⟶ x1 (λ x8 . 0) (setsum 0 (Inj0 0))) ⟶ (∀ x4 . ∀ x5 : (ι → ι) → ι → ι . ∀ x6 x7 . x1 (λ x8 . 0) (Inj1 (setsum (Inj1 (setsum 0 0)) (Inj1 0))) ⟶ False) ⟶ (∀ x4 x5 . ∀ x6 : ι → ι . ∀ x7 . In x5 (setsum (setsum 0 0) (Inj1 (setsum (Inj0 0) x4))) ⟶ x1 (λ x8 . x7) (Inj1 0) ⟶ x0 (λ x8 : ι → ι → (ι → ι) → ι . setsum (setsum x7 0) (x6 0)) 0 (setsum (Inj0 (setsum 0 (Inj0 0))) 0)) ⟶ (∀ x4 : ι → ι → ι . ∀ x5 . ∀ x6 : (((ι → ι) → ι) → (ι → ι) → ι) → ι → (ι → ι) → ι → ι . ∀ x7 . x0 (λ x8 : ι → ι → (ι → ι) → ι . 0) (setsum 0 (Inj0 (setsum x5 (Inj1 0)))) (setsum (Inj0 (setsum (setsum 0 0) (setsum 0 0))) (Inj0 (Inj1 0))) ⟶ In (Inj1 (Inj1 (Inj1 0))) (setsum 0 (setsum 0 (Inj1 0)))) ⟶ False) (proof)
Theorem 8e3ed.. : not (∀ x0 : ((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 → ο . ∀ x3 : (ι → (ι → ι) → ((ι → ι) → ι) → ι) → (((ι → ι) → ι) → ι) → ((ι → ι) → ι → ι → ι) → ο . (∀ x4 . ∀ x5 : (ι → ι → ι) → ι . ∀ x6 x7 . x3 (λ x8 . λ x9 : ι → ι . λ x10 : (ι → ι) → ι . 0) (λ x8 : (ι → ι) → ι . 0) (λ x8 : ι → ι . λ x9 x10 . Inj1 (setsum (Inj0 x9) 0))) ⟶ (∀ x4 x5 x6 x7 . x3 (λ x8 . λ x9 : ι → ι . λ x10 : (ι → ι) → ι . setsum (Inj1 0) (setsum (Inj0 (setsum 0 0)) x7)) (λ x8 : (ι → ι) → ι . x6) (λ x8 : ι → ι . λ x9 x10 . x10) ⟶ x3 (λ x8 . λ x9 : ι → ι . λ x10 : (ι → ι) → ι . Inj1 (x10 (λ x11 . setsum (setsum 0 0) 0))) (λ x8 : (ι → ι) → ι . 0) (λ x8 : ι → ι . λ x9 x10 . setsum x10 x7)) ⟶ (∀ x4 : (ι → (ι → ι) → ι) → ((ι → ι) → ι → ι) → ι → ι → ι . ∀ x5 x6 x7 . In (x4 (λ x8 . λ x9 : ι → ι . x6) (λ x8 : ι → ι . λ x9 . Inj1 (Inj0 (setsum 0 0))) (setsum 0 (setsum (setsum 0 0) 0)) x5) (setsum 0 (setsum (setsum (setsum 0 0) 0) x7)) ⟶ x0 (λ x8 : ι → ι → ι . x7) (Inj1 (setsum x5 (Inj1 x6))) ⟶ x2 (λ x8 . λ x9 : ι → ι → ι → ι . Inj1 0) (λ x8 . x5) (λ x8 . 0) x6 0) ⟶ (∀ x4 : (ι → ι → ι) → ι . ∀ x5 : ι → ((ι → ι) → ι) → (ι → ι) → ι → ι . ∀ x6 : ι → ι . ∀ x7 . In (x6 (Inj0 (setsum (Inj1 0) (Inj1 0)))) (setsum (x5 (x5 0 (λ x8 : ι → ι . setsum 0 0) (λ x8 . Inj1 0) (setsum 0 0)) (λ x8 : ι → ι . setsum (x6 0) 0) (λ x8 . x5 0 (λ x9 : ι → ι . 0) (λ x9 . x7) (setsum 0 0)) 0) (x4 (λ x8 x9 . Inj1 (setsum 0 0)))) ⟶ x2 (λ x8 . λ x9 : ι → ι → ι → ι . x6 (Inj1 x7)) (λ x8 . x8) (λ x8 . setsum 0 0) (x5 0 (λ x8 : ι → ι . x5 0 (λ x9 : ι → ι . setsum (setsum 0 0) (setsum 0 0)) (λ x9 . Inj1 (x6 0)) 0) (λ x8 . setsum (x5 (setsum 0 0) (λ x9 : ι → ι . Inj0 0) (λ x9 . Inj0 0) (Inj1 0)) (Inj1 0)) (x4 (λ x8 x9 . 0))) (x6 0) ⟶ x0 (λ x8 : ι → ι → ι . x5 (Inj1 0) (λ x9 : ι → ι . Inj0 0) Inj1 (Inj1 0)) (Inj1 (x6 (Inj0 x7)))) ⟶ (∀ x4 : ((ι → ι → ι) → ι → ι) → ι → ι . ∀ x5 x6 x7 . In x5 x5 ⟶ x1 (λ x8 . Inj1 x5) (λ x8 : (ι → ι → ι) → ι → ι . 0)) ⟶ (∀ x4 : ι → ι . ∀ x5 x6 . ∀ x7 : (ι → ι) → ι . In (Inj0 (setsum (Inj0 (Inj1 0)) 0)) (Inj0 (setsum (x4 (Inj1 0)) x6)) ⟶ x1 (λ x8 . 0) (λ x8 : (ι → ι → ι) → ι → ι . x5) ⟶ x3 (λ x8 . λ x9 : ι → ι . λ x10 : (ι → ι) → ι . setsum (Inj1 (setsum 0 0)) (Inj1 x8)) (λ x8 : (ι → ι) → ι . Inj1 (setsum (Inj1 0) 0)) (λ x8 : ι → ι . λ x9 x10 . x10)) ⟶ (∀ x4 : ((ι → ι → ι) → (ι → ι) → ι) → ((ι → ι) → ι) → ι → ι → ι . ∀ x5 . ∀ x6 : ι → ((ι → ι) → ι) → ι . ∀ x7 : ι → ι . In (Inj0 0) (setsum (x6 (setsum (setsum 0 0) (Inj1 0)) (λ x8 : ι → ι . x5)) (x6 0 (λ x8 : ι → ι . x5))) ⟶ x0 (λ x8 : ι → ι → ι . 0) (setsum 0 (Inj1 (setsum (Inj0 0) (x4 (λ x8 : ι → ι → ι . λ x9 : ι → ι . 0) (λ x8 : ι → ι . 0) 0 0))))) ⟶ (∀ x4 : ι → (ι → ι) → ι . ∀ x5 : ι → (ι → ι → ι) → ι . ∀ x6 : ι → ι → ι . ∀ x7 . x0 (λ x8 : ι → ι → ι . Inj1 (setsum (setsum (setsum 0 0) 0) (setsum (Inj0 0) 0))) 0 ⟶ x3 (λ x8 . λ x9 : ι → ι . λ x10 : (ι → ι) → ι . setsum 0 (x10 (λ x11 . setsum 0 0))) (λ x8 : (ι → ι) → ι . Inj0 (Inj1 (Inj1 (setsum 0 0)))) (λ x8 : ι → ι . λ x9 x10 . setsum (x8 0) x7)) ⟶ False) (proof)
