Proofgold Asset

Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known 3c838.. : ∀ 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 ⟶ x18 = x5 ⟶ x19 = x6 ⟶ x28) ⟶ (x16 = x3 ⟶ x17 = x4 ⟶ x20 = x5 ⟶ x21 = x6 ⟶ x28) ⟶ (x16 = x3 ⟶ x17 = x4 ⟶ x22 = x5 ⟶ x23 = x6 ⟶ x28) ⟶ (x16 = x3 ⟶ x17 = x4 ⟶ x24 = x5 ⟶ x25 = x6 ⟶ x28) ⟶ (x16 = x3 ⟶ x17 = x4 ⟶ x26 = x5 ⟶ x27 = x6 ⟶ x28) ⟶ (x18 = x3 ⟶ x19 = x4 ⟶ x16 = x5 ⟶ x17 = x6 ⟶ x28) ⟶ (x18 = x3 ⟶ x19 = x4 ⟶ x20 = x5 ⟶ x21 = x6 ⟶ x28) ⟶ (x18 = x3 ⟶ x19 = x4 ⟶ x22 = x5 ⟶ x23 = x6 ⟶ x28) ⟶ (x18 = x3 ⟶ x19 = x4 ⟶ x24 = x5 ⟶ x25 = x6 ⟶ x28) ⟶ (x18 = x3 ⟶ x19 = x4 ⟶ x26 = x5 ⟶ x27 = x6 ⟶ x28) ⟶ (x20 = x3 ⟶ x21 = x4 ⟶ x16 = x5 ⟶ x17 = x6 ⟶ x28) ⟶ (x20 = x3 ⟶ x21 = x4 ⟶ x18 = x5 ⟶ x19 = x6 ⟶ x28) ⟶ (x20 = x3 ⟶ x21 = x4 ⟶ x22 = x5 ⟶ x23 = x6 ⟶ x28) ⟶ (x20 = x3 ⟶ x21 = x4 ⟶ x24 = x5 ⟶ x25 = x6 ⟶ x28) ⟶ (x20 = x3 ⟶ x21 = x4 ⟶ x26 = x5 ⟶ x27 = x6 ⟶ x28) ⟶ (x22 = x3 ⟶ x23 = x4 ⟶ x16 = x5 ⟶ x17 = x6 ⟶ x28) ⟶ (x22 = x3 ⟶ x23 = x4 ⟶ x18 = x5 ⟶ x19 = x6 ⟶ x28) ⟶ (x22 = x3 ⟶ x23 = x4 ⟶ x20 = x5 ⟶ x21 = x6 ⟶ x28) ⟶ (x22 = x3 ⟶ x23 = x4 ⟶ x24 = x5 ⟶ x25 = x6 ⟶ x28) ⟶ (x22 = x3 ⟶ x23 = x4 ⟶ x26 = x5 ⟶ x27 = x6 ⟶ x28) ⟶ (x24 = x3 ⟶ x25 = x4 ⟶ x16 = x5 ⟶ x17 = x6 ⟶ x28) ⟶ (x24 = x3 ⟶ x25 = x4 ⟶ x18 = x5 ⟶ x19 = x6 ⟶ x28) ⟶ (x24 = x3 ⟶ x25 = x4 ⟶ x20 = x5 ⟶ x21 = x6 ⟶ x28) ⟶ (x24 = x3 ⟶ x25 = x4 ⟶ x22 = x5 ⟶ x23 = x6 ⟶ x28) ⟶ (x24 = x3 ⟶ x25 = x4 ⟶ x26 = x5 ⟶ x27 = x6 ⟶ x28) ⟶ (x26 = x3 ⟶ x27 = x4 ⟶ x16 = x5 ⟶ x17 = x6 ⟶ x28) ⟶ (x26 = x3 ⟶ x27 = x4 ⟶ x18 = x5 ⟶ x19 = x6 ⟶ x28) ⟶ (x26 = x3 ⟶ x27 = x4 ⟶ x20 = x5 ⟶ x21 = x6 ⟶ x28) ⟶ (x26 = x3 ⟶ x27 = x4 ⟶ x22 = x5 ⟶ x23 = x6 ⟶ x28) ⟶ (x26 = x3 ⟶ x27 = x4 ⟶ x24 = x5 ⟶ x25 = x6 ⟶ x28) ⟶ x28) ⟶ x15) ⟶ x15
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Theorem 712bb.. : ∀ 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 ⟶ x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 x2 ⟶ x1 x3 ⟶ x1 x4 ⟶ x1 x5 ⟶ (x2 = x3 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x4 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x5 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x7 ⟶ ∀ x8 : ο . x8) ⟶ ∀ x8 : ι → ι → ι → ι → ο . (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x3 x10 x2)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x4 x10 x2)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x5 x10 x2)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x6 x10 x2)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x7 x10 x2)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x6 x10 x3)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x7 x10 x3)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x6 x10 x4)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x7 x10 x4)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x6 x10 