Proofgold Signed Transaction

Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param u22 : ι
Param atleastp^atleastp : ι → ι → ο
Param ordsucc^ordsucc : ι → ι
Definition u1 := 1
Definition u2 := ordsucc u1
Definition u3 := ordsucc u2
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition 94591.. := λ x0 x1 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . λ x2 x3 . x0 (x1 x2 x2 x2 x3 x3 x3 x3 x2 x3 x3 x2 x3 x3 x3 x3 x2 x3 x2 x3 x3 x3 x3) (x1 x2 x2 x3 x2 x3 x3 x2 x3 x3 x3 x3 x2 x2 x3 x3 x3 x3 x3 x3 x3 x3 x2) (x1 x2 x3 x2 x2 x3 x2 x3 x3 x2 x3 x3 x3 x3 x3 x2 x3 x3 x3 x3 x3 x3 x2) (x1 x3 x2 x2 x2 x2 x3 x3 x3 x3 x2 x3 x3 x3 x2 x3 x3 x3 x2 x3 x3 x3 x3) (x1 x3 x3 x3 x2 x2 x2 x2 x3 x3 x3 x2 x3 x3 x3 x3 x2 x3 x3 x2 x3 x3 x3) (x1 x3 x3 x2 x3 x2 x2 x3 x2 x3 x3 x3 x2 x2 x3 x3 x3 x3 x3 x3 x3 x2 x3) (x1 x3 x2 x3 x3 x2 x3 x2 x2 x2 x3 x3 x3 x3 x3 x2 x3 x3 x3 x3 x3 x2 x3) (x1 x2 x3 x3 x3 x3 x2 x2 x2 x3 x2 x3 x3 x3 x2 x3 x3 x3 x3 x2 x3 x3 x3) (x1 x3 x3 x2 x3 x3 x3 x2 x3 x2 x3 x3 x2 x2 x2 x3 x3 x3 x3 x2 x3 x3 x3) (x1 x3 x3 x3 x2 x3 x3 x3 x2 x3 x2 x2 x3 x2 x3 x3 x2 x3 x3 x3 x3 x2 x3) (x1 x2 x3 x3 x3 x2 x3 x3 x3 x3 x2 x2 x3 x3 x2 x2 x3 x3 x3 x3 x3 x3 x3) (x1 x3 x2 x3 x3 x3 x2 x3 x3 x2 x3 x3 x2 x3 x3 x2 x2 x3 x2 x3 x3 x3 x3) (x1 x3 x2 x3 x3 x3 x2 x3 x3 x2 x2 x3 x3 x2 x3 x3 x3 x2 x3 x3 x2 x3 x3) (x1 x3 x3 x3 x2 x3 x3 x3 x2 x2 x3 x2 x3 x3 x2 x3 x3 x2 x3 x3 x2 x3 x3) (x1 x3 x3 x2 x3 x3 x3 x2 x3 x3 x3 x2 x2 x3 x3 x2 x3 x2 x3 x3 x2 x3 x3) (x1 x2 x3 x3 x3 x2 x3 x3 x3 x3 x2 x3 x2 x3 x3 x3 x2 x2 x3 x3 x2 x3 x3) (x1 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x2 x2 x2 x2 x2 x2 x3 x3 x3 x2) (x1 x2 x3 x3 x2 x3 x3 x3 x3 x3 x3 x3 x2 x3 x3 x3 x3 x2 x2 x2 x3 x2 x3) (x1 x3 x3 x3 x3 x2 x3 x3 x2 x2 x3 x3 x3 x3 x3 x3 x3 x3 x2 x2 x2 x3 x2) (x1 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x2 x2 x2 x2 x3 x3 x2 x2 x2 x3) (x1 x3 x3 x3 x3 x3 x2 x2 x3 x3 x2 x3 x3 x3 x3 x3 x3 x3 x2 x3 x2 x2 x2) (x1 x3 x2 x2 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x2 x3 x2 x3 x2 x2)
Param 55574.. : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι
Definition 0aea9.. := λ x0 x1 . x0 ∈ u22 ⟶ x1 ∈ u22 ⟶ 94591.. (55574.. x0) (55574.. x1) = λ x3 x4 . x3
Param ordinal^ordinal : ι → ο
Known 865bf.. : ∀ x0 . atleastp u3 x0 ⟶ (∀ x1 . x1 ∈ x0 ⟶ ordinal x1) ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 ∈ x3 ⟶ x3 ∈ x4 ⟶ x1) ⟶ x1
Param nat_p^nat_p : ι → ο
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known nat_p_trans^nat_p_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ nat_p x1
Known daa33.. : nat_p u22
Param binunion^binunion : ι → ι → ι
Param Sing^Sing : ι → ι
Definition SetAdjoin^SetAdjoin := λ x0 x1 . binunion x0 (Sing x1)
Param UPair^UPair : ι → ι → ι
Param setminus^setminus : ι → ι → ι
Param u17 : ι
Param u18 : ι
Param u19 : ι
Param u20 : ι
Param u21 : ι
Known dcb62.. : ∀ x0 . x0 ∈ setminus u22 u17 ⟶ ∀ x1 : ι → ο . x1 u17 ⟶ x1 u18 ⟶ x1 u19 ⟶ x1 u20 ⟶ x1 u21 ⟶ x1 x0
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Param TwoRamseyGraph_3_6_17 : ι → ι → ο
Known d6d79.. : ∀ x0 . x0 ⊆ u17 ⟶ atleastp u3 x0 ⟶ not (∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ TwoRamseyGraph_3_6_17 x1 x2)
Known 1dc5a.. : ∀ x0 x1 x2 . nIn x2 x1 ⟶ atleastp x0 x1 ⟶ atleastp (ordsucc x0) (binunion x1 (Sing x2))