Theorem 29cbb.. : not (∀ x0 : (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) → ο . ∀ x3 : (ι → ι) → ι → (((ι → ι) → ι → ι) → ι → ι) → ο . (∀ x4 : ι → ι → ι . ∀ x5 : ((ι → ι → ι) → ι → ι) → ι . ∀ x6 : (ι → ι) → ι . ∀ x7 . In (setsum 0 (setsum (x6 (λ x8 . Inj0 0)) 0)) (setsum (x6 (λ x8 . Inj0 (setsum 0 0))) (Inj0 (setsum (setsum 0 0) (x6 (λ x8 . 0))))) ⟶ x0 (λ x8 . setsum (Inj1 (Inj1 (setsum 0 0))) (x6 (λ x9 . Inj0 x8))) (x4 (Inj1 (Inj1 (Inj0 0))) (setsum 0 (Inj1 (x6 (λ x8 . 0))))) ⟶ x3 (λ x8 . x7) (x5 (λ x8 : ι → ι → ι . λ x9 . setsum 0 (Inj1 (x8 0 0)))) (λ x8 : (ι → ι) → ι → ι . λ x9 . setsum (Inj1 x7) (x6 (λ x10 . x10)))) ⟶ (∀ x4 : (((ι → ι) → ι) → ι) → ι . ∀ x5 : ι → (ι → ι) → ι . ∀ x6 : (((ι → ι) → ι) → ι) → ι → ι . ∀ x7 . In (x4 (λ x8 : (ι → ι) → ι . 0)) (Inj1 (setsum (Inj1 0) (setsum (Inj0 0) (setsum 0 0)))) ⟶ x3 (λ x8 . Inj0 0) (Inj0 (Inj0 (x5 0 (λ x8 . 0)))) (λ x8 : (ι → ι) → ι → ι . λ x9 . 0) ⟶ x0 (λ x8 . Inj0 (Inj0 (Inj1 x8))) (x6 (λ x8 : (ι → ι) → ι . x5 (setsum 0 0) (setsum (Inj0 0))) (setsum (x4 (λ x8 : (ι → ι) → ι . Inj1 0)) 0))) ⟶ (∀ x4 : (ι → ι → ι) → ι . ∀ x5 : ((ι → ι) → ι → ι → ι) → ((ι → ι) → ι) → ι . ∀ x6 : ((ι → ι) → ι) → ι → ι → ι → ι . ∀ x7 . x3 (λ x8 . Inj1 0) 0 (λ x8 : (ι → ι) → ι → ι . λ x9 . 0) ⟶ x2 (λ x8 : (((ι → ι) → ι) → ι → ι) → ι . 0) (λ x8 : ι → ι → ι . λ x9 : (ι → ι) → ι . setsum (x9 (λ x10 . x8 (setsum 0 0) 0)) (setsum 0 0)) (setsum (Inj1 (Inj0 (setsum 0 0))) (Inj0 0)) (λ x8 : ι → ι . x7)) ⟶ (∀ x4 x5 . ∀ x6 : ι → ι → ι → ι . ∀ x7 . x2 (λ x8 : (((ι → ι) → ι) → ι → ι) → ι . x5) (λ x8 : ι → ι → ι . λ x9 : (ι → ι) → ι . 0) (Inj1 (Inj0 0)) (λ x8 : ι → ι . setsum (setsum (Inj1 (x6 0 0 0)) (setsum x7 (x8 0))) x5) ⟶ x3 (λ x8 . Inj0 (Inj1 (setsum (Inj0 0) (setsum 0 0)))) (x6 0 0 0) (λ x8 : (ι → ι) → ι → ι . λ x9 . Inj0 (Inj1 0))) ⟶ (∀ x4 x5 x6 x7 . x3 (λ x8 . setsum (setsum (setsum (Inj1 0) 0) (Inj1 (setsum 0 0))) 0) (setsum (setsum (setsum 0 (setsum 0 0)) (setsum x5 x4)) (Inj1 0)) (λ x8 : (ι → ι) → ι → ι . λ x9 . x7) ⟶ x1 (λ x8 : (ι → ι → ι → ι) → ι . λ x9 . Inj1 x7) (λ x8 x9 . setsum x6 0) (λ x8 . 0)) ⟶ (∀ x4 : ι → ι . ∀ x5 x6 x7 . x1 (λ x8 : (ι → ι → ι → ι) → ι . λ x9 . Inj0 (Inj0 (setsum x7 (setsum 0 0)))) (λ x8 x9 . x7) (λ x8 . 0) ⟶ x3 (λ x8 . 0) 0 (λ x8 : (ι → ι) → ι → ι . setsum (setsum 0 (x8 (λ x9 . x7) 0)))) ⟶ (∀ x4 x5 . ∀ x6 : ι → ι . ∀ x7 . x0 (λ x8 . x5) (Inj0 0) ⟶ x0 (λ x8 . setsum x5 (Inj1 (Inj1 (x6 0)))) (setsum (Inj1 (setsum 0 (Inj1 0))) (setsum (x6 x7) (Inj1 (Inj0 0))))) ⟶ (∀ x4 . ∀ x5 : ((ι → ι) → ι → ι → ι) → (ι → ι) → ι . ∀ x6 : ι → ι → ι → ι → ι . ∀ x7 : ι → ι . x0 (λ x8 . setsum (setsum (setsum x8 (x6 0 0 0 0)) x8) (setsum 0 0)) (setsum (x5 (λ x8 : ι → ι . λ x9 x10 . setsum x9 (setsum 0 0)) (λ x8 . x8)) (setsum (x6 (setsum 0 0) (setsum 0 0) (Inj1 0) (x7 0)) (setsum 0 (x6 0 0 0 0)))) ⟶ False) ⟶ False) (proof)
Theorem 91710.. : not (∀ x0 : (ι → ι) → (ι → ι) → ι → ο . ∀ x1 : (ι → ι) → (ι → ι) → (ι → (ι → ι) → ι → ι) → ο . ∀ x2 : (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) → ο . (∀ x4 . ∀ x5 : (ι → ι) → ι → ι . ∀ x6 : (((ι → ι) → ι → ι) → ι) → ι . ∀ x7 . x2 (λ x8 . x7) (λ x8 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . x8 (λ x9 : ι → ι . λ x10 . setsum (x8 (λ x11 : ι → ι . λ x12 . setsum 0 0) (λ x11 . x8 (λ x12 : ι → ι . λ x13 . 0) (λ x12 . 0) 0) 0) (setsum (setsum 0 0) x10)) (λ x9 . Inj1 0) 0) ⟶ x3 (λ x8 : ι → ((ι → ι) → ι → ι) → (ι → ι) → ι . setsum (Inj0 (x6 (λ x9 : (ι → ι) → ι → ι . 0))) (Inj1 0)) (λ x8 : ι → (ι → ι) → ι . λ x9 x10 . x8 x7 (λ x11 . setsum (x8 (Inj0 0) (λ x12 . x11)) (Inj1 x11)))) ⟶ (∀ x4 : (((ι → ι) → ι) → (ι → ι) → ι) → (ι → ι → ι) → ι → ι → ι . ∀ x5 x6 x7 . In (Inj1 x5) (x4 (λ x8 : (ι → ι) → ι . λ x9 : ι → ι . Inj1 (setsum (setsum 0 0) 0)) (λ x8 x9 . x9) 0 (setsum 0 x6)) ⟶ x3 (λ x8 : ι → ((ι → ι) → ι → ι) → (ι → ι) → ι . setsum x6 x6) (λ x8 : ι → (ι → ι) → ι . λ x9 x10 . setsum (setsum (Inj1 (Inj1 0)) 0) (Inj0 (Inj0 x7))) ⟶ x1 (λ x8 . setsum 0 0) (λ x8 . setsum (setsum (setsum (Inj0 0) (Inj1 0)) x5) x7) (λ x8 . λ x9 : ι → ι . λ x10 . 