Proofgold Asset

Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known 16baa.. : ∀ x0 : ι → ο . ∀ x1 x2 : ι → ι → ι → ι → ο . (∀ x3 x4 x5 x6 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x0 x6 ⟶ ∀ x7 : ο . (x1 x3 x4 x5 x6 ⟶ x7) ⟶ (x2 x3 x4 x5 x6 ⟶ x7) ⟶ (x1 x5 x6 x3 x4 ⟶ x7) ⟶ x7) ⟶ (∀ x3 x4 x5 x6 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x0 x6 ⟶ x2 x3 x4 x5 x6 ⟶ x2 x5 x6 x3 x4) ⟶ ∀ x3 . x0 x3 ⟶ ∀ x4 . x0 x4 ⟶ ∀ x5 . x0 x5 ⟶ ∀ x6 . x0 x6 ⟶ ∀ x7 . x0 x7 ⟶ ∀ x8 . x0 x8 ⟶ ∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ ∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ not (x2 x3 x4 x5 x6) ⟶ not (x2 x3 x4 x7 x8) ⟶ not (x2 x3 x4 x9 x10) ⟶ not (x2 x3 x4 x11 x12) ⟶ not (x2 x3 x4 x13 x14) ⟶ not (x2 x5 x6 x7 x8) ⟶ not (x2 x5 x6 x9 x10) ⟶ not (x2 x5 x6 x11 x12) ⟶ not (x2 x5 x6 x13 x14) ⟶ not (x2 x7 x8 x9 x10) ⟶ not (x2 x7 x8 x11 x12) ⟶ not (x2 x7 x8 x13 x14) ⟶ not (x2 x9 x10 x11 x12) ⟶ not (x2 x9 x10 x13 x14) ⟶ not (x2 x11 x12 x13 x14) ⟶ ∀ x15 : ο . (∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ ∀ x20 . x0 x20 ⟶ ∀ x21 . x0 x21 ⟶ ∀ x22 . x0 x22 ⟶ ∀ x23 . x0 x23 ⟶ ∀ x24 . x0 x24 ⟶ ∀ x25 . x0 x25 ⟶ ∀ x26 . x0 x26 ⟶ ∀ x27 . x0 x27 ⟶ x1 x16 x17 x18 x19 ⟶ x1 x18 x19 x20 x21 ⟶ x1 x20 x21 x22 x23 ⟶ x1 x22 x23 x24 x25 ⟶ x1 x24 x25 x26 x27 ⟶ not (x2 x16 x17 x18 x19) ⟶ not (x2 x16 x17 x20 x21) ⟶ not (x2 x16 x17 x22 x23) ⟶ not (x2 x16 x17 x24 x25) ⟶ not (x2 x16 x17 x26 x27) ⟶ not (x2 x18 x19 x20 x21) ⟶ not (x2 x18 x19 x22 x23) ⟶ not (x2 x18 x19 x24 x25) ⟶ not (x2 x18 x19 x26 x27) ⟶ not (x2 x20 x21 x22 x23) ⟶ not (x2 x20 x21 x24 x25) ⟶ not (x2 x20 x21 x26 x27) ⟶ not (x2 x22 x23 x24 x25) ⟶ not (x2 x22 x23 x26 x27) ⟶ not (x2 x24 x25 x26 x27) ⟶ (∀ x28 : ο . (x16 = x3 ⟶ x17 = x4 ⟶ x28) ⟶ (x18 = x3 ⟶ x19 = x4 ⟶ x28) ⟶ (x20 = x3 ⟶ x21 = x4 ⟶ x28) ⟶ (x22 = x3 ⟶ x23 = x4 ⟶ x28) ⟶ (x24 = x3 ⟶ x25 = x4 ⟶ x28) ⟶ (x26 = x3 ⟶ x27 = x4 ⟶ x28) ⟶ x28) ⟶ x15) ⟶ x15
Param ap^ap : ι → ι → ι
Known 54331.. : ∀ x0 x1 : ι → ο . ∀ x2 x3 x4 x5 x6 x7 . (∀ x8 : ι → ο . x8 x2 ⟶ x8 x3 ⟶ x8 x4 ⟶ x8 x5 ⟶ ∀ x9 . x1 x9 ⟶ x8 x9) ⟶ x0 x2 ⟶ ∀ x8 x9 x10 : ι → ι . (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ x0 (ap (x8 x11) x12)) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ap (x8 x11) (ap (x8 x11) x12) = x12) ⟶ (∀ x11 . x0 x11 ⟶ ap (x8 x11) x2 = x3) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ x0 (ap (x9 x11) x12)) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ap (x9 x11) (ap (x9 x11) x12) = x12) ⟶ (∀ x11 . x0 x11 ⟶ ap (x9 x11) x2 = x4) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ x0 (ap (x10 x11) x12)) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ap (x10 x11) (ap (x10 x11) x12) = x12) ⟶ (∀ x11 . x0 x11 ⟶ ap (x10 x11) x2 = x5) ⟶ ∀ x11 : ι → ι → ι → ι → ο . (∀ x12 x13 x14 x15 . x0 x12 ⟶ x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x11 x12 x13 x14 x15 ⟶ x11 x12 (ap (x8 x12) x13) x14 (ap (x8 x14) x15)) ⟶ (∀ x12 x13 x14 x15 . x0 x12 ⟶ x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x11 x12 x13 x14 x15 ⟶ x11 x12 (ap (x9 x12) x13) x14 (ap (x9 x14) x15)) ⟶ (∀ x12 x13 x14 x15 . x0 x12 ⟶ x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x11 x12 x13 x14 x15 ⟶ x11 x12 (ap (x10 x12) x13) x14 (ap (x10 x14) x15)) ⟶ (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ ∀ x20 . x0 x20 ⟶ ∀ x21 . x0 x21 ⟶ ∀ x22 . x0 x22 ⟶ not (x11 x12 x2 x13 x14) ⟶ not (x11 x12 x2 x15 x16) ⟶ not (x11 x12 x2 x17 x18) ⟶ not (x11 x12 x2 x19 x20) ⟶ not (x11 x12 x2 x21 x22) ⟶ not (x11 x13 x14 x15 x16) ⟶ not (x11 x13 x14 x17 x18) ⟶ not (x11 x13 x14 x19 x20) ⟶ not (x11 x13 x14 x21 x22) ⟶ not (x11 x15 x16 x17 x18) ⟶ not (x11 x15 x16 x19 x20) ⟶ not (x11 x15 x16 x21 x22) ⟶ not (x11 x17 x18 x19 x20) ⟶ not (x11 x17 x18 x21 x22) ⟶ not (x11 x19 x20 x21 x22) ⟶ False) ⟶ ∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ ∀ x20 . x0 x20 ⟶ ∀ x21 . x0 x21 ⟶ ∀ x22 . x0 x22 ⟶ ∀ x23 . x1 x23 ⟶ not (x11 x12 x23 x13 x14) ⟶ not (x11 x12 x23 x15 x16) ⟶ not (x11 x12 x23 x17 x18) ⟶ not (x11 x12 x23 x19 x20) ⟶ not (x11 x12 x23 x21 x22) ⟶ not (x11 x13 x14 x15 x16) ⟶ not (x11 x13 x14 x17 x18) ⟶ not (x11 x13 x14 x19 x20) ⟶ not (x11 x13 x14 x21 x22) ⟶ not (x11 x15 x16 x17 x18) ⟶ not (x11 x15 x16 x19 x20) ⟶ not (x11 x15 x16 x21 x22) ⟶ not (x11 x17 x18 x19 x20) ⟶ not (x11 x17 x18 x21 x22) ⟶ not (x11 x19 x20 x21 x22) ⟶ False
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Theorem 017d9.. : ∀ x0 x1 : ι → ο . ∀ x2 x3 x4 x5 x6 x7 . (∀ x8 : ι → ο . x8 x2 ⟶ x8 x3 ⟶ x8 x4 ⟶ x8 x5 ⟶ x8 x6 ⟶ x8 x7 ⟶ ∀ x9 . x0 x9 ⟶ x8 x9) ⟶ (∀ x8 : ι → ο . x8 x2 ⟶ x8 x3 ⟶ x8 x4 ⟶ x8 x5 ⟶ ∀ x9 . x1 x9 ⟶ x8 x9) ⟶ x0 x2 ⟶ (∀ x8 . x0 x8 ⟶ ∀ x9 : ι → ο . (x1 x8 ⟶ x9 x8) ⟶ x9 x6 ⟶ x9 x7 ⟶ x9 x8) ⟶ ∀ x8 x9 x10 : ι → ι . (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ x0 (ap (x8 x11) x12)) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ap (x8 x11) (ap (x8 x11) x12) = x12) ⟶ (∀ x11 . x0 x11 ⟶ ap (x8 x11) x2 = x3) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ x0 (ap (x9 x11) x12)) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ap (x9 x11) (ap (x9 x11) x12) = x12) ⟶ (∀ x11 . x0 x11 ⟶ ap (x9 x11) x2 = x4) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ x0 (ap (x10 x11) x12)) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ap (x10 x11) (ap (x10 x11) x12) = x12) ⟶ (∀ x11 . x0 x11 ⟶ ap (x10 x11) x2 = x5) ⟶ ∀ x11 : ι → ι → ι → ι → ο . (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ not (x11 x12 x3 x13 x2)) ⟶ (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ not (x11 x12 x4 x13 x2)) ⟶ (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ not (x11 x12 x5 x13 x2)) ⟶ (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ not (x11 x12 x6 x13 x2)) ⟶ (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ not (x11 x12 x7 x13 x2)) ⟶ ∀ x12 : ι → ι → ι → ι → ο . (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ ∀ x17 : ο . (x11 x13 x14 x15 x16 ⟶ x17) ⟶ (x12 x13 x14 x15 x16 ⟶ x17) ⟶ (x11 x15 x16 x13 x14 ⟶ x17) ⟶ x17) ⟶ (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ x12 x13 x14 x15 x16 ⟶ x12 x15 x16 x13 x14) ⟶ (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ x12 x13 x14 x15 x16 ⟶ x12 x13 (ap (x8 x13) x14) x15 (ap (x8 x15) x16)) ⟶ (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ x12 x13 x14 x15 x16 ⟶ x12 x13 (ap (x9 x13) x14) x15 (ap (x9 x15) x16)) ⟶ (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ x12 x13 x14 x15 x16 ⟶ x12 x13 (ap (x10 x13) x14) x15 (ap (x10 x15) x16)) ⟶ (∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ ∀ x20 . x0 x20 ⟶ ∀ x21 . x0 x21 ⟶ ∀ x22 . x0 x22 ⟶ ∀ x23 . x0 x23 ⟶ x11 x13 x2 x14 x15 ⟶ x11 x14 x15 x16 x17 ⟶ x11 x16 x17 x18 x19 ⟶ x11 x18 x19 x20 x21 ⟶ x11 x20 x21 x22 x23 ⟶ not (x12 x13 x2 x14 x15) ⟶ not (x12 x13 x2 x16 x17) ⟶ not (x12 x13 x2 x18 x19) ⟶ not (x12 x13 x2 x20 x21) ⟶ not (x12 x13 x2 x22 x23) ⟶ not (x12 x14 x15 x16 x17) ⟶ not (x12 x14 x15 x18 x19) ⟶ not (x12 x14 x15 x20 x21) ⟶ not (x12 x14 x15 x22 x23) ⟶ not (x12 x16 x17 x18 x19) ⟶ not (x12 x16 x17 x20 x21) ⟶ not (x12 x16 x17 x22 x23) ⟶ not (x12 x18 x19 x20 x21) ⟶ not (x12 x18 x19 x22 x23) ⟶ not (x12 x20 x21 x22 x23) ⟶ False) ⟶ ∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ ∀ x20 . x0 x20 ⟶ ∀ x21 . x0 x21 ⟶ ∀ x22 . x0 x22 ⟶ ∀ x23 . x0 x23 ⟶ not (x12 x13 x2 x14 x15) ⟶ not (x12 x13 x2 x16 x17) ⟶ not (x12 x13 x2 x18 x19) ⟶ not (x12 x13 x2 x20 x21) ⟶ not (x12 x13 x2 x22 x23) ⟶ not (x12 x14 x15 x16 x17) ⟶ not (x12 x14 x15 x18 x19) ⟶ not (x12 x14 x15 x20 x21) ⟶ not (x12 x14 x15 x22 x23) ⟶ not (x12 x16 x17 x18 x19) ⟶ not (x12 x16 x17 x20 x21) ⟶ not (x12 x16 x17 x22 x23) ⟶ not (x12 x18 x19 x20 x21) ⟶ not (x12 x18 x19 x22 x23) ⟶ not (x12 x20 x21 x22 x23) ⟶ False (proof)
