Proofgold Address

Param nat_p^nat_p : ι → ο
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param equip^equip : ι → ι → ο
Param binintersect^binintersect : ι → ι → ι
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 ordsucc^ordsucc : ι → ι
Param setminus^setminus : ι → ι → ι
Param binunion^binunion : ι → ι → ι
Param Sing^Sing : ι → ι
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 SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
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 SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Theorem 94e32.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2) ⟶ ∀ x2 x3 . x2 ∈ x0 ⟶ ∀ x4 . nat_p x4 ⟶ (∀ x5 . x5 ∈ x0 ⟶ (x5 = x2 ⟶ ∀ x6 : ο . x6) ⟶ not (x1 x5 x2) ⟶ or (equip (binintersect (DirGraphOutNeighbors x0 x1 x5) (DirGraphOutNeighbors x0 x1 x2)) x4) (equip (binintersect (DirGraphOutNeighbors x0 x1 x5) (DirGraphOutNeighbors x0 x1 x2)) (ordsucc x4))) ⟶ (∀ x5 . x5 ∈ {x6 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x2) (Sing x2))|equip (binintersect (DirGraphOutNeighbors x0 x1 x6) (DirGraphOutNeighbors x0 x1 x2)) x4} ⟶ not (x1 x3 x5)) ⟶ (∀ x5 . x5 ∈ setminus (DirGraphOutNeighbors x0 x1 x3) (Sing x2) ⟶ not (x1 x2 x5)) ⟶ ∀ x5 . x5 ∈ setminus (DirGraphOutNeighbors x0 x1 x3) (Sing x2) ⟶ equip (binintersect (DirGraphOutNeighbors x0 x1 x5) (DirGraphOutNeighbors x0 x1 x2)) (ordsucc x4) (proof)
Param Subq^Subq : ι → ι → ο
Param atleastp^atleastp : ι → ι → ο
Param u3 : ι
Param u6 : ι
Theorem 1efd9.. : ∀ 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 x3 . x2 ∈ x0 ⟶ ∀ x4 . nat_p x4 ⟶ (∀ x5 . x5 ∈ x0 ⟶ (x5 = x2 ⟶ ∀ x6 : ο . x6) ⟶ not (x1 x5 x2) ⟶ or (equip (binintersect (DirGraphOutNeighbors x0 x1 x5) (DirGraphOutNeighbors x0 x1 x2)) x4) (equip (binintersect (DirGraphOutNeighbors x0 x1 x5) (DirGraphOutNeighbors x0 x1 x2)) (ordsucc x4))) ⟶ (∀ x5 . x5 ∈ {x6 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x2) (Sing x2))|equip (binintersect (DirGraphOutNeighbors x0 x1 x6) (DirGraphOutNeighbors x0 x1 x2)) x4} ⟶ not (x1 x3 x5)) ⟶ (∀ x5 . x5 ∈ setminus (DirGraphOutNeighbors x0 x1 x3) (Sing x2) ⟶ not (x1 x2 x5)) ⟶ ∀ x5 . x5 ∈ setminus (DirGraphOutNeighbors x0 x1 x3) (Sing x2) ⟶ equip (binintersect (DirGraphOutNeighbors x0 x1 x5) (DirGraphOutNeighbors x0 x1 x2)) (ordsucc x4) (proof)