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Proofgold Asset

asset id
0d1818580b9c945f484ba727240127d1b54a985f25a4c588ff95dc22ac33263c
asset hash
6beb91e4c28b305728bd5d9122a49fbe7a70e713b6594848e83e968455bb77a1
bday / block
34653
tx
82125..
preasset
doc published by Pr4zB..
Param 4402e.. : ι(ιιο) → ο
Param cf2df.. : ι(ιιο) → ο
Definition SubqSubq := λ x0 x1 . ∀ x2 . x2x0x2x1
Param setminussetminus : ιιι
Param SingSing : ιι
Definition FalseFalse := ∀ x0 : ο . x0
Definition notnot := λ x0 : ο . x0False
Definition 8b6ad.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 . ∀ x5 : ο . ((x1 = x2∀ x6 : ο . x6)(x1 = x3∀ x6 : ο . x6)(x2 = x3∀ x6 : ο . x6)(x1 = x4∀ x6 : ο . x6)(x2 = x4∀ x6 : ο . x6)(x3 = x4∀ x6 : ο . x6)not (x0 x1 x2)not (x0 x1 x3)not (x0 x2 x3)not (x0 x1 x4)not (x0 x2 x4)not (x0 x3 x4)x5)x5
Definition c5756.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 . ∀ x6 : ο . (8b6ad.. x0 x1 x2 x3 x4(x1 = x5∀ x7 : ο . x7)(x2 = x5∀ x7 : ο . x7)(x3 = x5∀ x7 : ο . x7)(x4 = x5∀ x7 : ο . x7)not (x0 x1 x5)not (x0 x2 x5)x0 x3 x5x0 x4 x5x6)x6
Definition 2de86.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 . ∀ x7 : ο . (c5756.. x0 x1 x2 x3 x4 x5(x1 = x6∀ x8 : ο . x8)(x2 = x6∀ x8 : ο . x8)(x3 = x6∀ x8 : ο . x8)(x4 = x6∀ x8 : ο . x8)(x5 = x6∀ x8 : ο . x8)not (x0 x1 x6)x0 x2 x6not (x0 x3 x6)x0 x4 x6not (x0 x5 x6)x7)x7
Definition 02f3e.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 . ∀ x8 : ο . (2de86.. x0 x1 x2 x3 x4 x5 x6(x1 = x7∀ x9 : ο . x9)(x2 = x7∀ x9 : ο . x9)(x3 = x7∀ x9 : ο . x9)(x4 = x7∀ x9 : ο . x9)(x5 = x7∀ x9 : ο . x9)(x6 = x7∀ x9 : ο . x9)x0 x1 x7x0 x2 x7x0 x3 x7x0 x4 x7not (x0 x5 x7)not (x0 x6 x7)x8)x8
Definition a5c5d.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 . ∀ x9 : ο . (02f3e.. x0 x1 x2 x3 x4 x5 x6 x7(x1 = x8∀ x10 : ο . x10)(x2 = x8∀ x10 : ο . x10)(x3 = x8∀ x10 : ο . x10)(x4 = x8∀ x10 : ο . x10)(x5 = x8∀ x10 : ο . x10)(x6 = x8∀ x10 : ο . x10)(x7 = x8∀ x10 : ο . x10)not (x0 x1 x8)not (x0 x2 x8)not (x0 x3 x8)x0 x4 x8not (x0 x5 x8)not (x0 x6 x8)not (x0 x7 x8)x9)x9
Definition b0749.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 x9 . ∀ x10 : ο . (a5c5d.. x0 x1 x2 x3 x4 x5 x6 x7 x8(x1 = x9∀ x11 : ο . x11)(x2 = x9∀ x11 : ο . x11)(x3 = x9∀ x11 : ο . x11)(x4 = x9∀ x11 : ο . x11)(x5 = x9∀ x11 : ο . x11)(x6 = x9∀ x11 : ο . x11)(x7 = x9∀ x11 : ο . x11)(x8 = x9∀ x11 : ο . x11)x0 x1 x9not (x0 x2 x9)x0 x3 x9not (x0 x4 x9)not (x0 x5 x9)x0 x6 x9not (x0 x7 x9)x0 x8 x9x10)x10
Definition f1c88.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 . ∀ x11 : ο . (b0749.. x0 x1 x2 x3 x4 x5 x6 x7 x8 x9(x1 = x10∀ x12 : ο . x12)(x2 = x10∀ x12 : ο . x12)(x3 = x10∀ x12 : ο . x12)(x4 = x10∀ x12 : ο . x12)(x5 = x10∀ x12 : ο . x12)(x6 = x10∀ x12 : ο . x12)(x7 = x10∀ x12 : ο . x12)(x8 = x10∀ x12 : ο . x12)(x9 = x10∀ x12 : ο . x12)x0 x1 x10x0 x2 x10not (x0 x3 x10)not (x0 x4 x10)x0 x5 x10not (x0 x6 x10)not (x0 x7 x10)x0 x8 x10not (x0 x9 x10)x11)x11
