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)}
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param setminus^setminus : ι → ι → ι
Param Sing^Sing : ι → ι
Param binunion^binunion : ι → 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
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param ordsucc^ordsucc : ι → ι
Definition u1 := 1
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
Param atleastp^atleastp : ι → ι → ο
Known 4fb58..^{Pigeonhole_not_atleastp_ordsucc} : ∀ x0 . nat_p x0 ⟶ not (atleastp (ordsucc x0) x0)
Known atleastp_tra^atleastp_tra : ∀ x0 x1 x2 . atleastp x0 x1 ⟶ atleastp x1 x2 ⟶ atleastp x0 x2
Known equip_atleastp^{equip_atleastp} : ∀ x0 x1 . equip x0 x1 ⟶ atleastp x0 x1
Param setsum^setsum : ι → ι → ι
Known equip_tra^equip_tra : ∀ x0 x1 x2 . equip x0 x1 ⟶ equip x1 x2 ⟶ equip x0 x2
Param add_nat^add_nat : ι → ι → ι
Known add_nat_0R^add_nat_0R : ∀ x0 . add_nat x0 0 = x0
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Known nat_0^nat_0 : nat_p 0
Known c88e0.. : ∀ x0 x1 x2 x3 . nat_p x0 ⟶ nat_p x1 ⟶ equip x0 x2 ⟶ equip x1 x3 ⟶ equip (add_nat x0 x1) (setsum x2 x3)
Known nat_1^nat_1 : nat_p 1
Known equip_ref^equip_ref : ∀ x0 . equip x0 x0
Known equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Known d778e.. : ∀ x0 x1 x2 x3 . equip x0 x2 ⟶ equip x1 x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ nIn x4 x1) ⟶ equip (binunion x0 x1) (setsum x2 x3)
Known 5169f..^equip_Sing_1 : ∀ x0 . equip (Sing x0) u1
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known Subq_atleastp^{Subq_atleastp} : ∀ x0 x1 . x0 ⊆ x1 ⟶ atleastp x0 x1
Known binunion_Subq_min^{binunion_Subq_min} : ∀ x0 x1 x2 . x0 ⊆ x2 ⟶ x1 ⊆ x2 ⟶ binunion x0 x1 ⊆ x2
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 set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known binintersectE^{binintersectE} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ and (x2 ∈ x0) (x2 ∈ x1)
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Known binintersectI^{binintersectI} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ x1 ⟶ x2 ∈ binintersect x0 x1
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
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
Known 6cd03.. : ∀ x0 x1 . x1 ∈ x0 ⟶ Sing x1 ⊆ x0
Known setminusE1^setminusE1 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ x2 ∈ x0
Known SepE2^SepE2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x1 x2
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Known neq_i_sym^neq_i_sym : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Theorem 9a487.