0)) ⟶ (∀ x4 x5 x6 x7 . In (setsum (Inj1 (Inj0 (Inj1 0))) x5) (Inj1 0) ⟶ x2 (λ x8 . setsum (Inj1 0) (Inj1 (setsum 0 (setsum 0 0)))) (λ x8 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . Inj1 0)) ⟶ (∀ x4 : ι → (ι → ι) → ι . ∀ x5 x6 x7 . x2 (λ x8 . Inj0 0) (λ x8 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . 0) ⟶ x0 (λ x8 . x8) (λ x8 . 0) (setsum (Inj1 (Inj1 x7)) (setsum (Inj0 (Inj0 0)) (Inj1 (setsum 0 0))))) ⟶ (∀ x4 : ι → ι → ι . ∀ x5 : (ι → ι → ι → ι) → (ι → ι → ι) → ι → ι . ∀ x6 . ∀ x7 : ι → ι . x1 (λ x8 . Inj0 (Inj0 (setsum (setsum 0 0) (setsum 0 0)))) (λ x8 . x8) (λ x8 . λ x9 : ι → ι . λ x10 . Inj0 (x9 0))) ⟶ (∀ x4 . ∀ x5 x6 : ι → ι → ι . ∀ x7 . In x7 (setsum x7 (x5 (x5 (x6 0 0) (setsum 0 0)) (x5 (x6 0 0) 0))) ⟶ x1 (λ x8 . x7) (λ x8 . 0) (λ x8 . λ x9 : ι → ι . λ x10 . x8) ⟶ x3 (λ x8 : ι → ((ι → ι) → ι → ι) → (ι → ι) → ι . Inj1 (Inj1 (setsum (setsum 0 0) (Inj0 0)))) (λ x8 : ι → (ι → ι) → ι . λ x9 x10 . setsum x9 x7)) ⟶ (∀ x4 : (((ι → ι) → ι) → ι) → ι . ∀ x5 : ((ι → ι → ι) → (ι → ι) → ι → ι) → (ι → ι → ι) → ι → ι . ∀ x6 : ι → ι → ι . ∀ x7 : (ι → ι) → ((ι → ι) → ι) → ι → ι → ι . x0 (λ x8 . Inj0 (Inj1 0)) (λ x8 . Inj0 (x7 (λ x9 . setsum 0 (Inj0 0)) (λ x9 : ι → ι . Inj1 0) x8 0)) (Inj0 (setsum 0 (x4 (λ x8 : (ι → ι) → ι . Inj1 0))))) ⟶ (∀ x4 . ∀ x5 : (((ι → ι) → ι) → (ι → ι) → ι) → ι → ι . ∀ x6 : (((ι → ι) → ι → ι) → ι) → ι . ∀ x7 : ((ι → ι → ι) → (ι → ι) → ι) → ι → ι → ι . In (x7 (λ x8 : ι → ι → ι . λ x9 : ι → ι . x8 (setsum (setsum 0 0) (setsum 0 0)) (Inj1 (x6 (λ x10 : (ι → ι) → ι → ι . 0)))) 0 0) (Inj1 (x6 (λ x8 : (ι → ι) → ι → ι . x6 (λ x9 : (ι → ι) → ι → ι . setsum 0 0)))) ⟶ x0 (λ x8 . 0) (λ x8 . 0) (Inj0 0) ⟶ x2 (λ x8 . setsum 0 (setsum (Inj1 0) (setsum (Inj0 0) (Inj0 0)))) (λ x8 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . x5 (λ x9 : (ι → ι) → ι . λ x10 : ι → ι . Inj1 (setsum (setsum 0 0) 0)) (x5 (λ x9 : (ι → ι) → ι . λ x10 : ι → ι . setsum 0 (x10 0)) (setsum (Inj0 0) (Inj0 0))))) ⟶ False) (proof)
Theorem 05d15.. : not (∀ x0 : ((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) → ((ι → ι) → ι) → (ι → ι) → ι) → (((ι → ι) → ι → ι) → (ι → ι) → ι) → ο . ∀ x2 : (ι → ι) → (ι → ι) → ο . ∀ x3 : (ι → ι) → ι → ο . (∀ x4 x5 x6 x7 . In (setsum (Inj0 0) 0) (Inj0 (Inj1 x5)) ⟶ x3 (λ x8 . setsum x8 (setsum (setsum (Inj0 0) 0) (setsum 0 x5))) x4) ⟶ (∀ x4 : ((ι → ι → ι) → ι → ι) → ι . ∀ x5 : ι → ι . ∀ x6 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ι → (ι → ι) → ι . ∀ x7 : ι → ι . x3 (λ x8 . setsum (Inj0 (Inj1 (x6 (λ x9 : ι → ι → ι . λ x10 : ι → ι . λ x11 . 0) 0 (λ x9 . 0)))) (Inj1 (setsum (setsum 0 0) (Inj0 0)))) (x5 0) ⟶ x1 (λ x8 : ι → ι . λ x9 x10 . λ x11 : ι → ι . λ x12 . Inj0 x9) (λ x8 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x9 : (ι → ι) → ι . λ x10 : ι → ι . x9 (λ x11 . setsum 0 (Inj1 0))) (λ x8 : (ι → ι) → ι → ι . λ x9 : ι → ι . Inj1 (setsum (setsum (x6 (λ x10 : ι → ι → ι . λ x11 : ι → ι . λ x12 . 0) 0 (λ x10 . 0)) (setsum 0 0)) 0))) ⟶ (∀ x4 : ι → ι → (ι → ι) → ι → ι . ∀ x5 x6 x7 . In (Inj1 (setsum 0 (Inj1 (x4 0 0 (λ x8 . 0) 0)))) (setsum (setsum (Inj0 (Inj1 0)) 0) x5) ⟶ x1 (λ x8 : ι → ι . λ x9 x10 . λ x11 : ι → ι . λ x12 . Inj1 0) (λ x8 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x9 : (ι → ι) → ι . λ x10 : ι → ι . Inj1 (x10 (setsum (x10 0) (setsum 0 0)))) (λ x8 : (ι → ι) → ι → ι . λ x9 : ι → ι . 0) ⟶ x2 (λ x8 . Inj1 x7) (setsum (Inj1 (Inj0 (setsum 0 0))))) ⟶ (∀ x4 : (ι → ι) → ι . ∀ x5 x6 . ∀ x7 : (ι → ι → ι) → (ι → ι → ι) → ι . In (x7 (λ x8 x9 . 0) (λ x8 x9 . Inj1 (setsum (x7 (λ x10 x11 . 0) (λ x10 x11 . 0)) (setsum 0 0)))) (Inj0 0) ⟶ x2 (λ x8 . 0) (λ x8 . 0) ⟶ x0 (λ x8 : ι → ι → (ι → ι) → ι → ι . Inj1 (Inj1 0)) (λ x8 . λ x9 x10 : ι → ι . λ x11 . setsum (Inj1 (Inj1 (setsum 0 0))) (x9 0)) (λ x8 . 0) (Inj1 (x4 (λ x8 . x8)))) ⟶ (∀ x4 . ∀ x5 : ι → ((ι → ι) → ι) → ι . ∀ x6 : ι → ι . ∀ x7 . x2 (λ x8 . 0) (λ x8 . 0) ⟶ x1 (λ x8 : ι → ι . λ x9 x10 . λ x11 : ι → ι . λ x12 . 0) (λ x8 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x9 : (ι → ι) → ι . λ x10 : ι → ι . 