Proofgold Asset

Theorem 4f733.. : ∀ x0 : ι → ο . ∀ x1 x2 : ι → ι → ι . (∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x1 x3 x4)) ⟶ (∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5)) ⟶ (∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 x3 (x1 x4 x5) = x1 (x2 x3 x4) (x2 x3 x5)) ⟶ ∀ x3 x4 x5 x6 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x0 x6 ⟶ x2 (x1 x3 x4) (x1 x5 x6) = x1 (x1 (x2 x3 x5) (x2 x3 x6)) (x1 (x2 x4 x5) (x2 x4 x6)) (proof)
Theorem be839.. : ∀ x0 : ι → ο . ∀ x1 x2 : ι → ι → ι . (∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x1 x3 x4)) ⟶ (∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5)) ⟶ (∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 x3 (x1 x4 x5) = x1 (x2 x3 x4) (x2 x3 x5)) ⟶ (∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x2 x3 x4)) ⟶ (∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 (x1 x3 x4) x5 = x1 x3 (x1 x4 x5)) ⟶ ∀ x3 x4 x5 x6 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x0 x6 ⟶ x2 (x1 x3 x4) (x1 x5 x6) = x1 (x2 x3 x5) (x1 (x2 x3 x6) (x1 (x2 x4 x5) (x2 x4 x6))) (proof)
Param ordsucc^ordsucc : ι → ι
Theorem 9f06e.. : ∀ x0 : ι → ο . ∀ x1 x2 : ι → ι → ι . (∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x1 x3 x4)) ⟶ (∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5)) ⟶ (∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 x3 (x1 x4 x5) = x1 (x2 x3 x4) (x2 x3 x5)) ⟶ (∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x2 x3 x4)) ⟶ (∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x2 x3 x4 = x2 x4 x3) ⟶ (∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 (x1 x3 x4) (x1 x4 x5) = x1 x3 (x1 (x2 2 x4) x5)) ⟶ ∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x2 (x1 x3 x4) (x1 x3 x4) = x1 (x2 x3 x3) (x1 (x2 2 (x2 x3 x4)) (x2 x4 x4)) (proof)
Theorem 75e74.. : ∀ x0 : ι → ο . ∀ x1 x2 : ι → ι → ι . (∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x1 x3 x4)) ⟶ (∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5)) ⟶ (∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 x3 (x1 x4 x5) = x1 (x2 x3 x4) (x2 x3 x5)) ⟶ ∀ x3 : ι → ι . (∀ x4 . x0 x4 ⟶ x0 (x3 x4)) ⟶ (∀ x4 x5 . x0 x4 ⟶ x0 x5 ⟶ x0 (x2 x4 x5)) ⟶ (∀ x4 x5 . x0 x4 ⟶ x0 x5 ⟶ x2 x4 x5 = x2 x5 x4) ⟶ (∀ x4 x5 . x0 x4 ⟶ x0 x5 ⟶ x2 x4 (x3 x5) = x3 (x2 x4 x5)) ⟶ (∀ x4 x5 x6 . x0 x4 ⟶ x0 x5 ⟶ x0 x6 ⟶ x1 (x1 x4 (x3 x5)) (x1 x5 x6) = x1 x4 x6) ⟶ ∀ x4 x5 . x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x4 x5) (x1 x4 (x3 x5)) = x1 (x2 x4 x4) (x3 (x2 x5 x5)) (proof)
Param nat_p^nat_p : ι → ο
Param add_nat^add_nat : ι → ι → ι
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
Known nat_ind^nat_ind : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 . nat_p x1 ⟶ x0 x1 ⟶ x0 (ordsucc x1)) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known add_nat_0R^add_nat_0R : ∀ x0 . add_nat x0 0 = x0
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known neq_ordsucc_0^{neq_ordsucc_0} : ∀ x0 . ordsucc x0 = 0 ⟶ ∀ x1 : ο . x1
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Known add_nat_SL^add_nat_SL : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ add_nat (ordsucc x0) x1 = ordsucc (add_nat x0 x1)
Known 7be16.. : ∀ x0 : ι → ο . x0 0 ⟶ x0 1 ⟶ (∀ x1 . nat_p x1 ⟶ x0 (ordsucc (ordsucc x1))) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known neq_0_ordsucc^{neq_0_ordsucc} : ∀ x0 . 0 = ordsucc x0 ⟶ ∀ x1 : ο . x1