x5)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x7 x10 x5)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x7 x10 x6)) ⟶ ∀ x9 : ι → ι → ι → ι → ο . (∀ x10 x11 x12 x13 . x0 x10 ⟶ x0 x11 ⟶ x0 x12 ⟶ x0 x13 ⟶ ∀ x14 : ο . (x8 x10 x11 x12 x13 ⟶ x14) ⟶ (x9 x10 x11 x12 x13 ⟶ x14) ⟶ (x8 x12 x13 x10 x11 ⟶ x14) ⟶ x14) ⟶ (∀ x10 x11 x12 x13 . x0 x10 ⟶ x0 x11 ⟶ x0 x12 ⟶ x0 x13 ⟶ x9 x10 x11 x12 x13 ⟶ x9 x12 x13 x10 x11) ⟶ ∀ x10 . x0 x10 ⟶ ∀ x11 . x0 x11 ⟶ ∀ x12 . x0 x12 ⟶ x1 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 ⟶ not (x9 x10 x2 x11 x12) ⟶ not (x9 x10 x2 x13 x14) ⟶ not (x9 x10 x2 x15 x16) ⟶ not (x9 x10 x2 x17 x18) ⟶ not (x9 x10 x2 x19 x20) ⟶ not (x9 x11 x12 x13 x14) ⟶ not (x9 x11 x12 x15 x16) ⟶ not (x9 x11 x12 x17 x18) ⟶ not (x9 x11 x12 x19 x20) ⟶ not (x9 x13 x14 x15 x16) ⟶ not (x9 x13 x14 x17 x18) ⟶ not (x9 x13 x14 x19 x20) ⟶ not (x9 x15 x16 x17 x18) ⟶ not (x9 x15 x16 x19 x20) ⟶ not (x9 x17 x18 x19 x20) ⟶ ∀ x21 : ο . (∀ x22 . x0 x22 ⟶ ∀ x23 . x0 x23 ⟶ x1 x23 ⟶ ∀ x24 . x0 x24 ⟶ ∀ x25 . x0 x25 ⟶ ∀ x26 . x0 x26 ⟶ ∀ x27 . x0 x27 ⟶ ∀ x28 . x0 x28 ⟶ ∀ x29 . x0 x29 ⟶ ∀ x30 . x0 x30 ⟶ ∀ x31 . x0 x31 ⟶ ∀ x32 . x0 x32 ⟶ x8 x22 x2 x24 x23 ⟶ x8 x24 x23 x25 x26 ⟶ x8 x25 x26 x27 x28 ⟶ x8 x27 x28 x29 x30 ⟶ x8 x29 x30 x31 x32 ⟶ not (x9 x22 x2 x24 x23) ⟶ not (x9 x22 x2 x25 x26) ⟶ not (x9 x22 x2 x27 x28) ⟶ not (x9 x22 x2 x29 x30) ⟶ not (x9 x22 x2 x31 x32) ⟶ not (x9 x24 x23 x25 x26) ⟶ not (x9 x24 x23 x27 x28) ⟶ not (x9 x24 x23 x29 x30) ⟶ not (x9 x24 x23 x31 x32) ⟶ not (x9 x25 x26 x27 x28) ⟶ not (x9 x25 x26 x29 x30) ⟶ not (x9 x25 x26 x31 x32) ⟶ not (x9 x27 x28 x29 x30) ⟶ not (x9 x27 x28 x31 x32) ⟶ not (x9 x29 x30 x31 x32) ⟶ x21) ⟶ x21 (proof)
Theorem 3dd41.. : ∀ 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 ⟶ x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 x2 ⟶ x1 x3 ⟶ x1 x4 ⟶ x1 x5 ⟶ (x2 = x3 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x4 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x5 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x7 ⟶ ∀ x8 : ο . x8) ⟶ ∀ x8 : ι → ι → ι → ι → ο . (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x3 x10 x2)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x4 x10 x2)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x5 x10 x2)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x6 x10 x2)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x7 x10 x2)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x6 x10 x3)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x7 x10 x3)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x6 x10 x4)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x7 x10 x4)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x6 x10 x5)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x7 x10 x5)) ⟶ (∀ x9 . x0 x9 ⟶ ∀ x10 . x0 