0) (λ x8 : (ι → ι) → ι → ι . λ x9 : ι → ι . setsum (setsum (setsum (setsum 0 0) (setsum 0 0)) (Inj0 0)) (Inj1 0))) ⟶ (∀ x4 . ∀ x5 : (ι → ι → ι) → ((ι → ι) → ι) → (ι → ι) → ι . ∀ x6 : ι → ι . ∀ x7 : ((ι → ι) → ι → ι → ι) → ι . x1 (λ x8 : ι → ι . λ x9 x10 . λ x11 : ι → ι . λ x12 . x11 (Inj1 (Inj0 x10))) (λ x8 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x9 : (ι → ι) → ι . λ x10 : ι → ι . 0) (λ x8 : (ι → ι) → ι → ι . λ x9 : ι → ι . 0) ⟶ x3 (λ x8 . 0) (Inj0 (Inj1 (x5 (λ x8 x9 . 0) (λ x8 : ι → ι . setsum 0 0) (λ x8 . 0))))) ⟶ (∀ x4 : ι → ι → ι . ∀ x5 : ι → (ι → ι) → ι . ∀ x6 : (((ι → ι) → ι → ι) → ι → ι) → ι → (ι → ι) → ι . ∀ x7 : ι → ι . In (Inj0 (Inj1 0)) (x5 (Inj0 (setsum 0 0)) (λ x8 . 0)) ⟶ x0 (λ x8 : ι → ι → (ι → ι) → ι → ι . 0) (λ x8 . λ x9 x10 : ι → ι . λ x11 . setsum (Inj0 (x10 (setsum 0 0))) (Inj0 x11)) (λ x8 . x6 (λ x9 : (ι → ι) → ι → ι . λ x10 . setsum (setsum (x7 0) (Inj0 0)) 0) 0 (λ x9 . setsum (setsum (Inj1 0) (setsum 0 0)) (setsum x9 (x6 (λ x10 : (ι → ι) → ι → ι . λ x11 . 0) 0 (λ x10 . 0))))) (setsum (setsum 0 (setsum (setsum 0 0) (setsum 0 0))) (Inj0 0))) ⟶ (∀ x4 : (((ι → ι) → ι → ι) → ι → ι → ι) → ι . ∀ x5 x6 . ∀ x7 : ((ι → ι → ι) → ι) → ι . x0 (λ x8 : ι → ι → (ι → ι) → ι → ι . Inj1 x6) (λ x8 . λ x9 x10 : ι → ι . λ x11 . 0) (λ x8 . Inj0 (setsum (Inj1 (setsum 0 0)) 0)) (Inj0 x5) ⟶ x2 (λ x8 . x6) (λ x8 . x7 (λ x9 : ι → ι → ι . setsum (setsum x6 (setsum 0 0)) 0))) ⟶ False) (proof)
Theorem 6993e.. : not (∀ x0 : ((((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) → 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) → ο . (∀ x4 x5 x6 . ∀ x7 : (ι → (ι → ι) → ι → ι) → ι . In (Inj0 0) x6 ⟶ x0 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ι . setsum (setsum (setsum (x7 (λ x9 . λ x10 : ι → ι . λ x11 . 0)) (x7 (λ x9 . λ x10 : ι → ι . λ x11 . 0))) (setsum 0 0)) x5) (setsum (Inj1 0) (x7 (λ x8 . λ x9 : ι → ι . λ x10 . Inj1 (Inj1 0)))) (Inj1 0) (λ x8 : ι → ι . x7 (λ x9 . λ x10 : ι → ι . λ x11 . 0)) (Inj0 x5) ⟶ x3 (λ x8 : ι → ι → ι . λ x9 . x8 (Inj0 0) (Inj0 0)) x5 (λ x8 : (ι → ι) → ι . 0)) ⟶ (∀ x4 : ι → ι . ∀ x5 : (ι → ι) → ι . ∀ x6 x7 . x3 (λ x8 : ι → ι → ι . λ x9 . Inj0 x6) (setsum (setsum (setsum (Inj0 0) (setsum 0 0)) 0) (setsum 0 (x5 (λ x8 . setsum 0 0)))) (λ x8 : (ι → ι) → ι . x8 (λ x9 . Inj1 (setsum 0 (setsum 0 0)))) ⟶ x1 (λ x8 x9 . λ x10 : ι → ι . x7) (λ x8 . setsum (Inj0 (setsum (Inj0 0) 0)) x7)) ⟶ (∀ x4 : (ι → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι . ∀ x5 : ι → ι . ∀ x6 : ι → ((ι → ι) → ι) → ι . ∀ x7 : ι → (ι → ι) → ι . x2 (λ x8 x9 . Inj1 0) (λ x8 : ι → ι . setsum 0 (x7 (Inj1 (setsum 0 0)) (λ x9 . Inj1 (setsum 0 0)))) (λ x8 : ι → ι → ι . λ x9 x10 . x7 (Inj0 x9) (λ x11 . x8 (setsum (Inj0 0) (Inj1 0)) x10)) 0 ⟶ x2 (λ x8 x9 . 0) (λ x8 : ι → ι . Inj0 0) (λ x8 : ι → ι → ι . λ x9 x10 . Inj1 (Inj0 (Inj0 (Inj1 0)))) (x7 (x4 (λ x8 x9 . setsum 0 (Inj0 0)) (λ x8 : ι → ι . λ x9 . x6 0 (λ x10 : ι → ι . x7 0 (λ x11 . 0))) (λ x8 . Inj1 0)) (λ x8 . Inj1 (Inj0 (Inj1 0))))) ⟶ (∀ x4 : ι → ι → (ι → ι) → ι → ι . ∀ x5 : ι → ι . ∀ x6 : ι → ι → ι → ι → ι . ∀ x7 : ι → (ι → ι → ι) → ι → ι → ι . x2 (λ x8 x9 . setsum x9 x9) (λ x8 : ι → ι . 0) (λ x8 : ι → ι → ι . λ x9 x10 . Inj0 (Inj0 (setsum 0 (x7 0 (λ x11 x12 . 0) 0 0)))) (Inj0 (Inj0 (Inj1 (setsum 0 0)))) ⟶ x1 (λ x8 x9 . λ x10 : ι → ι . Inj1 x8) (λ x8 . setsum (Inj0 (x5 (Inj0 0))) (x6 0 x8 x8 (Inj0 (setsum 0 0))))) ⟶ (∀ x4 . ∀ x5 : ι → ((ι → ι) → ι → ι) → (ι → ι) → ι . ∀ x6 x7 . x2 (λ x8 x9 . setsum 0 (Inj1 0)) (λ x8 : ι → ι . x6) (λ x8 : ι → ι → ι . λ x9 x10 . x9) 0 ⟶ x1 (λ x8 x9 . λ x10 : ι → ι . setsum (Inj0 (setsum (x10 0) (setsum 0 0))) (setsum (setsum x7 (Inj1 0)) 0)) (λ x8 . 0)) ⟶ (∀ x4 : (ι → ι) → ι . ∀ x5 . ∀ x6 : ι → ι . ∀ x7 : ι → (ι → ι) → ι . In (Inj1 0) (setsum 0 (x4 (λ x8 . x8))) ⟶ x1 (λ x8 x9 . λ x10 : ι → ι . x7 (Inj1 0) (λ x11 . 0)) (λ x8 . 