Known ordsucc_inj^ordsucc_inj : ∀ x0 x1 . ordsucc x0 = ordsucc x1 ⟶ x0 = x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known nat_inv^nat_inv : ∀ x0 . nat_p x0 ⟶ or (x0 = 0) (∀ x1 : ο . (∀ x2 . and (nat_p x2) (x0 = ordsucc x2) ⟶ x1) ⟶ x1)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Param ordinal^ordinal : ι → ο
Known ordinal_ordsucc_In^{ordinal_ordsucc_In} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Known In_2_3^In_2_3 : 2 ∈ 3
Theorem a7fcd.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ ordsucc x1 = add_nat x0 x0 ⟶ or (x0 ∈ x1) (and (x1 = 1) (x0 = 1)) (proof)
Param mul_nat^mul_nat : ι → ι → ι
Known mul_nat_SR^mul_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ mul_nat x0 (ordsucc x1) = add_nat x0 (mul_nat x0 x1)
Known nat_1^nat_1 : nat_p 1
Known mul_nat_1R^mul_nat_1R : ∀ x0 . mul_nat x0 1 = x0
Known nat_0^nat_0 : nat_p 0
Known cases_2^cases_2 : ∀ x0 . x0 ∈ 2 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 x0
Known mul_nat_0R^mul_nat_0R : ∀ x0 . mul_nat x0 0 = 0
Known In_1_4^In_1_4 : 1 ∈ 4
Known In_2_4^In_2_4 : 2 ∈ 4
Known mul_nat_SL^mul_nat_SL : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ mul_nat (ordsucc x0) x1 = add_nat (mul_nat x0 x1) x1
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known add_nat_Subq_L^{add_nat_Subq_R} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ x0 ⊆ add_nat x0 x1
Known add_nat_com^add_nat_com : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ add_nat x0 x1 = add_nat x1 x0
Known add_nat_p^add_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (add_nat x0 x1)
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Known mul_nat_p^mul_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (mul_nat x0 x1)
Theorem 87731.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ ordsucc (ordsucc x0) ⟶ ordsucc (mul_nat x1 x1) ∈ mul_nat (ordsucc (ordsucc x0)) (ordsucc (ordsucc x0)) (proof)
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known Subq_tra^Subq_tra : ∀ x0 x1 x2 . x0 ⊆ x1 ⟶ x1 ⊆ x2 ⟶ x0 ⊆ x2
Known f0633.. : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ x0 ⊆ x1 ⟶ ordsucc x0 ⊆ ordsucc x1
Theorem a52ed.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc (mul_nat x1 x1) ⊆ mul_nat x0 x0 (proof)
Theorem b889a.. : ∀ x0 . nat_p x0 ⟶ (x0 = 1 ⟶ ∀ x1 : ο . x1) ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc (mul_nat x1 x1) ∈ mul_nat x0 x0 (proof)
Known nat_ordsucc_in_ordsucc^{nat_ordsucc_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Theorem 85b3a..^add_nat_In_R : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . nat_p x2 ⟶ add_nat x1 x2 ∈ add_nat x0 x2 (proof)
Known nat_inv_impred^{nat_inv_impred} : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 . nat_p x1 ⟶ x0 (ordsucc x1)) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Theorem fe2c2.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ ∀ x2 . nat_p x2 ⟶ add_nat x0 x1 = ordsucc x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x1 ⟶ add_nat x3 x4 ∈ x2 (proof)
Param nat_primrec^nat_primrec : ι → (ι → ι → ι) → ι → ι
Param If_i^If_i : ο → ι → ι → ι
Definition 96eca.. := λ x0 . nat_primrec 0 (λ x1 x2 . If_i (ordsucc x2 ∈ x0) (ordsucc x2) 0)
Known nat_primrec_0^{nat_primrec_0} : ∀ x0 . ∀ x1 : ι → ι → ι . nat_primrec x0 x1 0 = x0