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known UPairE^UPairE : ∀ x0 x1 x2 . x0 ∈ UPair x1 x2 ⟶ or (x0 = x1) (x0 = x2)
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Param equip^equip : ι → ι → ο
Known equip_atleastp^{equip_atleastp} : ∀ x0 x1 . equip x0 x1 ⟶ atleastp x0 x1
Known 1c73f.. : ∀ x0 x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ (∀ x3 . x3 ∈ x0 ⟶ or (x3 = x1) (x3 = x2)) ⟶ equip 2 x0
Known UPairI1^UPairI1 : ∀ x0 x1 . x0 ∈ UPair x0 x1
Known UPairI2^UPairI2 : ∀ x0 x1 . x1 ∈ UPair x0 x1
Known 48f02.. : ∀ x0 . x0 ∈ u17 ⟶ ∀ x1 . x1 ∈ u17 ⟶ 0aea9.. x0 x1 ⟶ TwoRamseyGraph_3_6_17 x0 x1
Known nat_trans^nat_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known nat_17^nat_17 : nat_p u17
Param u4 : ι
Param u5 : ι
Param u6 : ι
Param u7 : ι
Param u8 : ι
Param u9 : ι
Param u10 : ι
Param u11 : ι
Param u12 : ι
Param u13 : ι
Param u14 : ι
Param u15 : ι
Param u16 : ι
Known 66f20.. : ∀ x0 . x0 ∈ u17 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 u5 ⟶ x1 u6 ⟶ x1 u7 ⟶ x1 u8 ⟶ x1 u9 ⟶ x1 u10 ⟶ x1 u11 ⟶ x1 u12 ⟶ x1 u13 ⟶ x1 u14 ⟶ x1 u15 ⟶ x1 u16 ⟶ x1 x0
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known 768c1.. : ((λ x1 x2 . x2) = λ x1 x2 . x1) ⟶ ∀ x0 : ο . x0
Known e86b0.. : 55574.. u17 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x18
Known aafc6.. : 55574.. u1 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x2
Known 617e2.. : u1 ∈ u22
Known 96b76.. : u17 ∈ u22
Known fa851.. : 55574.. u2 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x3
Known a7839.. : u2 ∈ u22
Known cases_3^cases_3 : ∀ x0 . x0 ∈ u3 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 x0
Known 9379b.. : 55574.. u3 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x4
Known 7410a.. : 55574.. 0 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x1
Known c34a2.. : 0 ∈ u22
Known 9018e.. : u3 ∈ u22
Known 5f4d4.. : 55574.. u4 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x5
Known 540e6.. : u4 ∈ u22
Known b535d.. : 55574.. u5 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x6
Known 8a085.. : u5 ∈ u22
Known 8ef56.. : 55574.. u6 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x7
Known 8f513.. : u6 ∈ u22
Known 151b0.. : 55574.. u7 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x8
Known 3224f.. : u7 ∈ u22
Known 9e99f.. : 55574.. u8 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x9
Known e5453.. : u8 ∈ u22
Known 896c4.. : 55574.. u9 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x10
Known 8413f.. : u9 ∈ u22
Known 89d98.. : 55574.. u10 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x11
Known abaf6.. : u10 ∈ u22
Known 83484.. : ∀ x0 . x0 ∈ u11 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 u5 ⟶ x1 u6 ⟶ x1 u7 ⟶ x1 u8 ⟶ x1 u9 ⟶ x1 u10 ⟶ x1 x0
Known 76683.. : 55574.. u11 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x12
Known 0158f.. : u11 ∈ u22
Known 2ab0d.. : 55574.. u12 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x13
Known 126ca.. : u12 ∈ u22
Known 0b155.. : 55574.. u13 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x14
Known 49a59.. : u13 ∈ u22
Known 38fc2.. : 55574.. u14 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x15
Known caae0.. : u14 ∈ u22
Known 134b9.. : 55574.. u15 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x16
Known 9c9ec.. : u15 ∈ u22