0) ⟶ x1 (λ x8 x9 . λ x10 : ι → ι . x7 (setsum (setsum x9 0) x9) (λ x11 . setsum 0 x9)) (setsum (setsum 0 (Inj1 (setsum 0 0))))) ⟶ (∀ x4 x5 x6 . ∀ x7 : ((ι → ι) → ι) → ι . In (Inj0 0) (setsum 0 (setsum (setsum (Inj0 0) (x7 (λ x8 : ι → ι . 0))) (setsum (setsum 0 0) (setsum 0 0)))) ⟶ x0 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ι . x8 (λ x9 : ι → ι → ι . λ x10 : ι → ι . λ x11 . 0) (λ x9 : ι → ι . λ x10 . Inj0 (Inj1 (setsum 0 0)))) (setsum 0 (Inj1 (setsum 0 (Inj0 0)))) (x7 (λ x8 : ι → ι . Inj1 (Inj1 x6))) (λ x8 : ι → ι . Inj0 (x8 x5)) 0 ⟶ x0 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ι . 0) (Inj0 (setsum x5 x6)) 0 (λ x8 : ι → ι . setsum (Inj1 (setsum 0 0)) (Inj0 x6)) 0) ⟶ (∀ x4 : (ι → ι → ι) → ι . ∀ x5 x6 . ∀ x7 : ι → ι → ι . In (setsum 0 0) (Inj1 (Inj1 (Inj0 (Inj0 0)))) ⟶ x0 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ι . Inj0 0) (Inj1 (setsum 0 (Inj1 (Inj0 0)))) x5 (λ x8 : ι → ι . setsum x6 x5) (setsum (setsum (setsum 0 0) 0) 0) ⟶ x3 (λ x8 : ι → ι → ι . λ x9 . Inj0 0) 0 (λ x8 : (ι → ι) → ι . setsum 0 (setsum (x7 (x7 0 0) (setsum 0 0)) (x7 (setsum 0 0) (setsum 0 0))))) ⟶ False) (proof)
Theorem 30bb5.. : not (∀ x0 : ((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) → 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 . ∀ x7 : (((ι → ι) → ι) → ι) → ι . x3 (λ x8 . λ x9 : (ι → ι → ι) → (ι → ι) → ι . λ x10 . setsum (setsum (setsum (Inj1 0) (Inj0 0)) (x9 (λ x11 x12 . x10) (λ x11 . Inj1 0))) (x7 (λ x11 : (ι → ι) → ι . x9 (λ x12 x13 . Inj0 0) (λ x12 . 0)))) (setsum (setsum (Inj1 (setsum 0 0)) 0) (Inj1 (Inj1 (setsum 0 0)))) (setsum (setsum (Inj0 (setsum 0 0)) (Inj1 0)) (Inj0 (Inj0 (setsum 0 0))))) ⟶ (∀ x4 . ∀ x5 : ι → (ι → ι → ι) → (ι → ι) → ι → ι . ∀ x6 x7 . x3 (λ x8 . λ x9 : (ι → ι → ι) → (ι → ι) → ι . λ x10 . setsum 0 (setsum (setsum x8 (Inj0 0)) (Inj1 (setsum 0 0)))) (Inj1 x7) (x5 x4 (λ x8 x9 . x7) (λ x8 . setsum x6 0) (x5 (x5 (setsum 0 0) (λ x8 x9 . Inj1 0) (λ x8 . Inj1 0) (Inj0 0)) (λ x8 x9 . Inj1 (setsum 0 0)) (λ x8 . setsum 0 0) 0)) ⟶ x1 (λ x8 . Inj1 0) (λ x8 : ι → ι → ι → ι . x5 (Inj0 (Inj0 0)) (λ x9 x10 . setsum 0 0) (λ x9 . x6) (setsum (x8 0 x7 (Inj0 0)) (x5 0 (λ x9 x10 . Inj1 0) (λ x9 . Inj1 0) (setsum 0 0))))) ⟶ (∀ x4 . ∀ x5 : (ι → ι) → ι . ∀ x6 : ι → ι . ∀ x7 . x2 (λ x8 : ι → ι → ι . λ x9 x10 x11 . 0) (λ x8 . x7) (x6 (setsum (Inj0 0) (setsum (setsum 0 0) 0)))) ⟶ (∀ x4 : ι → ι . ∀ x5 . ∀ x6 : ι → ι . ∀ x7 : (((ι → ι) → ι) → ι) → (ι → ι → ι) → ι . In (Inj0 (Inj1 (setsum (setsum 0 0) (setsum 0 0)))) (setsum 0 0) ⟶ x2 (λ x8 : ι → ι → ι . λ x9 x10 x11 . Inj0 (setsum (Inj0 (Inj0 0)) 0)) (λ x8 . x7 (λ x9 : (ι → ι) → ι . Inj1 (setsum 0 0)) (λ x9 x10 . Inj1 x9)) (setsum x5 0) ⟶ x1 (λ x8 . setsum 0 (x7 (λ x9 : (ι → ι) → ι . x6 (setsum 0 0)) (λ x9 x10 . Inj1 (setsum 0 0)))) (λ x8 : ι → ι → ι → ι . Inj0 (Inj0 (Inj0 (Inj0 0))))) ⟶ (∀ x4 : ((ι → ι) → (ι → ι) → ι → ι) → ι → ι . ∀ x5 x6 x7 . In (setsum (setsum (x4 (λ x8 x9 : ι → ι . λ x10 . 0) x5) (setsum (Inj0 0) x6)) 0) (Inj0 (setsum (Inj0 (Inj1 0)) x5)) ⟶ x3 (λ x8 . λ x9 : (ι → ι → ι) → (ι → ι) → ι . λ x10 . 0) 0 x7 ⟶ x1 (λ x8 . 0) (λ x8 : ι → ι → ι → ι . 0)) ⟶ (∀ x4 . ∀ x5 : ι → ι . ∀ x6 . ∀ x7 : (ι → ι) → ι → ι → ι . x1 (λ x8 . 0) (λ x8 : ι → ι → ι → ι . x6) ⟶ x0 (λ x8 : ι → ((ι → ι) → ι → ι) → ι → ι . x7 (λ x9 . Inj0 (Inj1 0)) (x8 (Inj1 (setsum 0 0)) (λ x9 : ι → ι . λ x10 . 0) (x8 (setsum 0 0) (λ x9 : ι → ι . λ x10 . x9 0) 0)) (x7 (λ x9 . setsum (setsum 0 0) (Inj1 0)) (Inj0 (setsum 0 0)) x6)) (λ x8 x9 . Inj0) (λ x8 : ι → ι → ι . Inj0 (setsum (setsum (Inj1 0) (setsum 0 0)) (setsum (Inj1 0) 0))) (λ x8 : ι → ι . x6) (λ x8 . x6)) ⟶ (∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι . ∀ x7 . In (setsum 0 (setsum (setsum (Inj0 0) (setsum 0 0)) 0)) (Inj1 (Inj1 x7)) ⟶ x0 (λ x8 : ι → ((ι → ι) → ι → ι) → ι → ι . 0) (λ x8 x9 x10 . Inj0 (Inj1 x8)) (λ x8 : ι → ι → ι . 0) (λ x8 : ι → ι . x6 (setsum 0 (Inj1 (x8 0)))) (λ x8 . Inj1 (Inj1 (Inj0 (x5 0 0))))) ⟶ (∀ x4 x5 x6 . ∀ x7 : ι → ((ι → ι) → ι) → ι → ι . x0 (λ x8 : ι → ((ι → ι) → ι → ι) → ι → ι . x8 (x8 (setsum (Inj1 0) (x8 0 (λ x9 : ι → ι . λ x10 . 