Proofgold Signed Transaction

Param ap^ap : ι → ι → ι
Param lam^Sigma : ι → (ι → ι) → ι
Param ordsucc^ordsucc : ι → ι
Param If_i^If_i : ο → ι → ι → ι
Definition u17_to_Church17_buggy := λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . ap (lam 17 (λ x18 . If_i (x18 = 0) x1 (If_i (x18 = 1) x18 (If_i (x18 = 2) x3 (If_i (x18 = 3) x4 (If_i (x18 = 4) x5 (If_i (x18 = 5) x6 (If_i (x18 = 6) x7 (If_i (x18 = 7) x8 (If_i (x18 = 8) x9 (If_i (x18 = 9) x10 (If_i (x18 = 10) x11 (If_i (x18 = 11) x12 (If_i (x18 = 12) x13 (If_i (x18 = 13) x14 (If_i (x18 = 14) x15 (If_i (x18 = 15) x16 x17))))))))))))))))) 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 u1 := 1
Definition u2 := ordsucc u1
Definition u3 := ordsucc u2
Definition u4 := ordsucc u3
Definition u5 := ordsucc u4
Definition u6 := ordsucc u5
Definition u7 := ordsucc u6
Definition u8 := ordsucc u7
Definition u9 := ordsucc u8
Definition u10 := ordsucc u9
Definition u11 := ordsucc u10
Definition u12 := ordsucc u11
Definition u13 := ordsucc u12
Definition u14 := ordsucc u13
Definition u15 := ordsucc u14
Definition u16 := ordsucc u15
Definition u17 := ordsucc u16
Definition TwoRamseyGraph_3_6_17_buggy := λ x0 x1 . x0 ∈ u17 ⟶ x1 ∈ u17 ⟶ TwoRamseyGraph_3_6_Church17 (u17_to_Church17_buggy x0) (u17_to_Church17_buggy x1) = λ x3 x4 . x3
Param Church17_p : (ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι) → ο
Definition False^False := ∀ x0 : ο . x0
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param atleastp^atleastp : ι → ι → ο
Definition not^not := λ x0 : ο . x0 ⟶ False
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
Theorem a48ed.. : (∀ x0 . x0 ∈ u17 ⟶ Church17_p (u17_to_Church17_buggy x0)) ⟶ (∀ x0 . x0 ∈ u17 ⟶ ∀ x1 . x1 ∈ u17 ⟶ u17_to_Church17_buggy x0 = u17_to_Church17_buggy x1 ⟶ x0 = x1) ⟶ (∀ x0 x1 x2 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church17_p x0 ⟶ Church17_p x1 ⟶ Church17_p x2 ⟶ (x0 = x1 ⟶ ∀ x3 : ο . x3) ⟶ (x0 = x2 ⟶ ∀ x3 : ο . x3) ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ (TwoRamseyGraph_3_6_Church17 x0 x1 = λ x4 x5 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x0 x2 = λ x4 x5 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x1 x2 = λ x4 x5 . x4) ⟶ False) ⟶ ∀ x0 . x0 ⊆ u17 ⟶ atleastp u3 x0 ⟶ not (∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ TwoRamseyGraph_3_6_17_buggy x1 x2) (proof)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition TwoRamseyProp_atleastp := λ x0 x1 x2 . ∀ x3 : ι → ι → ο . (∀ x4 x5 . x3 x4 x5 ⟶ x3 x5 x4) ⟶ or (∀ x4 : ο . (∀ x5 . and (x5 ⊆ x2) (and (atleastp x0 x5) (∀ x6 . x6 ∈ x5 ⟶ ∀ x7 . x7 ∈ x5 ⟶ (x6 = x7 ⟶ ∀ x8 : ο . x8) ⟶ x3 x6 x7)) ⟶ x4) ⟶ x4) (∀ x4 : ο . (∀ x5 . and (x5 ⊆ x2) (and (atleastp x1 x5) (∀ x6 . x6 ∈ x5 ⟶ ∀ x7 . x7 ∈ x5 ⟶ (x6 = x7 ⟶ ∀ x8 : ο . x8) ⟶ not (x3 x6 x7))) ⟶ x4) ⟶ x4)
Theorem 6e358.. : (∀ x0 . x0 ∈ u17 ⟶ Church17_p (u17_to_Church17_buggy x0)) ⟶ (∀ x0 . x0 ∈ u17 ⟶ ∀ x1 . x1 ∈ u17 ⟶ u17_to_Church17_buggy x0 = u17_to_Church17_buggy x1 ⟶ x0 = x1) ⟶ (∀ x0 x1 x2 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church17_p x0 ⟶ Church17_p x1 ⟶ Church17_p x2 ⟶ (x0 = x1 ⟶ ∀ x3 : ο . x3) ⟶ (x0 = x2 ⟶ ∀ x3 : ο . x3) ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ (TwoRamseyGraph_3_6_Church17 x0 x1 = λ x4 x5 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x0 x2 = λ x4 x5 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x1 x2 = λ x4 x5 . x4) ⟶ False) ⟶ (∀ x0 x1 . TwoRamseyGraph_3_6_17_buggy x0 x1 ⟶ TwoRamseyGraph_3_6_17_buggy x1 x0) ⟶ (∀ x0 . x0 ⊆ u17 ⟶ atleastp u6 x0 ⟶ not (∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ not (TwoRamseyGraph_3_6_17_buggy x1 x2))) ⟶ not (TwoRamseyProp_atleastp 3 6 17) (proof)
Param TwoRamseyProp^{TwoRamseyProp} : ι → ι → ι → ο
Known TwoRamseyProp_atleastp_atleastp : ∀ x0 x1 x2 x3 x4 . TwoRamseyProp x0 x2 x4 ⟶ atleastp x1 x0 ⟶ atleastp x3 x2 ⟶ TwoRamseyProp_atleastp x1 x3 x4
Known atleastp_ref : ∀ x0 . atleastp x0 x0
Theorem b410b.. : (∀ x0 . x0 ∈ u17 ⟶ Church17_p (u17_to_Church17_buggy x0)) ⟶ (∀ x0 . x0 ∈ u17 ⟶ ∀ x1 . x1 ∈ u17 ⟶ u17_to_Church17_buggy x0 = u17_to_Church17_buggy x1 ⟶ x0 = x1) ⟶ (∀ x0 x1 x2 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church17_p x0 ⟶ Church17_p x1 ⟶ Church17_p x2 ⟶ (x0 = x1 ⟶ ∀ x3 : ο . x3) ⟶ (x0 = x2 ⟶ ∀ x3 : ο . x3) ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ (TwoRamseyGraph_3_6_Church17 x0 x1 = λ x4 x5 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x0 x2 = λ x4 x5 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x1 x2 = λ x4 x5 . x4) ⟶ False) ⟶ (∀ x0 x1 . TwoRamseyGraph_3_6_17_buggy x0 x1 ⟶ TwoRamseyGraph_3_6_17_buggy x1 x0) ⟶ (∀ x0 . x0 ⊆ u17 ⟶ atleastp u6 x0 ⟶ not (∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ not (TwoRamseyGraph_3_6_17_buggy x1 x2))) ⟶ not (TwoRamseyProp 3 6 17) (proof)