Proofgold Signed Transaction

Param ordsucc^ordsucc : ι → ι
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 u18 := ordsucc u17
Definition u19 := ordsucc u18
Definition u20 := ordsucc u19
Definition u21 := ordsucc u20
Definition u22 := ordsucc u21
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Known 179f3.. : 0 ∈ u21
Theorem c34a2.. : 0 ∈ u22 (proof)
Known 07fdb.. : u1 ∈ u21
Theorem 617e2.. : u1 ∈ u22 (proof)
Known c25ea.. : u2 ∈ u21
Theorem a7839.. : u2 ∈ u22 (proof)
Known 0750b.. : u3 ∈ u21
Theorem 9018e.. : u3 ∈ u22 (proof)
Known 701a9.. : u4 ∈ u21
Theorem 540e6.. : u4 ∈ u22 (proof)
Param ap^ap : ι → ι → ι
Param lam^Sigma : ι → (ι → ι) → ι
Param If_i^If_i : ο → ι → ι → ι
Known 48efb.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 . x3 ∈ x1 ⟶ ap (lam x1 (λ x5 . If_i (x5 = x3) x0 (x2 (ordsucc x3) x5))) x3 = x0
Theorem cc49b.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 . ap (lam 22 (λ x23 . If_i (x23 = 0) x0 (If_i (x23 = 1) x1 (If_i (x23 = 2) x2 (If_i (x23 = 3) x3 (If_i (x23 = 4) x4 (If_i (x23 = 5) x5 (If_i (x23 = 6) x6 (If_i (x23 = 7) x7 (If_i (x23 = 8) x8 (If_i (x23 = 9) x9 (If_i (x23 = 10) x10 (If_i (x23 = 11) x11 (If_i (x23 = 12) x12 (If_i (x23 = 13) x13 (If_i (x23 = 14) x14 (If_i (x23 = 15) x15 (If_i (x23 = 16) x16 (If_i (x23 = 17) x17 (If_i (x23 = 18) x18 (If_i (x23 = 19) x19 (If_i (x23 = 20) x20 x21)))))))))))))))))))))) 0 = x0 (proof)
Known d21a1.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 x4 . (x4 = x3 ⟶ ∀ x5 : ο . x5) ⟶ ap (lam x1 (λ x6 . If_i (x6 = x3) x0 (x2 (ordsucc x3) x6))) x4 = ap (lam x1 (x2 (ordsucc x3))) x4
Known neq_1_0^neq_1_0 : u1 = 0 ⟶ ∀ x0 : ο . x0
Theorem 7045e.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 . ap (lam 22 (λ x23 . If_i (x23 = 0) x0 (If_i (x23 = 1) x1 (If_i (x23 = 2) x2 (If_i (x23 = 3) x3 (If_i (x23 = 4) x4 (If_i (x23 = 5) x5 (If_i (x23 = 6) x6 (If_i (x23 = 7) x7 (If_i (x23 = 8) x8 (If_i (x23 = 9) x9 (If_i (x23 = 10) x10 (If_i (x23 = 11) x11 (If_i (x23 = 12) x12 (If_i (x23 = 13) x13 (If_i (x23 = 14) x14 (If_i (x23 = 15) x15 (If_i (x23 = 16) x16 (If_i (x23 = 17) x17 (If_i (x23 = 18) x18 (If_i (x23 = 19) x19 (If_i (x23 = 20) x20 x21)))))))))))))))))))))) u1 = x1 (proof)
Known neq_2_0^neq_2_0 : u2 = 0 ⟶ ∀ x0 : ο . x0
Known neq_2_1^neq_2_1 : u2 = u1 ⟶ ∀ x0 : ο . x0
Theorem 7132f.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 . ap (lam 22 (λ x23 . If_i (x23 = 0) x0 (If_i (x23 = 1) x1 (If_i (x23 = 2) x2 (If_i (x23 = 3) x3 (If_i (x23 = 4) x4 (If_i (x23 = 5) x5 (If_i (x23 = 6) x6 (If_i (x23 = 7) x7 (If_i (x23 = 8) x8 (If_i (x23 = 9) x9 (If_i (x23 = 10) x10 (If_i (x23 = 11) x11 (If_i (x23 = 12) x12 (If_i (x23 = 13) x13 (If_i (x23 = 14) x14 (If_i (x23 = 15) x15 (If_i (x23 = 16) x16 (If_i (x23 = 17) x17 (If_i (x23 = 18) x18 (If_i (x23 = 19) x19 (If_i (x23 = 20) x20 x21)))))))))))))))))))))) u2 = x2 (proof)