Theorem 58e84.. : ∀ x0 x1 : ι → ο . ∀ x2 x3 x4 x5 x6 x7 . (∀ x8 : ι → ο . x8 x2 ⟶ x8 x3 ⟶ x8 x4 ⟶ x8 x5 ⟶ x8 x6 ⟶ x8 x7 ⟶ ∀ x9 . x0 x9 ⟶ x8 x9) ⟶ (∀ x8 : ι → ο . x8 x2 ⟶ x8 x3 ⟶ x8 x4 ⟶ x8 x5 ⟶ ∀ x9 . x1 x9 ⟶ x8 x9) ⟶ x0 x2 ⟶ (∀ x8 . x0 x8 ⟶ ∀ x9 : ι → ο . (x1 x8 ⟶ x9 x8) ⟶ x9 x6 ⟶ x9 x7 ⟶ x9 x8) ⟶ ∀ x8 x9 x10 : ι → ι . (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ x0 (ap (x8 x11) x12)) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ap (x8 x11) (ap (x8 x11) x12) = x12) ⟶ (∀ x11 . x0 x11 ⟶ ap (x8 x11) x2 = x3) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ x0 (ap (x9 x11) x12)) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ap (x9 x11) (ap (x9 x11) x12) = x12) ⟶ (∀ x11 . x0 x11 ⟶ ap (x9 x11) x2 = x4) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ x0 (ap (x10 x11) x12)) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ap (x10 x11) (ap (x10 x11) x12) = x12) ⟶ (∀ x11 . x0 x11 ⟶ ap (x10 x11) x2 = x5) ⟶ ∀ x11 : ι → ι → ι → ι → ο . (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ not (x11 x12 x3 x13 x2)) ⟶ (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ not (x11 x12 x4 x13 x2)) ⟶ (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ not (x11 x12 x5 x13 x2)) ⟶ (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ not (x11 x12 x6 x13 x2)) ⟶ (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ not (x11 x12 x7 x13 x2)) ⟶ ∀ x12 : ι → ι → ι → ι → ο . (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ ∀ x17 : ο . (x11 x13 x14 x15 x16 ⟶ x17) ⟶ (x12 x13 x14 x15 x16 ⟶ x17) ⟶ (x11 x15 x16 x13 x14 ⟶ x17) ⟶ x17) ⟶ (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ x12 x13 x14 x15 x16 ⟶ x12 x15 x16 x13 x14) ⟶ (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ x12 x13 x14 x15 x16 ⟶ x12 x13 (ap (x8 x13) x14) x15 (ap (x8 x15) x16)) ⟶ (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ x12 x13 x14 x15 x16 ⟶ x12 x13 (ap (x9 x13) x14) x15 (ap (x9 x15) x16)) ⟶ (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ x12 x13 x14 x15 x16 ⟶ x12 x13 (ap (x10 x13) x14) x15 (ap (x10 x15) x16)) ⟶ (∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ ∀ x20 . x0 x20 ⟶ ∀ x21 . x0 x21 ⟶ ∀ x22 . x0 x22 ⟶ ∀ x23 . x0 x23 ⟶ x11 x13 x2 x14 x15 ⟶ x11 x14 x15 x16 x17 ⟶ x11 x16 x17 x18 x19 ⟶ x11 x18 x19 x20 x21 ⟶ x11 x20 x21 x22 x23 ⟶ not (x12 x13 x2 x14 x15) ⟶ not (x12 x13 x2 x16 x17) ⟶ not (x12 x13 x2 x18 x19) ⟶ not (x12 x13 x2 x20 x21) ⟶ not (x12 x13 x2 x22 x23) ⟶ not (x12 x14 x15 x16 x17) ⟶ not (x12 x14 x15 x18 x19) ⟶ not (x12 x14 x15 x20 x21) ⟶ not (x12 x14 x15 x22 x23) ⟶ not (x12 x16 x17 x18 x19) ⟶ not (x12 x16 x17 x20 x21) ⟶ not (x12 x16 x17 x22 x23) ⟶ not (x12 x18 x19 x20 x21) ⟶ not (x12 x18 x19 x22 x23) ⟶ not (x12 x20 x21 x22 x23) ⟶ False) ⟶ (∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ ∀ x20 . x0 x20 ⟶ ∀ x21 . x0 x21 ⟶ ∀ x22 . x0 x22 ⟶ ∀ x23 . x0 x23 ⟶ ∀ x24 . x0 x24 ⟶ not (x1 x14) ⟶ not (x1 x16) ⟶ not (x1 x18) ⟶ not (x1 x20) ⟶ not (x1 x22) ⟶ not (x1 x24) ⟶ x11 x13 