Proofgold Signed Transaction

Param lam^Sigma : ι → (ι → ι) → ι
Definition setprod^setprod := λ x0 x1 . lam x0 (λ x2 . x1)
Param real^real : ι
Param ad280.. : ι → ι → ι
Param ap^ap : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
Definition e523d.. := {ad280.. (ap x0 0) (ap x0 1)|x0 ∈ setprod real real}
Param If_i^If_i : ο → ι → ι → ι
Known tuple_2_0_eq^tuple_2_0_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 0 = x0
Known tuple_2_1_eq^tuple_2_1_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 1 = x1
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Known tuple_2_setprod^{tuple_2_setprod} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 ⟶ lam 2 (λ x4 . If_i (x4 = 0) x2 x3) ∈ setprod x0 x1
Theorem d30b7.. : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ ad280.. x0 x1 ∈ e523d.. (proof)
Known ReplE_impred^ReplE_impred : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ prim5 x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3
Known ap0_Sigma^ap0_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 0 ∈ x0
Known ap1_Sigma^ap1_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 1 ∈ x1 (ap x2 0)
Theorem 37b9f.. : ∀ x0 . x0 ∈ e523d.. ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ real ⟶ ∀ x3 . x3 ∈ real ⟶ x0 = ad280.. x2 x3 ⟶ x1) ⟶ x1 (proof)
Param c7ce4.. : ι → ο
Param SNo^SNo : ι → ο
Known e4080.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ c7ce4.. (ad280.. x0 x1)
Known real_SNo^real_SNo : ∀ x0 . x0 ∈ real ⟶ SNo x0
Theorem 8a1f4.. : ∀ x0 . x0 ∈ e523d.. ⟶ c7ce4.. x0 (proof)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known 43bcd.. : ∀ x0 . ad280.. x0 0 = x0
Known real_0^real_0 : 0 ∈ real
Theorem cf956.. : real ⊆ e523d.. (proof)
Theorem 539a5.. : 0 ∈ e523d.. (proof)
Known real_1^real_1 : 1 ∈ real
Theorem b8ac1.. : 1 ∈ e523d.. (proof)
Definition 8d0f8.. := ad280.. 0 1
Theorem 3e36c.. : 8d0f8.. ∈ e523d.. (proof)
Param 28f8d.. : ι → ι
Known 1c01b.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ 28f8d.. (ad280.. x0 x1) = x0
Theorem 1d845.. : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ 28f8d.. (ad280.. x0 x1) = x0 (proof)
Param d634d.. : ι → ι
Known 5b520.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ d634d.. (ad280.. x0 x1) = x1
Theorem 7419a.. : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ d634d.. (ad280.. x0 x1) = x1 (proof)
Theorem 2c805.. : ∀ x0 . x0 ∈ e523d.. ⟶ 28f8d.. x0 ∈ real (proof)
Theorem 5adde.. : ∀ x0 . x0 ∈ e523d.. ⟶ d634d.. x0 ∈ real (proof)
Known 371c6.. : ∀ x0 x1 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ 28f8d.. x0 = 28f8d.. x1 ⟶ d634d.. x0 = d634d.. x1 ⟶ x0 = x1
Theorem b03c2.. : ∀ x0 . x0 ∈ e523d.. ⟶ ∀ x1 . x1 ∈ e523d.. ⟶ 28f8d.. x0 = 28f8d.. x1 ⟶ d634d.. x0 = d634d.. x1 ⟶ x0 = x1 (proof)
Param minus_SNo^minus_SNo : ι → ι