Theorem aff59.. : ∀ x0 . 96eca.. x0 0 = 0 (proof)
Known nat_primrec_S^{nat_primrec_S} : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . nat_p x2 ⟶ nat_primrec x0 x1 (ordsucc x2) = x1 x2 (nat_primrec x0 x1 x2)
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Theorem ac69f.. : ∀ x0 x1 . nat_p x1 ⟶ ordsucc (96eca.. x0 x1) ∈ x0 ⟶ 96eca.. x0 (ordsucc x1) = ordsucc (96eca.. x0 x1) (proof)
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Theorem b82e6.. : ∀ x0 x1 . nat_p x1 ⟶ nIn (ordsucc (96eca.. x0 x1)) x0 ⟶ 96eca.. x0 (ordsucc x1) = 0 (proof)
Param setminus^setminus : ι → ι → ι
Param omega^omega : ι
Param Sing^Sing : ι → ι
Known setminusE^setminusE : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ and (x2 ∈ x0) (nIn x2 x1)
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known nat_0_in_ordsucc^{nat_0_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ 0 ∈ ordsucc x0
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Theorem 64b8f.. : ∀ x0 . x0 ∈ setminus omega (Sing 0) ⟶ ∀ x1 . nat_p x1 ⟶ 96eca.. x0 x1 ∈ x0 (proof)
Param SNo^SNo : ι → ο
Param minus_SNo^minus_SNo : ι → ι
Param add_SNo^add_SNo : ι → ι → ι
Known minus_add_SNo_distr_3^{minus_add_SNo_distr_3} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ minus_SNo (add_SNo x0 (add_SNo x1 x2)) = add_SNo (minus_SNo x0) (add_SNo (minus_SNo x1) (minus_SNo x2))
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known minus_add_SNo_distr^{minus_add_SNo_distr} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ minus_SNo (add_SNo x0 x1) = add_SNo (minus_SNo x0) (minus_SNo x1)
Theorem 9112b.. : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ minus_SNo (add_SNo x0 (add_SNo x1 (add_SNo x2 x3))) = add_SNo (minus_SNo x0) (add_SNo (minus_SNo x1) (add_SNo (minus_SNo x2) (minus_SNo x3))) (proof)
Param int^int : ι
Param mul_SNo^mul_SNo : ι → ι → ι
Definition divides_int^divides_int := λ x0 x1 . and (and (x0 ∈ int) (x1 ∈ int)) (∀ x2 : ο . (∀ x3 . and (x3 ∈ int) (mul_SNo x0 x3 = x1) ⟶ x2) ⟶ x2)
Known minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known SNo_0^SNo_0 : SNo 0
Known divides_int_0^{divides_int_0} : ∀ x0 . x0 ∈ int ⟶ divides_int x0 0
Known nat_p_int^nat_p_int : ∀ x0 . nat_p x0 ⟶ x0 ∈ int
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known add_SNo_minus_R2'^{add_SNo_minus_R2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (add_SNo x0 (minus_SNo x1)) x1 = x0
Known nat_p_SNo^nat_p_SNo : ∀ x0 . nat_p x0 ⟶ SNo x0
Known divides_int_add_SNo^{divides_int_add_SNo} : ∀ x0 x1 x2 . divides_int x0 x1 ⟶ divides_int x0 x2 ⟶ divides_int x0 (add_SNo x1 x2)
Known ordinal_ordsucc_In_eq^{ordinal_ordsucc_In_eq} : ∀ x0 x1 . ordinal x0 ⟶ x1 ∈ x0 ⟶ or (ordsucc x1 ∈ x0) (x0 = ordsucc x1)
Known divides_int_ref^{divides_int_ref} : ∀ x0 . x0 ∈ int ⟶ divides_int x0 x0
Known ordinal_ordsucc_SNo_eq^{ordinal_ordsucc_SNo_eq} : ∀ x0 . ordinal x0 ⟶ ordsucc x0 = add_SNo 1 x0
Known SNo_1^SNo_1 : SNo 1
Known add_SNo_com_4_inner_mid^{add_SNo_com_4_inner_mid} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo (add_SNo x0 x1) (add_SNo x2 x3) = add_SNo (add_SNo x0 x2) (add_SNo x1 x3)
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known add_SNo_minus_SNo_rinv^{add_SNo_minus_SNo_rinv} : ∀ x0 . SNo x0 ⟶ add_SNo x0 (minus_SNo x0) = 0