0) 0)) (λ x9 : ι → ι . λ x10 . Inj0 (x8 0 (λ x11 : ι → ι . λ x12 . 0) 0)) (setsum 0 0)) (λ x9 : ι → ι . λ x10 . 0) (setsum (Inj1 x6) x6)) (λ x8 x9 x10 . Inj1 (setsum (Inj0 (Inj1 0)) x9)) (λ x8 : ι → ι → ι . 0) (λ x8 : ι → ι . x8 x6) (λ x8 . setsum x8 (Inj1 (setsum x5 (setsum 0 0)))) ⟶ x0 (λ x8 : ι → ((ι → ι) → ι → ι) → ι → ι . x8 0 (λ x9 : ι → ι . λ x10 . Inj0 (setsum 0 0)) (setsum x6 (Inj1 (Inj1 0)))) (λ x8 x9 x10 . 0) (λ x8 : ι → ι → ι . Inj1 (setsum x5 0)) (λ x8 : ι → ι . 0) (λ x8 . setsum 0 0)) ⟶ False) (proof)
Theorem 6c860.. : not (∀ x0 : (ι → ι → ι) → ι → ο . ∀ x1 : (ι → ι) → ι → ο . ∀ x2 : (ι → ι → ι) → ((ι → ι) → (ι → ι → ι) → ι) → ι → ο . ∀ x3 : (ι → ι → ι → ι) → ι → ο . (∀ x4 : ι → ι → ι . ∀ x5 : ι → ι . ∀ x6 : (ι → ι) → ι → (ι → ι) → ι . ∀ x7 . x2 (λ x8 x9 . setsum 0 (setsum (setsum x8 (x6 (λ x10 . 0) 0 (λ x10 . 0))) x7)) (λ x8 : ι → ι . λ x9 : ι → ι → ι . x6 (λ x10 . Inj0 (setsum 0 0)) (setsum (Inj1 0) (setsum (Inj1 0) (x8 0))) (λ x10 . 0)) x7 ⟶ x3 (λ x8 x9 x10 . x9) 0) ⟶ (∀ x4 x5 . ∀ x6 : (ι → (ι → ι) → ι → ι) → ι . ∀ x7 . In (Inj0 0) (Inj1 (setsum 0 (setsum 0 (setsum 0 0)))) ⟶ x3 (λ x8 x9 x10 . Inj1 (Inj0 0)) (Inj1 0) ⟶ x1 (λ x8 . 0) (Inj1 (Inj0 (Inj0 (Inj0 0))))) ⟶ (∀ x4 x5 x6 . ∀ x7 : ι → ι . In (Inj0 0) (Inj0 (Inj0 (Inj1 (Inj0 0)))) ⟶ x2 (λ x8 x9 . 0) (λ x8 : ι → ι . λ x9 : ι → ι → ι . setsum (Inj1 (setsum (Inj0 0) 0)) x6) 0) ⟶ (∀ x4 : ι → ι . ∀ x5 : (ι → ι → ι) → ι → ι . ∀ x6 . ∀ x7 : ι → (ι → ι → ι) → (ι → ι) → ι → ι . In (Inj0 (setsum 0 (setsum 0 (Inj0 0)))) x6 ⟶ x2 (λ x8 x9 . setsum 0 x6) (λ x8 : ι → ι . λ x9 : ι → ι → ι . Inj0 (x9 0 0)) (setsum (x5 (λ x8 x9 . Inj1 0) (x5 (λ x8 x9 . x9) (setsum 0 0))) 0) ⟶ x0 (λ x8 x9 . setsum (setsum x9 (setsum (setsum 0 0) 0)) x6) (setsum (setsum (x7 (Inj1 0) (λ x8 x9 . setsum 0 0) (λ x8 . setsum 0 0) 0) (setsum (Inj0 0) (setsum 0 0))) (setsum (Inj1 (Inj1 0)) (setsum (x4 0) (Inj0 0))))) ⟶ (∀ x4 : (((ι → ι) → ι → ι) → ι → ι) → ι → ι → ι → ι . ∀ x5 : ι → (ι → ι) → ι . ∀ x6 x7 . x1 (λ x8 . 0) (Inj1 0)) ⟶ (∀ x4 x5 x6 x7 . x1 (λ x8 . 0) x5 ⟶ x1 (λ x8 . x6) 0) ⟶ (∀ x4 x5 x6 x7 . In (Inj1 (Inj0 (Inj1 0))) (Inj1 (Inj1 0)) ⟶ x0 (λ x8 x9 . x7) (setsum 0 0)) ⟶ (∀ x4 . ∀ x5 x6 : ι → ι . ∀ x7 . x0 (λ x8 x9 . 0) (x5 (Inj0 (Inj0 0))) ⟶ x3 (λ x8 x9 x10 . 0) 0) ⟶ False) (proof)
Theorem 1b96e.. : not (∀ x0 : ((((ι → 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) → ι → ι) → ι → ι → ι) → ι → ι) → ο . (∀ x4 : ι → ι → (ι → ι) → ι . ∀ x5 . ∀ x6 : ι → ι . ∀ x7 . x0 (λ x8 : ((ι → ι) → ι → ι → ι) → ι . λ x9 : ι → ι → ι → ι . setsum 0 (x9 (setsum (setsum 0 0) (setsum 0 0)) (Inj1 0) 0)) (λ x8 . x7) ⟶ x3 (λ x8 . Inj0 (x6 (setsum (Inj0 0) (x6 0)))) (λ x8 : ((ι → ι) → ι → ι) → ι → ι → ι . λ x9 . x7)) ⟶ (∀ x4 : (ι → ι) → ι . ∀ x5 : ι → ((ι → ι) → ι) → ι → ι → ι . ∀ x6 . ∀ x7 : (ι → ι) → ι → ι . x3 (λ x8 . setsum (Inj0 0) (Inj0 0)) (λ x8 : ((ι → ι) → ι → ι) → ι → ι → ι . λ x9 . 0) ⟶ x1 (λ x8 . λ x9 : ι → ι . 0) (Inj1 0)) ⟶ (∀ x4 : ((ι → ι → ι) → (ι → ι) → ι) → ι . ∀ x5 : (ι → (ι → ι) → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι . ∀ x6 x7 . In (Inj1 0) (setsum x7 (Inj0 (setsum 0 (x4 (λ x8 : ι → ι → ι . λ x9 : ι → ι . 0))))) ⟶ x2 (λ x8 . setsum (Inj1 0) (Inj0 0)) (Inj1 (setsum 0 0)) (λ x8 : (ι → ι) → ι → ι . setsum (setsum 0 (Inj1 (Inj0 0))) (setsum x6 0)) ⟶ x2 (λ x8 . 0) x7 (λ x8 : (ι → ι) → ι → ι . 0)) ⟶ (∀ x4 . ∀ x5 : ι → ι → (ι → ι) → ι → ι . ∀ x6 x7 . In x7 (Inj1 (setsum (setsum (setsum 0 0) (setsum 0 0)) (x5 (setsum 0 0) (setsum 0 0) (λ x8 . 0) 0))) ⟶ x2 (λ x8 . Inj1 x6) (x5 (Inj0 (setsum x6 0)) x7 (λ x8 . x7) x4) (λ x8 : (ι → ι) → ι → ι . Inj1 x7) ⟶ x0 (λ x8 : ((ι → ι) → ι → ι → ι) → ι . λ x9 : ι → ι → ι → ι . 