Definition df3c6.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 . ∀ x12 : ο . (f1c88.. x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10(x1 = x11∀ x13 : ο . x13)(x2 = x11∀ x13 : ο . x13)(x3 = x11∀ x13 : ο . x13)(x4 = x11∀ x13 : ο . x13)(x5 = x11∀ x13 : ο . x13)(x6 = x11∀ x13 : ο . x13)(x7 = x11∀ x13 : ο . x13)(x8 = x11∀ x13 : ο . x13)(x9 = x11∀ x13 : ο . x13)(x10 = x11∀ x13 : ο . x13)not (x0 x1 x11)x0 x2 x11x0 x3 x11not (x0 x4 x11)not (x0 x5 x11)not (x0 x6 x11)not (x0 x7 x11)x0 x8 x11not (x0 x9 x11)not (x0 x10 x11)x12)x12
Definition 91ad9.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 . ∀ x13 : ο . (df3c6.. x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11(x1 = x12∀ x14 : ο . x14)(x2 = x12∀ x14 : ο . x14)(x3 = x12∀ x14 : ο . x14)(x4 = x12∀ x14 : ο . x14)(x5 = x12∀ x14 : ο . x14)(x6 = x12∀ x14 : ο . x14)(x7 = x12∀ x14 : ο . x14)(x8 = x12∀ x14 : ο . x14)(x9 = x12∀ x14 : ο . x14)(x10 = x12∀ x14 : ο . x14)(x11 = x12∀ x14 : ο . x14)x0 x1 x12not (x0 x2 x12)not (x0 x3 x12)not (x0 x4 x12)x0 x5 x12x0 x6 x12not (x0 x7 x12)not (x0 x8 x12)not (x0 x9 x12)not (x0 x10 x12)x0 x11 x12x13)x13
Definition andand := λ x0 x1 : ο . ∀ x2 : ο . (x0x1x2)x2
Definition nInnIn := λ x0 x1 . not (x0x1)
Known setminusEsetminusE : ∀ x0 x1 x2 . x2setminus x0 x1and (x2x0) (nIn x2 x1)
Definition oror := λ x0 x1 : ο . ∀ x2 : ο . (x0x2)(x1x2)x2
Known xmxm : ∀ x0 : ο . or x0 (not x0)
Known FalseEFalseE : False∀ x0 : ο . x0
Known 53a3c.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 . x2x0∀ x3 . x3x0not (x1 x2 x3)not (x1 x3 x2))cf2df.. x0 x1∀ x2 . x2x0∀ x3 . x3x0∀ x4 . x4x0∀ x5 . x5x0∀ x6 . x6x0(x2 = x3∀ x7 : ο . x7)(x2 = x4∀ x7 : ο . x7)(x3 = x4∀ x7 : ο . x7)(x2 = x5∀ x7 : ο . x7)(x3 = x5∀ x7 : ο . x7)(x4 = x5∀ x7 : ο . x7)(x2 = x6∀ x7 : ο . x7)(x3 = x6∀ x7 : ο . x7)(x4 = x6∀ x7 : ο . x7)(x5 = x6∀ x7 : ο . x7)not (x1 x2 x3)not (x1 x2 x4)not (x1 x3 x4)not (x1 x2 x5)not (x1 x3 x5)not (x1 x4 x5)not (x1 x2 x6)not (x1 x3 x6)not (x1 x4 x6)not (x1 x5 x6)False
Known 61345.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 . x2x0∀ x3 . x3x0x1 x2 x3x1 x3 x2)4402e.. x0 x1∀ x2 . x2x0∀ x3 . x3x0∀ x4 . x4x0(x2 = x3∀ x5 : ο . x5)(x2 = x4∀ x5 : ο . x5)(x3 = x4∀ x5 : ο . x5)x1 x2 x3x1 x2 x4x1 x3 x4False
Known Subq_traSubq_tra : ∀ x0 x1 x2 . x0x1x1x2x0x2
Known setminus_Subqsetminus_Subq : ∀ x0 x1 . setminus x0 x1x0
Known SingISingI : ∀ x0 . x0Sing x0
Theorem 5b508.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . (∀ x3 . x3x1∀ x4 . x4x1x2 x3 x4x2 x4 x3)4402e.. x1 x2cf2df.. x1 x2∀ x3 . x3x1x0setminus x1 (Sing x3)∀ x4 . x4x0∀ x5 . x5x0∀ x6 . x6x0∀ x7 . x7x0∀ x8 . x8x0∀ x9 . x9x0∀ x10 . x10x0∀ x11 . x11x0∀ x12 . x12x0∀ x13 . x13x0∀ x14 . x14x0∀ x15 . x15x091ad9.. x2 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15∀ x16 : ο . x16 (proof)