x10 ⟶ not (x8 x9 x7 x10 x6)) ⟶ ∀ x9 : ι → ι → ι → ι → ο . (∀ x10 x11 x12 x13 . x0 x10 ⟶ x0 x11 ⟶ x0 x12 ⟶ x0 x13 ⟶ ∀ x14 : ο . (x8 x10 x11 x12 x13 ⟶ x14) ⟶ (x9 x10 x11 x12 x13 ⟶ x14) ⟶ (x8 x12 x13 x10 x11 ⟶ x14) ⟶ x14) ⟶ (∀ x10 x11 x12 x13 . x0 x10 ⟶ x0 x11 ⟶ x0 x12 ⟶ x0 x13 ⟶ x9 x10 x11 x12 x13 ⟶ x9 x12 x13 x10 x11) ⟶ ∀ x10 x11 x12 : ι → ι → ι . (∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ x0 (x10 x13 x14)) ⟶ (∀ x13 . x0 x13 ⟶ ∀ x14 . x1 x14 ⟶ x1 (x10 x13 x14)) ⟶ (∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ x10 x13 (x10 x13 x14) = x14) ⟶ (∀ x13 . x0 x13 ⟶ x10 x13 x2 = x3) ⟶ (∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ x0 (x11 x13 x14)) ⟶ (∀ x13 . x0 x13 ⟶ ∀ x14 . x1 x14 ⟶ x1 (x11 x13 x14)) ⟶ (∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ x11 x13 (x11 x13 x14) = x14) ⟶ (∀ x13 . x0 x13 ⟶ x11 x13 x2 = x4) ⟶ (∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ x0 (x12 x13 x14)) ⟶ (∀ x13 . x0 x13 ⟶ ∀ x14 . x1 x14 ⟶ x1 (x12 x13 x14)) ⟶ (∀ x13 . x0 x13 ⟶ ∀ x14 . x0 x14 ⟶ x12 x13 (x12 x13 x14) = x14) ⟶ (∀ x13 . x0 x13 ⟶ x12 x13 x2 = x5) ⟶ (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ not (x9 x13 x14 x15 x16) ⟶ not (x9 x13 (x10 x13 x14) x15 (x10 x15 x16))) ⟶ (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ not (x9 x13 x14 x15 x16) ⟶ not (x9 x13 (x11 x13 x14) x15 (x11 x15 x16))) ⟶ (∀ x13 x14 x15 x16 . x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x0 x16 ⟶ not (x9 x13 x14 x15 x16) ⟶ not (x9 x13 (x12 x13 x14) x15 (x12 x15 x16))) ⟶ ∀ x13 . x1 x13 ⟶ ∀ x14 . x0 x14 ⟶ ∀ x15 . x0 x15 ⟶ ∀ x16 . x0 x16 ⟶ x1 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 (x9 x14 x13 x15 x16) ⟶ not (x9 x14 x13 x17 x18) ⟶ not (x9 x14 x13 x19 x20) ⟶ not (x9 x14 x13 x21 x22) ⟶ not (x9 x14 x13 x23 x24) ⟶ not (x9 x15 x16 x17 x18) ⟶ not (x9 x15 x16 x19 x20) ⟶ not (x9 x15 x16 x21 x22) ⟶ not (x9 x15 x16 x23 x24) ⟶ not (x9 x17 x18 x19 x20) ⟶ not (x9 x17 x18 x21 x22) ⟶ not (x9 x17 x18 x23 x24) ⟶ not (x9 x19 x20 x21 x22) ⟶ not (x9 x19 x20 x23 x24) ⟶ not (x9 x21 x22 x23 x24) ⟶ ∀ x25 : ο . (∀ x26 . x0 x26 ⟶ ∀ x27 . x0 x27 ⟶ x1 x27 ⟶ ∀ x28 . x0 x28 ⟶ ∀ x29 . x0 x29 ⟶ ∀ x30 . x0 x30 ⟶ ∀ x31 . x0 x31 ⟶ ∀ x32 . x0 x32 ⟶ ∀ x33 . x0 x33 ⟶ ∀ x34 . x0 x34 ⟶ ∀ x35 . x0 x35 ⟶ ∀ x36 . x0 x36 ⟶ x8 x26 x2 x28 x27 ⟶ x8 x28 x27 x29 x30 ⟶ x8 x29 x30 x31 x32 ⟶ x8 x31 x32 x33 x34 ⟶ x8 x33 x34 x35 x36 ⟶ not (x9 x26 x2 x28 x27) ⟶ not (x9 x26 x2 x29 x30) ⟶ not (x9 x26 x2 x31 x32) ⟶ not (x9 x26 x2 x33 x34) ⟶ not (x9 x26 x2 x35 x36) ⟶ not (x9 x28 x27 x29 x30) ⟶ not (x9 x28 x27 x31 x32) ⟶ not (x9 x28 x27 x33 x34) ⟶ not (x9 x28 x27 x35 x36) ⟶ not (x9 x29 x30 x31 x32) ⟶ not (x9 x29 x30 x33 x34) ⟶ not (x9 x29 x30 x35 x36) ⟶ not (x9 x31 x32 x33 x34) ⟶ not (x9 x31 x32 x35 x36) ⟶ not (x9 x33 x34 x35 x36) ⟶ x25) ⟶ x25 (proof)