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2) ⟶ ∀ x2 x3 . nat_p x2 ⟶ ∀ x4 x5 . x5 ∈ DirGraphOutNeighbors x0 x1 x4 ⟶ ∀ x6 x7 x8 . x6 ⊆ x0 ⟶ x7 ⊆ x0 ⟶ x8 ⊆ x0 ⟶ x7 = setminus (DirGraphOutNeighbors x0 x1 x5) (Sing x4) ⟶ x8 = setminus {x10 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x4) (Sing x4))|equip (binintersect (DirGraphOutNeighbors x0 x1 x10) (DirGraphOutNeighbors x0 x1 x4)) x3} x7 ⟶ equip x6 x2 ⟶ (∀ x9 . x9 ∈ x6 ⟶ x9 = x5 ⟶ ∀ x10 : ο . x10) ⟶ (∀ x9 . x9 ∈ x6 ⟶ nIn x9 x7) ⟶ (∀ x9 . x9 ∈ x6 ⟶ nIn x9 x8) ⟶ (∀ x9 . x9 ∈ x6 ⟶ nIn x9 (DirGraphOutNeighbors x0 x1 x4)) ⟶ (∀ x9 . x9 ∈ x6 ⟶ nIn x9 (DirGraphOutNeighbors x0 x1 x5)) ⟶ (∀ x9 . x9 ∈ x6 ⟶ ∀ x10 . x10 ∈ x6 ⟶ (x9 = x10 ⟶ ∀ x11 : ο . x11) ⟶ ∀ x11 . x11 ∈ binintersect (DirGraphOutNeighbors x0 x1 x9) (DirGraphOutNeighbors x0 x1 x10) ⟶ x11 ∈ x6) ⟶ ∀ x9 : ι → ι . (∀ x10 . x10 ∈ x6 ⟶ x9 x10 ∈ x7) ⟶ (∀ x10 . x10 ∈ x6 ⟶ x1 x10 (x9 x10)) ⟶ (∀ x10 . x10 ∈ x7 ⟶ ∀ x11 : ο . (∀ x12 . and (x12 ∈ x6) (x9 x12 = x10) ⟶ x11) ⟶ x11) ⟶ (∀ x10 . x10 ∈ x8 ⟶ or (equip (binintersect (DirGraphOutNeighbors x0 x1 x10) (DirGraphOutNeighbors x0 x1 x5)) u1) (equip (binintersect (DirGraphOutNeighbors x0 x1 x10) (DirGraphOutNeighbors x0 x1 x5)) x3)) ⟶ equip {x10 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x5) (Sing x5))|equip (binintersect (DirGraphOutNeighbors x0 x1 x10) (DirGraphOutNeighbors x0 x1 x5)) u1} x2 ⟶ x8 ⊆ {x10 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x5) (Sing x5))|equip (binintersect (DirGraphOutNeighbors x0 x1 x10) (DirGraphOutNeighbors x0 x1 x5)) x3} (proof)
Param u3 : ι
Param u6 : ι
Theorem 18a4e.. : ∀ 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 . nat_p x2 ⟶ ∀ x4 x5 . x5 ∈ DirGraphOutNeighbors x0 x1 x4 ⟶ ∀ x6 x7 x8 . x6 ⊆ x0 ⟶ x7 ⊆ x0 ⟶ x8 ⊆ x0 ⟶ x7 = setminus (DirGraphOutNeighbors x0 x1 x5) (Sing x4) ⟶ x8 = setminus {x10 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x4) (Sing x4))|equip (binintersect (DirGraphOutNeighbors x0 x1 x10) (DirGraphOutNeighbors x0 x1 x4)) x3} x7 ⟶ equip x6 x2 ⟶ (∀ x9 . x9 ∈ x6 ⟶ x9 = x5 ⟶ ∀ x10 : ο . x10) ⟶ (∀ x9 . x9 ∈ x6 ⟶ nIn x9 x7) ⟶ (∀ x9 . x9 ∈ x6 ⟶ nIn x9 x8) ⟶ (∀ x9 . x9 ∈ x6 ⟶ nIn x9 (DirGraphOutNeighbors x0 x1 x4)) ⟶ (∀ x9 . x9 ∈ x6 ⟶ nIn x9 (DirGraphOutNeighbors x0 x1 x5)) ⟶ (∀ x9 . x9 ∈ x6 ⟶ ∀ x10 . x10 ∈ x6 ⟶ (x9 = x10 ⟶ ∀ x11 : ο . x11) ⟶ ∀ x11 . x11 ∈ binintersect (DirGraphOutNeighbors x0 x1 x9) (DirGraphOutNeighbors x0 x1 x10) ⟶ x11 ∈ x6) ⟶ ∀ x9 : ι → ι . (∀ x10 . x10 ∈ x6 ⟶ x9 x10 ∈ x7) ⟶ (∀ x10 . x10 ∈ x6 ⟶ x1 x10 (x9 x10)) ⟶ (∀ x10 . x10 ∈ x7 ⟶ ∀ x11 : ο . (∀ x12 . and (x12 ∈ x6) (x9 x12 = x10) ⟶ x11) ⟶ x11) ⟶ (∀ x10 . x10 ∈ x8 ⟶ or (equip (binintersect (DirGraphOutNeighbors x0 x1 x10) (DirGraphOutNeighbors x0 x1 x5)) u1) (equip (binintersect (DirGraphOutNeighbors x0 x1 x10) (DirGraphOutNeighbors x0 x1 x5)) x3)) ⟶ equip {x10 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x5) (Sing x5))|equip (binintersect (DirGraphOutNeighbors x0 x1 x10) (DirGraphOutNeighbors x0 x1 x5)) u1} x2 ⟶ x8 ⊆ {x10 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x5) (Sing x5))|equip (binintersect (DirGraphOutNeighbors x0 x1 x10) (DirGraphOutNeighbors x0 x1 x5)) x3} (proof)