Proofgold Address

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
Theorem 3c6b1.. : ∀ x0 : ι → ο . ∀ x1 x2 x3 x4 x5 x6 . x0 x1 ⟶ ∀ x7 : ι → ι . (∀ x8 . x0 x8 ⟶ ∀ x9 . x0 x9 ⟶ x0 (ap (x7 x8) x9)) ⟶ (∀ x8 . x0 x8 ⟶ ∀ x9 . x0 x9 ⟶ ap (x7 x8) (ap (x7 x8) x9) = x9) ⟶ (∀ x8 . x0 x8 ⟶ ap (x7 x8) x1 = x2) ⟶ ∀ x8 : ι → ι → ι → ι → ο . (∀ x9 x10 x11 x12 . x0 x9 ⟶ x0 x10 ⟶ x0 x11 ⟶ x0 x12 ⟶ x8 x9 x10 x11 x12 ⟶ x8 x9 (ap (x7 x9) x10) x11 (ap (x7 x11) x12)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ ∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ not (x8 x9 x1 x10 x11) ⟶ not (x8 x9 x1 x12 x13) ⟶ not (x8 x9 x1 x14 x15) ⟶ not (x8 x9 x1 x16 x17) ⟶ not (x8 x9 x1 x18 x19) ⟶ not (x8 x10 x11 x12 x13) ⟶ not (x8 x10 x11 x14 x15) ⟶ not (x8 x10 x11 x16 x17) ⟶ not (x8 x10 x11 x18 x19) ⟶ not (x8 x12 x13 x14 x15) ⟶ not (x8 x12 x13 x16 x17) ⟶ not (x8 x12 x13 x18 x19) ⟶ not (x8 x14 x15 x16 x17) ⟶ not (x8 x14 x15 x18 x19) ⟶ not (x8 x16 x17 x18 x19) ⟶ False) ⟶ ∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ ∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ not (x8 x9 x2 x10 x11) ⟶ not (x8 x9 x2 x12 x13) ⟶ not (x8 x9 x2 x14 x15) ⟶ not (x8 x9 x2 x16 x17) ⟶ not (x8 x9 x2 x18 x19) ⟶ not (x8 x10 x11 x12 x13) ⟶ not (x8 x10 x11 x14 x15) ⟶ not (x8 x10 x11 x16 x17) ⟶ not (x8 x10 x11 x18 x19) ⟶ not (x8 x12 x13 x14 x15) ⟶ not (x8 x12 x13 x16 x17) ⟶ not (x8 x12 x13 x18 x19) ⟶ not (x8 x14 x15 x16 x17) ⟶ not (x8 x14 x15 x18 x19) ⟶ not (x8 x16 x17 x18 x19) ⟶ False (proof)
Theorem f8a5d.. : ∀ x0 : ι → ο . ∀ x1 x2 x3 x4 x5 x6 . x0 x1 ⟶ ∀ x7 : ι → ι . (∀ x8 . x0 x8 ⟶ ∀ x9 . x0 x9 ⟶ x0 (ap (x7 x8) x9)) ⟶ (∀ x8 . x0 x8 ⟶ ∀ x9 . x0 x9 ⟶ ap (x7 x8) (ap (x7 x8) x9) = x9) ⟶ (∀ x8 . x0 x8 ⟶ ap (x7 x8) x1 = x3) ⟶ ∀ x8 : ι → ι → ι → ι → ο . (∀ x9 x10 x11 x12 . x0 x9 ⟶ x0 x10 ⟶ x0 x11 ⟶ x0 x12 ⟶ x8 x9 x10 x11 x12 ⟶ x8 x9 (ap (x7 x9) x10) x11 (ap (x7 x11) x12)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ ∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ not (x8 x9 x1 x10 x11) ⟶ not (x8 x9 x1 x12 x13) ⟶ not (x8 x9 x1 x14 x15) ⟶ not (x8 x9 x1 x16 x17) ⟶ not (x8 x9 x1 x18 x19) ⟶ not (x8 x10 x11 x12 x13) ⟶ not (x8 x10 x11 x14 x15) ⟶ not (x8 x10 x11 x16 x17) ⟶ not (x8 x10 x11 x18 x19) ⟶ not (x8 x12 x13 x14 x15) ⟶ not (x8 x12 x13 x16 x17) ⟶ not (x8 x12 x13 x18 x19) ⟶ not (x8 x14 x15 x16 x17) ⟶ not (x8 x14 x15 x18 x19) ⟶ not (x8 x16 x17 x18 x19) ⟶ False) ⟶ ∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ ∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ not (x8 x9 x3 x10 x11) ⟶ not (x8 x9 x3 x12 x13) ⟶ not (x8 x9 x3 x14 x15) ⟶ not (x8 x9 x3 x16 x17) ⟶ not (x8 x9 x3 x18 x19) ⟶ not (x8 x10 x11 x12 x13) ⟶ not (x8 x10 x11 x14 x15) ⟶ not (x8 x10 x11 x16 x17) ⟶ not (x8 x10 x11 x18 x19) ⟶ not (x8 x12 x13 x14 x15) ⟶ not (x8 x12 x13 x16 x17) ⟶ not (x8 x12 x13 x18 x19) ⟶ not (x8 x14 x15 x16 x17) ⟶ not (x8 x14 x15 x18 x19) ⟶ not (x8 x16 x17 x18 x19) ⟶ False (proof)
Theorem 37836.. : ∀ x0 : ι → ο . ∀ x1 x2 x3 x4 x5 x6 . x0 x1 ⟶ ∀ x7 : ι → ι . (∀ x8 . x0 x8 ⟶ ∀ x9 . x0 x9 ⟶ x0 (ap (x7 x8) x9)) ⟶ (∀ x8 . x0 x8 ⟶ ∀ x9 . x0 x9 ⟶ ap (x7 x8) (ap (x7 x8) x9) = x9) ⟶ (∀ x8 . x0 x8 ⟶ ap (x7 x8) x1 = x4) ⟶ ∀ x8 : ι → ι → ι → ι → ο . (∀ x9 x10 x11 x12 . x0 x9 ⟶ x0 x10 ⟶ x0 x11 ⟶ x0 x12 ⟶ x8 x9 x10 x11 x12 ⟶ x8 x9 (ap (x7 x9) x10) x11 (ap (x7 x11) x12)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ ∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ not (x8 x9 x1 x10 x11) ⟶ not (x8 x9 x1 x12 x13) ⟶ not (x8 x9 x1 x14 x15) ⟶ not (x8 x9 x1 x16 x17) ⟶ not (x8 x9 x1 x18 x19) ⟶ not (x8 x10 x11 x12 x13) ⟶ not (x8 x10 x11 x14 x15) ⟶ not (x8 x10 x11 x16 x17) ⟶ not (x8 x10 x11 x18 x19) ⟶ not (x8 x12 x13 x14 x15) ⟶ not (x8 x12 x13 x16 x17) ⟶ not (x8 x12 x13 x18 x19) ⟶ not (x8 x14 x15 x16 x17) ⟶ not (x8 x14 x15 x18 x19) ⟶ not (x8 x16 x17 x18 x19) ⟶ False) ⟶ ∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ ∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ ∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ ∀ x17 . x0 x17 ⟶ ∀ x18 . x0 x18 ⟶ ∀ x19 . x0 x19 ⟶ not (x8 x9 x4 x10 x11) ⟶ not (x8 x9 x4 x12 x13) ⟶ not (x8 x9 x4 x14 x15) ⟶ not (x8 x9 x4 x16 x17) ⟶ not (x8 x9 x4 x18 x19) ⟶ not (x8 x10 x11 x12 x13) ⟶ not (x8 x10 x11 x14 x15) ⟶ not (x8 x10 x11 x16 x17) ⟶ not (x8 x10 x11 x18 x19) ⟶ not (x8 x12 x13 x14 x15) ⟶ not (x8 x12 x13 x16 x17) ⟶ not (x8 x12 x13 x18 x19) ⟶ not (x8 x14 x15 x16 x17) ⟶ not (x8 x14 x15 x18 x19) ⟶ not (x8 x16 x17 x18 x19) ⟶ False (proof)