0) (λ x8 . x8)) ⟶ (∀ x4 : (((ι → ι) → ι) → ι → ι → ι) → ι → (ι → ι) → ι → ι . ∀ x5 : (ι → ι) → ι . ∀ x6 : (((ι → ι) → ι) → ι → ι) → ι → (ι → ι) → ι . ∀ x7 : ι → ι . In (setsum (setsum (x6 (λ x8 : (ι → ι) → ι . λ x9 . setsum 0 0) (setsum 0 0) (λ x8 . setsum 0 0)) (Inj0 (setsum 0 0))) (setsum 0 (Inj0 (Inj1 0)))) (x6 (λ x8 : (ι → ι) → ι . x7) 0 (λ x8 . 0)) ⟶ x1 (λ x8 . λ x9 : ι → ι . x9 0) (Inj1 (setsum (x7 (Inj0 0)) (x7 (setsum 0 0))))) ⟶ (∀ x4 . ∀ x5 : ((ι → ι) → ι → ι → ι) → ι . ∀ x6 x7 . In (setsum x4 x7) (Inj1 (setsum (setsum (setsum 0 0) 0) x4)) ⟶ x1 (λ x8 . λ x9 : ι → ι . 0) 0 ⟶ x3 (λ x8 . x7) (λ x8 : ((ι → ι) → ι → ι) → ι → ι → ι . λ x9 . setsum 0 x6)) ⟶ (∀ x4 : (ι → ι → ι) → (ι → ι) → ι . ∀ x5 . ∀ x6 : ι → ι → ι → ι → ι . ∀ x7 . x0 (λ x8 : ((ι → ι) → ι → ι → ι) → ι . λ x9 : ι → ι → ι → ι . 0) (λ x8 . 0) ⟶ x0 (λ x8 : ((ι → ι) → ι → ι → ι) → ι . λ x9 : ι → ι → ι → ι . Inj0 (setsum x7 x7)) (λ x8 . Inj1 (Inj1 0))) ⟶ (∀ x4 x5 x6 : ι → ι . ∀ x7 . x0 (λ x8 : ((ι → ι) → ι → ι → ι) → ι . λ x9 : ι → ι → ι → ι . x7) (λ x8 . Inj1 0) ⟶ In x7 (x4 (setsum 0 (x4 (Inj0 0))))) ⟶ False) (proof)
Theorem c13a6.. : 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) → (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 → ι) → ι → ι → ι) → ι → ο . (∀ x4 x5 . ∀ x6 : (ι → ι) → (ι → ι) → ι . ∀ x7 : ι → ι . In (x7 0) x4 ⟶ x0 (λ x8 . λ x9 : (ι → ι → ι) → ι . Inj1 (x6 (λ x10 . Inj0 0) (λ x10 . x10))) (λ x8 : ((ι → ι) → ι) → ι → ι → ι . 0) ⟶ x3 (λ x8 : (ι → (ι → ι) → ι) → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι → ι . λ x11 x12 . Inj1 0) 0) ⟶ (∀ x4 . ∀ x5 : ι → ((ι → ι) → ι → ι) → ι . ∀ x6 . ∀ x7 : ι → ι . In (setsum 0 (Inj1 x6)) (Inj1 (Inj1 (x5 (Inj1 0) (λ x8 : ι → ι . λ x9 . 0)))) ⟶ x3 (λ x8 : (ι → (ι → ι) → ι) → (ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι → ι . λ x11 x12 . Inj1 0) x4 ⟶ x2 (λ x8 : ((ι → ι) → ι) → ι . setsum (setsum (setsum (setsum 0 0) 0) (setsum (setsum 0 0) (Inj0 0))) 0) 0 (λ x8 . 0) (λ x8 . setsum (Inj0 (Inj1 (x5 0 (λ x9 : ι → ι . λ x10 . 0)))) 0)) ⟶ (∀ x4 . ∀ x5 x6 : ι → ι . ∀ x7 : (ι → (ι → ι) → ι → ι) → ι . x0 (λ x8 . λ x9 : (ι → ι → ι) → ι . setsum 0 0) (λ x8 : ((ι → ι) → ι) → ι → ι → ι . x7 (λ x9 . λ x10 : ι → ι . λ x11 . setsum (setsum (setsum 0 0) (Inj0 0)) (setsum (Inj1 0) (setsum 0 0)))) ⟶ x2 (λ x8 : ((ι → ι) → ι) → ι . x6 0) 0 (λ x8 . 0) (λ x8 . 0)) ⟶ (∀ x4 : ((ι → ι) → ι) → ι → ι . ∀ x5 : (ι → ι) → ι . ∀ x6 : (ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . ∀ x7 . x2 (λ x8 : ((ι → ι) → ι) → ι . x8 (λ x9 : ι → ι . x7)) 0 (λ x8 . Inj0 (x6 (λ x9 . setsum x8 (Inj0 0)) (λ x9 : ι → ι . λ x10 . setsum x7 (setsum 0 0)) (λ x9 . 0) 0)) (λ x8 . setsum x8 (Inj0 0)) ⟶ False) ⟶ (∀ x4 : ι → ι . ∀ x5 : ι → ι → ι . ∀ x6 : (((ι → ι) → ι) → ι) → (ι → ι) → ι → ι → ι . ∀ x7 : ι → (ι → ι → ι) → (ι → ι) → ι → ι . x2 (λ x8 : ((ι → ι) → ι) → ι . Inj0 (setsum (x8 (λ x9 : ι → ι . Inj1 0)) 0)) 0 (λ x8 . 0) (λ x8 . Inj1 (Inj0 (setsum 0 (setsum 0 0)))) ⟶ x1 (λ x8 x9 . setsum 0 0) (λ x8 : ((ι → ι) → ι) → ι . λ x9 . Inj1 (setsum (setsum (x6 (λ x10 : (ι → ι) → ι . 0) (λ x10 . 0) 0 0) (setsum 0 0)) (setsum 0 (x8 (λ x10 : ι → ι . 0))))) (λ x8 . Inj0 (setsum (x6 (λ x9 : (ι → ι) → ι . x9 (λ x10 . 0)) (λ x9 . 0) 0 0) (x6 (λ x9 : (ι → ι) → ι . setsum 0 0) (λ x9 . x7 0 (λ x10 x11 . 0) (λ x10 . 0) 0) (x5 0 0) 0))) (λ x8 . setsum (Inj0 (setsum 0 0)) (Inj1 0)) (Inj1 (x5 (x7 (Inj0 0) (λ x8 x9 . setsum 0 0) (λ x8 . setsum 0 0) (x5 0 0)) (Inj1 (setsum 0 0))))) ⟶ (∀ x4 : ((ι → ι → ι) → ι) → ι . ∀ x5 : ι → ((ι → ι) → ι) → ι . ∀ x6 : ι → ι . ∀ x7 . x1 (λ x8 x9 . 0) (λ x8 : ((ι → ι) → ι) → ι . λ x9 . Inj0 (x6 0)) (λ x8 . 0) (λ x8 . x7) (x6 0) ⟶ x2 (λ x8 : ((ι → ι) → ι) → ι . 0) 0 (λ x8 . Inj1 (x5 x8 (λ x9 : ι → ι . 0))) (λ x8 . setsum (Inj0 x8) (x6 0))) ⟶ (∀ x4 : (((ι → ι) → ι) → ι → ι) → (ι → ι) → ι → ι . ∀ x5 x6 x7 . In (Inj0 x7) (Inj0 x5) ⟶ x2 (λ x8 : ((ι → ι) → ι) → ι . x5) x7 (λ x8 . Inj0 (setsum (Inj1 (setsum 0 0)) 0)) (λ x8 . 0) ⟶ x0 (λ x8 . λ x9 : (ι → ι → ι) → ι . x9 (λ x10 x11 . x9 (λ x12 x13 . x10))) (λ x8 : ((ι → ι) → ι) → ι → ι → ι . Inj0 0)) ⟶ (∀ x4 . ∀ x5 : (ι → ι) → ι . ∀ x6 . ∀ x7 : ((ι → ι) → ι) → ι . In (Inj1 0) (Inj1 (setsum 0 0)) ⟶ x0 (λ x8 . λ x9 : (ι → ι → ι) → ι . Inj0 (setsum 0 (setsum (Inj0 0) (x9 (λ x10 x11 . 0))))) (λ x8 : ((ι → ι) → ι) → ι → ι → ι . Inj1 0) ⟶ x0 (λ x8 . λ x9 : (ι → ι → ι) → ι . Inj1 0) (λ x8 : ((ι → ι) → ι) → ι → ι → ι . setsum 0 (x5 (λ x9 . 0)))) ⟶ False) (proof)
