Proofgold Signed Transaction

Definition 84660.. := λ x0 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . ∀ x1 : (ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι) → ο . x1 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 . x2) ⟶ x1 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 . x3) ⟶ x1 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 . x4) ⟶ x1 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 . x5) ⟶ x1 x0
Theorem 3badb.. : 84660.. (λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x0) (proof)
Theorem 89411.. : 84660.. (λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x1) (proof)
Theorem f1b43.. : 84660.. (λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x2) (proof)
Theorem 3a574.. : 84660.. (λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x3) (proof)
Param ordsucc^ordsucc : ι → ι
Definition u1 := 1
Definition u2 := ordsucc u1
Definition u3 := ordsucc u2
Definition u4 := ordsucc u3
Param u17_to_Church17 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι
Known cases_4^cases_4 : ∀ x0 . x0 ∈ u4 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 u1 ⟶ x1 u2 ⟶ x1 u3 ⟶ x1 x0
Known c5926.. : u17_to_Church17 0 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x1
Known b0ce1.. : u17_to_Church17 u1 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x2
Known e8ec5.. : u17_to_Church17 u2 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x3
Known 1ef08.. : u17_to_Church17 u3 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x4
Theorem 308d2.. : ∀ x0 . x0 ∈ u4 ⟶ 84660.. (u17_to_Church17 x0) (proof)
Param Church17_p : (ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι) → ο
Known e70c8.. : Church17_p (λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x0)
Known 1b7f9.. : Church17_p (λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x1)
Known 25b64.. : Church17_p (λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x2)
Known 9e7eb.. : Church17_p (λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x3)
Theorem 08c30.. : ∀ x0 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . 84660.. x0 ⟶ Church17_p x0 (proof)
Definition Church17_lt8 := λ x0 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . ∀ x1 : (ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι) → ο . x1 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 . x2) ⟶ x1 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 . x3) ⟶ x1 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 . x4) ⟶ x1 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 . x5) ⟶ x1 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 . x6) ⟶ x1 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 . x7) ⟶ x1 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 . x8) ⟶ x1 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 . x9) ⟶ x1 x0
Definition TwoRamseyGraph_3_6_Church17 := λ x0 x1 : 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) (x1 x2 x2 x3 x2 x3 x3 x2 x3 x3 x3 x3 x2 x2 x3 x3 x3 x3) (x1 x2 x3 x2 x2 x3 x2 x3 x3 x2 x3 x3 x3 x3 x3 x2 x3 x3) (x1 x3 x2 x2 x2 x2 x3 x3 x3 x3 x2 x3 x3 x3 x2 x3 x3 x3) (x1 x3 x3 x3 x2 x2 x2 x2 x3 x3 x3 x2 x3 x3 x3 x3 x2 x3) (x1 x3 x3 x2 x3 x2 x2 x3 x2 x3 x3 x3 x2 x2 x3 x3 x3 x3) (x1 x3 x2 x3 x3 x2 x3 x2 x2 x2 x3 x3 x3 x3 x3 x2 x3 x3) (x1 x2 x3 x3 x3 x3 x2 x2 x2 x3 x2 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) (x1 x3 x3 x3 x2 x3 x3 x3 x2 x3 x2 x2 x3 x2 x3 x3 x2 x3) (x1 x2 x3 x3 x3 x2 x3 x3 x3 x3 x2 x2 x3 x3 x2 x2 x3 x3) (x1 x3 x2 x3 x3 x3 x2 x3 x3 x2 x3 x3 x2 x3 x3 x2 x2 x3) (x1 x3 x2 x3 x3 x3 x2 x3 x3 x2 x2 x3 x3 x2 x3 x3 x3 x2) (x1 x3 x3 x3 x2 x3 x3 x3 x2 x2 x3 x2 x3 x3 x2 x3 x3 x2) (x1 x3 x3 x2 x3 x3 x3 x2 x3 x3 x3 x2 x2 x3 x3 x2 x3 x2) (x1 x2 x3 x3 x3 x2 x3 x3 x3 x3 x2 x3 x2 x3 x3 x3 x2 x2) (x1 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x2 x2 x2 x2 x2)
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known 768c1.. : ((λ x1 x2 . x2) = λ x1 x2 . x1) ⟶ ∀ x0 : ο . x0
Theorem 9ff71.. : ∀ x0 x1 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church17_lt8 x0 ⟶ Church17_lt8 x1 ⟶ (TwoRamseyGraph_3_6_Church17 x0 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x11) = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x0 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x12) = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x1 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x11) = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x1 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x12) = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x0 x1 = λ x3 x4 . x4) ⟶ ∀ x2 : ο . (84660.. x0 ⟶ x2) ⟶ (84660.. x1 ⟶ x2) ⟶ x2 (proof)