x14 x15 x16 ⟶ x11 x15 x16 x17 x18 ⟶ x11 x17 x18 x19 x20 ⟶ x11 x19 x20 x21 x22 ⟶ x11 x21 x22 x23 x24 ⟶ not (x12 x13 x14 x15 x16) ⟶ not (x12 x13 x14 x17 x18) ⟶ not (x12 x13 x14 x19 x20) ⟶ not (x12 x13 x14 x21 x22) ⟶ not (x12 x13 x14 x23 x24) ⟶ not (x12 x15 x16 x17 x18) ⟶ not (x12 x15 x16 x19 x20) ⟶ not (x12 x15 x16 x21 x22) ⟶ not (x12 x15 x16 x23 x24) ⟶ not (x12 x17 x18 x19 x20) ⟶ not (x12 x17 x18 x21 x22) ⟶ not (x12 x17 x18 x23 x24) ⟶ not (x12 x19 x20 x21 x22) ⟶ not (x12 x19 x20 x23 x24) ⟶ not (x12 x21 x22 x23 x24) ⟶ False) ⟶ ∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ ∀ x20 . x0 x20 ⟶ ∀ x21 . x0 x21 ⟶ ∀ x22 . x0 x22 ⟶ ∀ x23 . x0 x23 ⟶ ∀ x24 . x0 x24 ⟶ x11 x13 x14 x15 x16 ⟶ x11 x15 x16 x17 x18 ⟶ x11 x17 x18 x19 x20 ⟶ x11 x19 x20 x21 x22 ⟶ x11 x21 x22 x23 x24 ⟶ not (x12 x13 x14 x15 x16) ⟶ not (x12 x13 x14 x17 x18) ⟶ not (x12 x13 x14 x19 x20) ⟶ not (x12 x13 x14 x21 x22) ⟶ not (x12 x13 x14 x23 x24) ⟶ not (x12 x15 x16 x17 x18) ⟶ not (x12 x15 x16 x19 x20) ⟶ not (x12 x15 x16 x21 x22) ⟶ not (x12 x15 x16 x23 x24) ⟶ not (x12 x17 x18 x19 x20) ⟶ not (x12 x17 x18 x21 x22) ⟶ not (x12 x17 x18 x23 x24) ⟶ not (x12 x19 x20 x21 x22) ⟶ not (x12 x19 x20 x23 x24) ⟶ not (x12 x21 x22 x23 x24) ⟶ False (proof)
Theorem 5b8ac.. : ∀ x0 x1 : ι → ο . ∀ x2 x3 x4 x5 x6 x7 . (∀ x8 : ι → ο . x8 x2 ⟶ x8 x3 ⟶ x8 x4 ⟶ x8 x5 ⟶ x8 x6 ⟶ x8 x7 ⟶ ∀ x9 . x0 x9 ⟶ x8 x9) ⟶ (∀ x8 : ι → ο . x8 x2 ⟶ x8 x3 ⟶ x8 x4 ⟶ x8 x5 ⟶ ∀ x9 . x1 x9 ⟶ x8 x9) ⟶ x0 x2 ⟶ (∀ x8 . x0 x8 ⟶ ∀ x9 : ι → ο . (x1 x8 ⟶ x9 x8) ⟶ x9 x6 ⟶ x9 x7 ⟶ x9 x8) ⟶ ∀ x8 x9 x10 : ι → ι . (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ x0 (ap (x8 x11) x12)) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ap (x8 x11) (ap (x8 x11) x12) = x12) ⟶ (∀ x11 . x0 x11 ⟶ ap (x8 x11) x2 = x3) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ x0 (ap (x9 x11) x12)) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ap (x9 x11) (ap (x9 x11) x12) = x12) ⟶ (∀ x11 . x0 x11 ⟶ ap (x9 x11) x2 = x4) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ x0 (ap (x10 x11) x12)) ⟶ (∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ap (x10 x11) (ap (x10 x11) x12) = x12) ⟶ (∀ x11 . x0 x11 ⟶ ap (x10 x11) x2 = x5) ⟶ ∀ x11 : ι → ι → ι → ι → ο . (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ not (x11 x12 x3 x13 x2)) ⟶ (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ not (x11 x12 x4 x13 x2)) ⟶ (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ not (x11 x12 x5 x13 x2)) ⟶ (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ not (x11 x12 x6 x13 x2)) ⟶ (∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ not (x11 x12 x7 x13 x2)) ⟶ ∀ x12 : ι → ι → ι → ι → ο . (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ ∀ x17 : ο . (x11 x13 x14 x15 x16 ⟶ x17) ⟶ (x12 x13 x14 x15 x16 ⟶ x17) ⟶ (x11 x15 x16 x13 x14 ⟶ x17) ⟶ x17) ⟶ (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ x12 x13 x14 x15 x16 ⟶ x12 x15 x16 x13 x14) ⟶ (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ x12 x13 x14 x15 x16 ⟶ x12 x13 (ap (x8 x13) x14) x15 (ap (x8 x15) x16)) ⟶ (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ x12 x13 x14 x15 x16 ⟶ x12 x13 (ap (x9 x13) x14) x15 (ap (x9 x15) x16)) ⟶ (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ x12 x13 x14 x15 x16 ⟶ x12 x13 (ap (x10 x13) x14) x15 (ap (x10 x15) x16)) ⟶ (∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ ∀ x20 . x0 x20 ⟶ ∀ x21 . x0 x21 ⟶ ∀ x22 . x0 x22 ⟶ ∀ x23 . x0 x23 ⟶ x11 x13 x2 x14 x15 ⟶ x11 x14 x15 x16 x17 ⟶ x11 x16 x17 x18 x19 ⟶ x11 x18 x19 x20 x21 ⟶ x11 x20 x21 x22 x23 ⟶ not (x12 x13 x2 x14 x15) ⟶ not (x12 x13 x2 x16 x17) ⟶ not (x12 x13 x2 x18 x19) ⟶ not (x12 x13 x2 x20 x21) ⟶ not (x12 x13 x2 x22 x23) ⟶ not (x12 x14 x15 x16 x17) ⟶ not (x12 x14 x15 x18 x19) ⟶ not (x12 x14 x15 x20 x21) ⟶ not (x12 x14 x15 x22 x23) ⟶ not (x12 x16 x17 x18 x19) ⟶ not (x12 x16 x17 x20 x21) ⟶ not (x12 x16 x17 x22 x23) ⟶ not (x12 x18 x19 x20 x21) ⟶ not (x12 x18 x19 x22 x23) ⟶ not (x12 x20 x21 x22 x23) ⟶ False) ⟶ (∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ ∀ x20 . x0 x20 ⟶ ∀ x21 . x0 x21 ⟶ ∀ x22 . x0 x22 ⟶ ∀ x23 . x0 x23 ⟶ ∀ x24 . x0 x24 ⟶ not (x1 x14) ⟶ not (x1 x16) ⟶ not (x1 x18) ⟶ not (x1 x20) ⟶ not (x1 x22) ⟶ not (x1 x24) ⟶ x11 x13 x14 x15 x16 ⟶ x11 x15 x16 x17 x18 ⟶ x11 x17 x18 x19 x20 ⟶ x11 x19 x20 x21 x22 ⟶ x11 x21 x22 x23 x24 ⟶ not (x12 x13 x14 x15 x16) ⟶ not (x12 x13 x14 x17 x18) ⟶ not (x12 x13 x14 x19 x20) ⟶ not (x12 x13 x14 x21 x22) ⟶ not (x12 x13 x14 x23 x24) ⟶ not (x12 x15 x16 x17 x18) ⟶ not (x12 x15 x16 x19 x20) ⟶ not (x12 x15 x16 x21 x22) ⟶ not (x12 x15 x16 x23 x24) ⟶ not (x12 x17 x18 x19 x20) ⟶ not (x12 x17 x18 x21 x22) ⟶ not (x12 x17 x18 x23 x24) ⟶ not (x12 x19 x20 x21 x22) ⟶ not (x12 x19 x20 x23 x24) ⟶ not (x12 x21 x22 x23 x24) ⟶ False) ⟶ ∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ ∀ x20 . x0 x20 ⟶ ∀ x21 . x0 x21 ⟶ ∀ x22 . x0 x22 ⟶ ∀ x23 . x0 x23 ⟶ ∀ x24 . x0 x24 ⟶ not (x12 x13 x14 x15 x16) ⟶ not (x12 x13 x14 x17 x18) ⟶ not (x12 x13 x14 x19 x20) ⟶ not (x12 x13 x14 x21 x22) ⟶ not (x12 x13 x14 x23 x24) ⟶ not (x12 x15 x16 x17 x18) ⟶ not (x12 x15 x16 x19 x20) ⟶ not (x12 x15 x16 x21 x22) ⟶ not (x12 x15 x16 x23 x24) ⟶ not (x12 x17 x18 x19 x20) ⟶ not (x12 x17 x18 x21 x22) ⟶ not (x12 x17 x18 x23 x24) ⟶ not (x12 x19 x20 x21 x22) ⟶ not (x12 x19 x20 x23 x24) ⟶ not (x12 x21 x22 x23 x24) ⟶ False (proof)