Theorem 29cb8.. : 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 → ο . ∀ x2 : (ι → ι) → (((ι → ι → ι) → ι) → ι) → ι → ο . ∀ x3 : (ι → (ι → ι) → ι) → ι → ι → ο . (∀ x4 : (ι → ι) → ι . ∀ x5 : ι → ι . ∀ x6 : ((ι → ι) → ι → ι) → ι . ∀ x7 : (ι → ι) → ι . In (Inj1 0) (Inj1 (x7 (λ x8 . setsum (Inj0 0) (Inj1 0)))) ⟶ x1 (λ x8 . Inj1 0) (λ x8 . λ x9 : ι → ι → ι . setsum (setsum (setsum (Inj1 0) 0) 0) (setsum (setsum (setsum 0 0) 0) (setsum 0 (x9 0 0)))) (λ x8 : ι → ι → ι . 0) (Inj0 (Inj0 (setsum 0 (x5 0)))) ⟶ x3 (λ x8 . λ x9 : ι → ι . x6 (λ x10 : ι → ι . λ x11 . x10 (Inj0 (x9 0)))) (x6 (λ x8 : ι → ι . λ x9 . x6 (λ x10 : ι → ι . λ x11 . setsum (setsum 0 0) (Inj1 0)))) (setsum (Inj0 0) 0)) ⟶ (∀ x4 . ∀ x5 : ((ι → ι → ι) → ι) → ι . ∀ x6 : ((ι → ι → ι) → ι) → ι → ι . ∀ x7 . x3 (λ x8 . λ x9 : ι → ι . setsum (setsum x8 (setsum (Inj1 0) (setsum 0 0))) 0) (x5 (λ x8 : ι → ι → ι . x6 (λ x9 : ι → ι → ι . setsum 0 (setsum 0 0)) x7)) (x5 (λ x8 : ι → ι → ι . Inj0 0)) ⟶ In (Inj0 (setsum (setsum 0 0) (Inj0 (setsum 0 0)))) (Inj0 0)) ⟶ (∀ x4 : (ι → ι) → ((ι → ι) → ι) → ι . ∀ x5 x6 x7 . In (setsum (setsum 0 (setsum (x4 (λ x8 . 0) (λ x8 : ι → ι . 0)) x6)) (setsum (x4 (λ x8 . Inj1 0) (λ x8 : ι → ι . setsum 0 0)) 0)) (Inj1 (Inj1 (setsum (Inj1 0) 0))) ⟶ x3 (λ x8 . λ x9 : ι → ι . 0) (Inj0 (setsum x5 x5)) 0 ⟶ x2 (λ x8 . 0) (λ x8 : (ι → ι → ι) → ι . Inj0 (x8 (λ x9 x10 . 0))) x6) ⟶ (∀ x4 . ∀ x5 : (ι → ι → ι → ι) → ι → ι . ∀ x6 . ∀ x7 : (ι → ι) → ι → ι . In (setsum (setsum (setsum (x7 (λ x8 . 0) 0) 0) (x7 (λ x8 . Inj1 0) (setsum 0 0))) x6) (Inj0 x6) ⟶ x2 (λ x8 . 0) (λ x8 : (ι → ι → ι) → ι . x5 (λ x9 x10 x11 . setsum (Inj1 x10) (Inj0 (Inj0 0))) (x5 (λ x9 x10 x11 . Inj0 0) 0)) (Inj1 (setsum (setsum x6 (Inj1 0)) (Inj0 (Inj0 0)))) ⟶ x0 (λ x8 : ι → ι → (ι → ι) → ι → ι . x7 (λ x9 . 0) (setsum 0 0)) (setsum (Inj0 0) x6) (λ x8 . setsum (Inj0 (Inj0 (Inj0 0))) (setsum (x5 (λ x9 x10 x11 . Inj1 0) (setsum 0 0)) (setsum 0 0))) (Inj0 x6)) ⟶ (∀ x4 : ι → ι . ∀ x5 : ι → ((ι → ι) → ι) → (ι → ι) → ι . ∀ x6 . ∀ x7 : ((ι → ι) → ι) → ι . x0 (λ x8 : ι → ι → (ι → ι) → ι → ι . setsum (setsum (x5 (Inj1 0) (λ x9 : ι → ι . setsum 0 0) (λ x9 . setsum 0 0)) (x7 (λ x9 : ι → ι . setsum 0 0))) (setsum (setsum 0 (setsum 0 0)) (setsum (x5 0 (λ x9 : ι → ι . 0) (λ x9 . 0)) (x8 0 0 (λ x9 . 0) 0)))) x6 (λ x8 . Inj1 (Inj1 (x5 (x5 0 (λ x9 : ι → ι . 0) (λ x9 . 0)) (λ x9 : ι → ι . x8) (λ x9 . Inj1 0)))) 0 ⟶ x1 (λ x8 . setsum (setsum 0 0) 0) (λ x8 . λ x9 : ι → ι → ι . x9 x8 (x9 (Inj1 (setsum 0 0)) (Inj1 (x9 0 0)))) (λ x8 : ι → ι → ι . x6) (Inj1 0)) ⟶ (∀ x4 x5 x6 . ∀ x7 : (((ι → ι) → ι → ι) → ι) → ι . x1 (λ x8 . x7 (λ x9 : (ι → ι) → ι → ι . Inj1 (x7 (λ x10 : (ι → ι) → ι → ι . x10 (λ x11 . 0) 0)))) (λ x8 . λ x9 : ι → ι → ι . Inj0 0) (λ x8 : ι → ι → ι . Inj1 (setsum 0 x6)) (setsum 0 (setsum (Inj1 0) 0)) ⟶ In (setsum 0 (setsum 0 (Inj1 (setsum 0 0)))) (Inj0 (setsum (Inj1 (Inj1 0)) (Inj0 (setsum 0 0))))) ⟶ (∀ x4 . ∀ x5 : ι → ι . ∀ x6 x7 . x1 (λ x8 . 0) (λ x8 . λ x9 : ι → ι → ι . x7) (λ x8 : ι → ι → ι . x5 (setsum x7 (setsum (Inj1 0) 0))) x7 ⟶ x0 (λ x8 : ι → ι → (ι → ι) → ι → ι . Inj0 (setsum (setsum x6 x6) x7)) 0 (λ x8 . x8) (Inj0 x7)) ⟶ (∀ x4 x5 : ι → ι . ∀ x6 : ((ι → ι) → ι → ι) → ι . ∀ x7 . x0 (λ x8 : ι → ι → (ι → ι) → ι → ι . setsum (setsum (Inj1 (x5 0)) 0) (x6 (λ x9 : ι → ι . λ x10 . Inj0 0))) (Inj1 x7) (λ x8 . x6 (λ x9 : ι → ι . λ x10 . x9 0)) x7 ⟶ In (x6 (λ x8 : ι → ι . λ x9 . setsum (Inj0 (Inj0 0)) (Inj0 (x8 0)))) (Inj1 (setsum (x4 (setsum 0 0)) (setsum (x6 (λ x8 : ι → ι . λ x9 . 0)) (Inj0 0))))) ⟶ False) (proof)

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