Theorem 3f26c.. : ∀ x0 x1 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church17_lt8 x0 ⟶ Church17_lt8 x1 ⟶ (TwoRamseyGraph_3_6_Church17 x0 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x11) = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x0 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x13) = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x1 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x11) = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x1 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x13) = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x0 x1 = λ x3 x4 . x4) ⟶ ∀ x2 : ο . (84660.. x0 ⟶ x2) ⟶ (84660.. x1 ⟶ x2) ⟶ x2 (proof)
Theorem c3d94.. : ∀ x0 x1 x2 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church17_lt8 x0 ⟶ Church17_lt8 x1 ⟶ Church17_lt8 x2 ⟶ (TwoRamseyGraph_3_6_Church17 x0 x1 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 x0 x2 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 x1 x2 = λ x4 x5 . x5) ⟶ ∀ x3 : ο . (84660.. x0 ⟶ x3) ⟶ (84660.. x1 ⟶ x3) ⟶ (84660.. x2 ⟶ x3) ⟶ x3 (proof)
Known 51a81.. : Church17_p (λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4)
Known e224e.. : Church17_p (λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x5)
Known 5d397.. : Church17_p (λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x6)
Known 3b0d1.. : Church17_p (λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x7)
Theorem 27a8a.. : ∀ x0 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church17_lt8 x0 ⟶ Church17_p x0 (proof)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param u8 : ι
Param u17 : ι
Param nat_p^nat_p : ι → ο
Known nat_trans^nat_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known nat_17^nat_17 : nat_p u17
Known 6a4e9.. : u8 ∈ u17
Theorem 078d9.. : u8 ⊆ u17 (proof)
Param u5 : ι
Param u6 : ι
Param u7 : 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
Definition Church17_to_u17 := λ x0 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . x0 0 u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 u13 u14 u15 u16
Known In_0_4^In_0_4 : 0 ∈ 4
Known In_1_4^In_1_4 : 1 ∈ 4
Known In_2_4^In_2_4 : 2 ∈ 4
Known In_3_4^In_3_4 : 3 ∈ 4
Theorem 68886.. : ∀ x0 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . 84660.. x0 ⟶ Church17_to_u17 x0 ∈ u4 (proof)
Known 495f5.. : ∀ x0 . x0 ∈ u17 ⟶ Church17_to_u17 (u17_to_Church17 x0) = x0
Theorem b4098.. : ∀ x0 . x0 ∈ u17 ⟶ 84660.. (u17_to_Church17 x0) ⟶ x0 ∈ u4 (proof)
Param atleastp^atleastp : ι → ι → ο
Definition not^not := λ x0 : ο . x0 ⟶ False
Param TwoRamseyGraph_3_6_17 : ι → ι → ο
Param and^and : ο → ο → ο
Known f03aa.. : ∀ x0 . atleastp 3 x0 ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ (x2 = x3 ⟶ ∀ x5 : ο . x5) ⟶ (x2 = x4 ⟶ ∀ x5 : ο . x5) ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x1) ⟶ x1
Known 67bc1.. : ∀ x0 . x0 ∈ u8 ⟶ Church17_lt8 (u17_to_Church17 x0)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known d8d63.. : ∀ x0 x1 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church17_p x0 ⟶ Church17_p x1 ⟶ or (TwoRamseyGraph_3_6_Church17 x0 x1 = λ x3 x4 . x3) (TwoRamseyGraph_3_6_Church17 x0 x1 = λ x3 x4 . x4)
Known d0d1e.. : ∀ x0 . x0 ∈ u17 ⟶ ∀ x1 . x1 ∈ u17 ⟶ (TwoRamseyGraph_3_6_Church17 (u17_to_Church17 x0) (u17_to_Church17 x1) = λ x3 x4 . x3) ⟶ TwoRamseyGraph_3_6_17 x0 x1
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem 1e4e8.. : ∀ x0 . x0 ⊆ u8 ⟶ atleastp u3 x0 ⟶ (∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ not (TwoRamseyGraph_3_6_17 x1 x2)) ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ x0) (x2 ∈ u4) ⟶ x1) ⟶ x1 (proof)
Known e8716.. : ∀ x0 . atleastp u2 x0 ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x1) ⟶ x1
Known e7def.. : Church17_p (λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x8)
Known 48ba7.. : u17_to_Church17 u8 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x9
Known a8b9a.. : Church17_p (λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x9)
Known fd1a6.. : u9 ∈ u17
Known a3fb1.. : u17_to_Church17 u9 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x10
Theorem a2937.. : ∀ x0 . x0 ⊆ u8 ⟶ atleastp u2 x0 ⟶ (∀ x1 . x1 ∈ x0 ⟶ not (TwoRamseyGraph_3_6_17 x1 u8)) ⟶ (∀ x1 . x1 ∈ x0 ⟶ not (TwoRamseyGraph_3_6_17 x1 u9)) ⟶ (∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ not (TwoRamseyGraph_3_6_17 x1 x2)) ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ x0) (x2 ∈ u4) ⟶ x1) ⟶ x1 (proof)
Known 4f699.. : Church17_p (λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x10)
Known e886d.. : u10 ∈ u17
Known d7087.. : u17_to_Church17 u10 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x11
Theorem c0db6.. : ∀ x0 . x0 ⊆ u8 ⟶ atleastp u2 x0 ⟶ (∀ x1 . x1 ∈ x0 ⟶ not (TwoRamseyGraph_3_6_17 x1 u8)) ⟶ (∀ x1 . x1 ∈ x0 ⟶ not (TwoRamseyGraph_3_6_17 x1 u10)) ⟶ (∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ not (TwoRamseyGraph_3_6_17 x1 x2)) ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ x0) (x2 ∈ u4) ⟶ x1) ⟶ x1 (proof)