Definition b1848.. := λ x0 . ad280.. (minus_SNo (28f8d.. x0)) (minus_SNo (d634d.. x0))
Known real_minus_SNo^{real_minus_SNo} : ∀ x0 . x0 ∈ real ⟶ minus_SNo x0 ∈ real
Theorem a0a29.. : ∀ x0 . x0 ∈ e523d.. ⟶ b1848.. x0 ∈ e523d.. (proof)
Param add_SNo^add_SNo : ι → ι → ι
Definition a0628.. := λ x0 x1 . ad280.. (add_SNo (28f8d.. x0) (28f8d.. x1)) (add_SNo (d634d.. x0) (d634d.. x1))
Known real_add_SNo^real_add_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ add_SNo x0 x1 ∈ real
Theorem 2c51c.. : ∀ x0 . x0 ∈ e523d.. ⟶ ∀ x1 . x1 ∈ e523d.. ⟶ a0628.. x0 x1 ∈ e523d.. (proof)
Param mul_SNo^mul_SNo : ι → ι → ι
Definition 22598.. := λ x0 x1 . ad280.. (add_SNo (mul_SNo (28f8d.. x0) (28f8d.. x1)) (minus_SNo (mul_SNo (d634d.. x0) (d634d.. x1)))) (add_SNo (mul_SNo (28f8d.. x0) (d634d.. x1)) (mul_SNo (d634d.. x0) (28f8d.. x1)))
Known real_mul_SNo^real_mul_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ mul_SNo x0 x1 ∈ real
Theorem 4e788.. : ∀ x0 . x0 ∈ e523d.. ⟶ ∀ x1 . x1 ∈ e523d.. ⟶ 22598.. x0 x1 ∈ e523d.. (proof)
Theorem 7f5c4.. : ∀ x0 . x0 ∈ real ⟶ 28f8d.. x0 = x0 (proof)
Theorem 6ab69.. : ∀ x0 . x0 ∈ real ⟶ d634d.. x0 = 0 (proof)
Known mul_SNo_zeroL^{mul_SNo_zeroL} : ∀ x0 . SNo x0 ⟶ mul_SNo 0 x0 = 0
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known SNo_1^SNo_1 : SNo 1
Known minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Known SNo_0^SNo_0 : SNo 0
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Theorem 41c17.. : ∀ x0 . x0 ∈ real ⟶ 22598.. 8d0f8.. x0 = ad280.. 0 x0 (proof)
Theorem 6f070.. : ∀ x0 . x0 ∈ real ⟶ 28f8d.. (22598.. 8d0f8.. x0) = 0 (proof)
Theorem 5bda1.. : ∀ x0 . x0 ∈ real ⟶ d634d.. (22598.. 8d0f8.. x0) = x0 (proof)
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Theorem 7c38f.. : ∀ x0 . x0 ∈ e523d.. ⟶ x0 = a0628.. (28f8d.. x0) (22598.. 8d0f8.. (d634d.. x0)) (proof)
Param div_SNo^div_SNo : ι → ι → ι
Param exp_SNo_nat^exp_SNo_nat : ι → ι → ι
Definition 41fb9.. := λ x0 . ad280.. (div_SNo (28f8d.. x0) (add_SNo (exp_SNo_nat (28f8d.. x0) 2) (exp_SNo_nat (d634d.. x0) 2))) (minus_SNo (div_SNo (d634d.. x0) (add_SNo (exp_SNo_nat (28f8d.. x0) 2) (exp_SNo_nat (d634d.. x0) 2))))
Known real_div_SNo^real_div_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ div_SNo x0 x1 ∈ real
Known exp_SNo_nat_2^{exp_SNo_nat_2} : ∀ x0 . SNo x0 ⟶ exp_SNo_nat x0 2 = mul_SNo x0 x0
Known eb0da.. : ∀ x0 . c7ce4.. x0 ⟶ SNo (d634d.. x0)
Known 85533.. : ∀ x0 . c7ce4.. x0 ⟶ SNo (28f8d.. x0)
Theorem 66311.. : ∀ x0 . x0 ∈ e523d.. ⟶ 41fb9.. x0 ∈ e523d.. (proof)
Definition 74066.. := λ x0 x1 . 22598.. x0 (41fb9.. x1)
Theorem 4d97a.. : ∀ x0 . x0 ∈ e523d.. ⟶ ∀ x1 . x1 ∈ e523d.. ⟶ 74066.. x0 x1 ∈ e523d.. (proof)
Param Sep^Sep : ι → (ι → ο) → ι
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Theorem 9e5a7.. : real = {x1 ∈ e523d..|28f8d.. x1 = x1} (proof)