Known neq_3_0^neq_3_0 : u3 = 0 ⟶ ∀ x0 : ο . x0
Known neq_3_1^neq_3_1 : u3 = u1 ⟶ ∀ x0 : ο . x0
Known neq_3_2^neq_3_2 : u3 = u2 ⟶ ∀ x0 : ο . x0
Theorem c0b4c.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 . ap (lam 22 (λ x23 . If_i (x23 = 0) x0 (If_i (x23 = 1) x1 (If_i (x23 = 2) x2 (If_i (x23 = 3) x3 (If_i (x23 = 4) x4 (If_i (x23 = 5) x5 (If_i (x23 = 6) x6 (If_i (x23 = 7) x7 (If_i (x23 = 8) x8 (If_i (x23 = 9) x9 (If_i (x23 = 10) x10 (If_i (x23 = 11) x11 (If_i (x23 = 12) x12 (If_i (x23 = 13) x13 (If_i (x23 = 14) x14 (If_i (x23 = 15) x15 (If_i (x23 = 16) x16 (If_i (x23 = 17) x17 (If_i (x23 = 18) x18 (If_i (x23 = 19) x19 (If_i (x23 = 20) x20 x21)))))))))))))))))))))) u3 = x3 (proof)
Known neq_4_0^neq_4_0 : u4 = 0 ⟶ ∀ x0 : ο . x0
Known neq_4_1^neq_4_1 : u4 = u1 ⟶ ∀ x0 : ο . x0
Known neq_4_2^neq_4_2 : u4 = u2 ⟶ ∀ x0 : ο . x0
Known neq_4_3^neq_4_3 : u4 = u3 ⟶ ∀ x0 : ο . x0
Theorem 2bb23.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 . ap (lam 22 (λ x23 . If_i (x23 = 0) x0 (If_i (x23 = 1) x1 (If_i (x23 = 2) x2 (If_i (x23 = 3) x3 (If_i (x23 = 4) x4 (If_i (x23 = 5) x5 (If_i (x23 = 6) x6 (If_i (x23 = 7) x7 (If_i (x23 = 8) x8 (If_i (x23 = 9) x9 (If_i (x23 = 10) x10 (If_i (x23 = 11) x11 (If_i (x23 = 12) x12 (If_i (x23 = 13) x13 (If_i (x23 = 14) x14 (If_i (x23 = 15) x15 (If_i (x23 = 16) x16 (If_i (x23 = 17) x17 (If_i (x23 = 18) x18 (If_i (x23 = 19) x19 (If_i (x23 = 20) x20 x21)))))))))))))))))))))) u4 = x4 (proof)
Definition 55574.. := λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 . ap (lam 22 (λ x23 . If_i (x23 = 0) x1 (If_i (x23 = 1) x2 (If_i (x23 = 2) x3 (If_i (x23 = 3) x4 (If_i (x23 = 4) x5 (If_i (x23 = 5) x6 (If_i (x23 = 6) x7 (If_i (x23 = 7) x8 (If_i (x23 = 8) x9 (If_i (x23 = 9) x10 (If_i (x23 = 10) x11 (If_i (x23 = 11) x12 (If_i (x23 = 12) x13 (If_i (x23 = 13) x14 (If_i (x23 = 14) x15 (If_i (x23 = 15) x16 (If_i (x23 = 16) x17 (If_i (x23 = 17) x18 (If_i (x23 = 18) x19 (If_i (x23 = 19) x20 (If_i (x23 = 20) x21 x22)))))))))))))))))))))) x0
Theorem 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 (proof)
Theorem 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 (proof)
Theorem 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 (proof)
Theorem 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 (proof)
Theorem 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 (proof)