Proofgold Address

Param nat_p^nat_p : ι → ο
Param Sep^Sep : ι → (ι → ο) → ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition DirGraphOutNeighbors := λ x0 . λ x1 : ι → ι → ο . λ x2 . {x3 ∈ x0|and (x2 = x3 ⟶ ∀ x4 : ο . x4) (x1 x2 x3)}
Param setminus^setminus : ι → ι → ι
Param binunion^binunion : ι → ι → ι
Param Sing^Sing : ι → ι
Param equip^equip : ι → ι → ο
Param binintersect^binintersect : ι → ι → ι
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known setminusE^setminusE : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ and (x2 ∈ x0) (nIn x2 x1)
Known binintersectE^{binintersectE} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ and (x2 ∈ x0) (x2 ∈ x1)
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known SepE2^SepE2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x1 x2
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known setminusE2^setminusE2 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ nIn x2 x1
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Known cfabd.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2) ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ DirGraphOutNeighbors x0 x1 x2 ⟶ x2 ∈ DirGraphOutNeighbors x0 x1 x3
Theorem f38da.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2) ⟶ ∀ x2 . nat_p x2 ⟶ ∀ x3 x4 . x3 ∈ x0 ⟶ x4 ∈ DirGraphOutNeighbors x0 x1 x3 ⟶ (∀ x5 . x5 ∈ {x6 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x6) (DirGraphOutNeighbors x0 x1 x3)) x2} ⟶ not (x1 x4 x5)) ⟶ ∀ x5 : ι → ι . (∀ x6 . x6 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x4) (Sing x4)) ⟶ x5 x6 ∈ binintersect (DirGraphOutNeighbors x0 x1 x6) (DirGraphOutNeighbors x0 x1 x4)) ⟶ ∀ x6 . x6 ∈ {x7 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x7) (DirGraphOutNeighbors x0 x1 x3)) x2} ⟶ ∀ x7 : ο . (∀ x8 . and (x8 ∈ setminus (DirGraphOutNeighbors x0 x1 x4) (Sing x3)) (x1 x6 x8) ⟶ x7) ⟶ x7 (proof)
Param Subq^Subq : ι → ι → ο
Param atleastp^atleastp : ι → ι → ο
Param u3 : ι
Param u6 : ι
Theorem b51da.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2) ⟶ (∀ x2 . x2 ⊆ x0 ⟶ atleastp u3 x2 ⟶ not (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x1 x3 x4)) ⟶ (∀ x2 . x2 ⊆ x0 ⟶ atleastp u6 x2 ⟶ not (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x1 x3 x4))) ⟶ ∀ x2 . nat_p x2 ⟶ ∀ x3 x4 . x3 ∈ x0 ⟶ x4 ∈ DirGraphOutNeighbors x0 x1 x3 ⟶ (∀ x5 . x5 ∈ {x6 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x6) (DirGraphOutNeighbors x0 x1 x3)) x2} ⟶ not (x1 x4 x5)) ⟶ ∀ x5 : ι → ι . (∀ x6 . x6 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x4) (Sing x4)) ⟶ x5 x6 ∈ binintersect (DirGraphOutNeighbors x0 x1 x6) (DirGraphOutNeighbors x0 x1 x4)) ⟶ (∀ x6 . x6 ∈ {x7 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x7) (DirGraphOutNeighbors x0 x1 x3)) x2} ⟶ ∀ x7 . x7 ∈ DirGraphOutNeighbors x0 x1 x3 ⟶ x1 x7 x6 ⟶ x7 = x5 x6) ⟶ ∀ x6 . x6 ∈ {x7 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x7) (DirGraphOutNeighbors x0 x1 x3)) x2} ⟶ ∀ x7 : ο . (∀ x8 . and (x8 ∈ setminus (DirGraphOutNeighbors x0 x1 x4) (Sing x3)) (x1 x6 x8) ⟶ x7) ⟶ x7 (proof)