Known 660da.. : ∀ x0 . x0 ∈ u16 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 u5 ⟶ x1 u6 ⟶ x1 u7 ⟶ x1 u8 ⟶ x1 u9 ⟶ x1 u10 ⟶ x1 u11 ⟶ x1 u12 ⟶ x1 u13 ⟶ x1 u14 ⟶ x1 u15 ⟶ x1 x0
Known b8157.. : 55574.. u16 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x17
Known d63b1.. : u16 ∈ u22
Known f9732.. : ∀ x0 . x0 ∈ u18 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 u5 ⟶ x1 u6 ⟶ x1 u7 ⟶ x1 u8 ⟶ x1 u9 ⟶ x1 u10 ⟶ x1 u11 ⟶ x1 u12 ⟶ x1 u13 ⟶ x1 u14 ⟶ x1 u15 ⟶ x1 u16 ⟶ x1 u17 ⟶ x1 x0
Known bb555.. : 55574.. u18 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x19
Known 7ba92.. : u18 ∈ u22
Known cases_4^cases_4 : ∀ x0 . x0 ∈ u4 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 x0
Known cases_7^cases_7 : ∀ x0 . x0 ∈ u7 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 u5 ⟶ x1 u6 ⟶ x1 x0
Known cases_8^cases_8 : ∀ x0 . x0 ∈ u8 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 u5 ⟶ x1 u6 ⟶ x1 u7 ⟶ x1 x0
Known 27a71.. : ∀ x0 . x0 ∈ u19 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 u5 ⟶ x1 u6 ⟶ x1 u7 ⟶ x1 u8 ⟶ x1 u9 ⟶ x1 u10 ⟶ x1 u11 ⟶ x1 u12 ⟶ x1 u13 ⟶ x1 u14 ⟶ x1 u15 ⟶ x1 u16 ⟶ x1 u17 ⟶ x1 u18 ⟶ x1 x0
Known 1435b.. : 55574.. u19 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x20
Known 91e6c.. : u19 ∈ u22
Known 866c8.. : ∀ x0 . x0 ∈ u12 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 u5 ⟶ x1 u6 ⟶ x1 u7 ⟶ x1 u8 ⟶ x1 u9 ⟶ x1 u10 ⟶ x1 u11 ⟶ x1 x0
Known 6de8d.. : ∀ x0 . x0 ∈ u13 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 u5 ⟶ x1 u6 ⟶ x1 u7 ⟶ x1 u8 ⟶ x1 u9 ⟶ x1 u10 ⟶ x1 u11 ⟶ x1 u12 ⟶ x1 x0
Known dca77.. : ∀ x0 . x0 ∈ u14 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 u5 ⟶ x1 u6 ⟶ x1 u7 ⟶ x1 u8 ⟶ x1 u9 ⟶ x1 u10 ⟶ x1 u11 ⟶ x1 u12 ⟶ x1 u13 ⟶ x1 x0
Known f3498.. : ∀ x0 . x0 ∈ u15 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 u5 ⟶ x1 u6 ⟶ x1 u7 ⟶ x1 u8 ⟶ x1 u9 ⟶ x1 u10 ⟶ x1 u11 ⟶ x1 u12 ⟶ x1 u13 ⟶ x1 u14 ⟶ x1 x0
Known 6ab99.. : ∀ x0 . x0 ∈ u20 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 u5 ⟶ x1 u6 ⟶ x1 u7 ⟶ x1 u8 ⟶ x1 u9 ⟶ x1 u10 ⟶ x1 u11 ⟶ x1 u12 ⟶ x1 u13 ⟶ x1 u14 ⟶ x1 u15 ⟶ x1 u16 ⟶ x1 u17 ⟶ x1 u18 ⟶ x1 u19 ⟶ x1 x0
Known 54789.. : 55574.. u20 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x21
Known 5e08e.. : u20 ∈ u22
Known cases_5^cases_5 : ∀ x0 . x0 ∈ u5 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 x0
Known cases_6^cases_6 : ∀ x0 . x0 ∈ u6 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 u5 ⟶ x1 x0
Known cases_9^cases_9 : ∀ x0 . x0 ∈ u9 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 u5 ⟶ x1 u6 ⟶ x1 u7 ⟶ x1 u8 ⟶ x1 x0
Known a039e.. : ∀ x0 . x0 ∈ u21 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 u4 ⟶ x1 u5 ⟶ x1 u6 ⟶ x1 u7 ⟶ x1 u8 ⟶ x1 u9 ⟶ x1 u10 ⟶ x1 u11 ⟶ x1 u12 ⟶ x1 u13 ⟶ x1 u14 ⟶ x1 u15 ⟶ x1 u16 ⟶ x1 u17 ⟶ x1 u18 ⟶ x1 u19 ⟶ x1 u20 ⟶ x1 x0
Known cases_1^cases_1 : ∀ x0 . x0 ∈ u1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Known 667cd.. : 55574.. u21 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . x22
Known 01ee3.. : u21 ∈ u22
Known cases_2^cases_2 : ∀ x0 . x0 ∈ u2 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 x0
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Theorem 907a2.. : ∀ x0 . x0 ⊆ u22 ⟶ atleastp u3 x0 ⟶ not (∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ 0aea9.. x1 x2) (proof)