Proofgold Address

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param omega^omega : ι
Param mul_nat^mul_nat : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
Definition even_nat^even_nat := λ x0 . and (x0 ∈ omega) (∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (x0 = mul_nat 2 x2) ⟶ x1) ⟶ x1)
Definition divides_nat^divides_nat := λ x0 x1 . and (and (x0 ∈ omega) (x1 ∈ omega)) (∀ x2 : ο . (∀ x3 . and (x3 ∈ omega) (mul_nat x0 x3 = x1) ⟶ x2) ⟶ x2)
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Param nat_p^nat_p : ι → ο
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_2^nat_2 : nat_p 2
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem b2900.. : ∀ x0 . even_nat x0 ⟶ divides_nat 2 x0 (proof)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Definition prime_nat^prime_nat := λ x0 . and (and (x0 ∈ omega) (1 ∈ x0)) (∀ x1 . x1 ∈ omega ⟶ divides_nat x1 x0 ⟶ or (x1 = 1) (x1 = x0))
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known neq_2_1^neq_2_1 : 2 = 1 ⟶ ∀ x0 : ο . x0
Theorem 447ce.. : ∀ x0 . prime_nat x0 ⟶ even_nat x0 ⟶ x0 = 2 (proof)
Param int^int : ι
Param add_SNo^add_SNo : ι → ι → ι
Param mul_SNo^mul_SNo : ι → ι → ι
Known c6211.. : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ ∀ x2 . x2 ∈ int ⟶ ∀ x3 . x3 ∈ int ⟶ ∀ x4 . x4 ∈ int ⟶ ∀ x5 . x5 ∈ int ⟶ x0 = add_SNo (mul_SNo x2 x2) (add_SNo (mul_SNo x3 x3) (add_SNo (mul_SNo x4 x4) (mul_SNo x5 x5))) ⟶ ∀ x6 . x6 ∈ int ⟶ ∀ x7 . x7 ∈ int ⟶ ∀ x8 . x8 ∈ int ⟶ ∀ x9 . x9 ∈ int ⟶ x1 = add_SNo (mul_SNo x6 x6) (add_SNo (mul_SNo x7 x7) (add_SNo (mul_SNo x8 x8) (mul_SNo x9 x9))) ⟶ ∀ x10 : ο . (∀ x11 . x11 ∈ omega ⟶ ∀ x12 . x12 ∈ omega ⟶ ∀ x13 . x13 ∈ omega ⟶ ∀ x14 . x14 ∈ omega ⟶ mul_SNo x0 x1 = add_SNo (mul_SNo x11 x11) (add_SNo (mul_SNo x12 x12) (add_SNo (mul_SNo x13 x13) (mul_SNo x14 x14))) ⟶ x10) ⟶ x10
Param odd_nat^odd_nat : ι → ο
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known nat_complete_ind^{nat_complete_ind} : ∀ x0 : ι → ο . (∀ x1 . nat_p x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known 054a4.. : ∀ x0 : ι → ο . x0 0 ⟶ x0 1 ⟶ x0 2 ⟶ (∀ x1 . nat_p x1 ⟶ x0 (ordsucc (ordsucc (ordsucc x1)))) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known nat_0^nat_0 : nat_p 0
Param SNo^SNo : ι → ο
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known SNo_0^SNo_0 : SNo 0
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known nat_1^nat_1 : nat_p 1
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Known SNo_1^SNo_1 : SNo 1
Known add_SNo_1_1_2^{add_SNo_1_1_2} : add_SNo 1 1 = 2
Known SNo_2^SNo_2 : SNo 2
Definition not^not := λ x0 : ο . x0 ⟶ False
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known even_nat_or_odd_nat^{even_nat_or_odd_nat} : ∀ x0 . nat_p x0 ⟶ or (even_nat x0) (odd_nat x0)
Known neq_ordsucc_0^{neq_ordsucc_0} : ∀ x0 . ordsucc x0 = 0 ⟶ ∀ x1 : ο . x1
Known ordsucc_inj^ordsucc_inj : ∀ x0 x1 . ordsucc x0 = ordsucc x1 ⟶ x0 = x1
Known b8d2f.. : ∀ x0 . x0 ∈ omega ⟶ not (prime_nat x0) ⟶ 1 ∈ x0 ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ 1 ∈ x2 ⟶ 1 ∈ x3 ⟶ x0 = mul_nat x2 x3 ⟶ x1) ⟶ x1
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known mul_nat_mul_SNo^{mul_nat_mul_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat x0 x1 = mul_SNo x0 x1
Known Subq_omega_int^{Subq_omega_int} : omega ⊆ int
Known nat_p_trans^nat_p_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ nat_p x1
Known nat_ordsucc_in_ordsucc^{nat_ordsucc_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Known nat_0_in_ordsucc^{nat_0_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ 0 ∈ ordsucc x0
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Theorem 8ff04.. : (∀ x0 . prime_nat x0 ⟶ odd_nat x0 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (∀ x3 : ο . (∀ x4 . and (x4 ∈ omega) (∀ x5 : ο . (∀ x6 . and (x6 ∈ omega) (∀ x7 : ο . (∀ x8 . and (x8 ∈ omega) (x0 = add_SNo (mul_SNo x2 x2) (add_SNo (mul_SNo x4 x4) (add_SNo (mul_SNo x6 x6) (mul_SNo x8 x8)))) ⟶ x7) ⟶ x7) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1) ⟶ ∀ x0 . x0 ∈ omega ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (∀ x3 : ο . (∀ x4 . and (x4 ∈ omega) (∀ x5 : ο . (∀ x6 . and (x6 ∈ omega) (∀ x7 : ο . (∀ x8 . and (x8 ∈ omega) (x0 = add_SNo (mul_SNo x2 x2) (add_SNo (mul_SNo x4 x4) (add_SNo (mul_SNo x6 x6) (mul_SNo x8 x8)))) ⟶ x7) ⟶ x7) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1 (proof)