Known ordinal_SNo^ordinal_SNo : ∀ x0 . ordinal x0 ⟶ SNo x0
Known ordinal_Hered^{ordinal_Hered} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordinal x1
Theorem b106b.. : ∀ x0 . x0 ∈ setminus omega (Sing 0) ⟶ ∀ x1 . nat_p x1 ⟶ divides_int x0 (add_SNo x1 (minus_SNo (96eca.. x0 x1))) (proof)
Theorem f495c.. : ∀ x0 . x0 ∈ setminus omega (Sing 0) ⟶ ∀ x1 . nat_p x1 ⟶ 96eca.. x0 x1 = 0 ⟶ divides_int x0 x1 (proof)
Known add_SNo_assoc^{add_SNo_assoc} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo (add_SNo x0 x1) x2
Known add_SNo_minus_L2^{add_SNo_minus_L2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (minus_SNo x0) (add_SNo x0 x1) = x1
Theorem 8c422.. : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo (add_SNo x0 (minus_SNo x1)) (add_SNo x1 x2) = add_SNo x0 x2 (proof)
Known divides_int_diff_SNo_rev : ∀ x0 x1 . x1 ∈ int ⟶ ∀ x2 . x2 ∈ int ⟶ divides_int x0 (add_SNo x1 (minus_SNo x2)) ⟶ divides_int x0 (add_SNo x2 (minus_SNo x1))
Known nat_p_trans^nat_p_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ nat_p x1
Theorem 826f0.. : ∀ x0 . x0 ∈ setminus omega (Sing 0) ⟶ ∀ x1 x2 . nat_p x1 ⟶ nat_p x2 ⟶ 96eca.. x0 x1 = 96eca.. x0 x2 ⟶ divides_int x0 (add_SNo x1 (minus_SNo x2)) (proof)
Known mul_SNo_distrR^{mul_SNo_distrR} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo (add_SNo x0 x1) x2 = add_SNo (mul_SNo x0 x2) (mul_SNo x1 x2)
Known mul_SNo_distrL^{mul_SNo_distrL} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (add_SNo x1 x2) = add_SNo (mul_SNo x0 x1) (mul_SNo x0 x2)
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Known mul_SNo_minus_distrR^{mul_minus_SNo_distrR} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 (minus_SNo x1) = minus_SNo (mul_SNo x0 x1)
Theorem ee3a2.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo (add_SNo x0 x1) (add_SNo x0 (minus_SNo x1)) = add_SNo (mul_SNo x0 x0) (minus_SNo (mul_SNo x1 x1)) (proof)
Param SNoLt^SNoLt : ι → ι → ο
Known SNoLt_trichotomy_or_impred^{SNoLt_trichotomy_or_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 : ο . (SNoLt x0 x1 ⟶ x2) ⟶ (x0 = x1 ⟶ x2) ⟶ (SNoLt x1 x0 ⟶ x2) ⟶ x2
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known mul_SNo_neg_neg^{mul_SNo_neg_neg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 0 ⟶ SNoLt x1 0 ⟶ SNoLt 0 (mul_SNo x0 x1)
Known mul_SNo_neg_pos^{mul_SNo_neg_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 0 ⟶ SNoLt 0 x1 ⟶ SNoLt (mul_SNo x0 x1) 0
Known mul_SNo_pos_neg^{mul_SNo_pos_neg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt x1 0 ⟶ SNoLt (mul_SNo x0 x1) 0
Known mul_SNo_pos_pos^{mul_SNo_pos_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ SNoLt 0 (mul_SNo x0 x1)
Theorem c8949.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = 0 ⟶ or (x0 = 0) (x1 = 0) (proof)
Param divides_nat^divides_nat : ι → ι → ο
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))
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known In_0_1^In_0_1 : 0 ∈ 1
Theorem 59ab3.. : ∀ x0 . prime_nat x0 ⟶ x0 ∈ setminus omega (Sing 0) (proof)
Param PNoLt^PNoLt : ι → (ι → ο) → ι → (ι → ο) → ο
Param PNoEq_^PNoEq_ : ι → (ι → ο) → (ι → ο) → ο
Definition PNoLe^PNoLe := λ x0 . λ x1 : ι → ο . λ x2 . λ x3 : ι → ο . or (PNoLt x0 x1 x2 x3) (and (x0 = x2) (PNoEq_ x0 x1 x3))
Param SNoLev^SNoLev : ι → ι
Definition SNoLe^SNoLe := λ x0 x1 . PNoLe (SNoLev x0) (λ x2 . x2 ∈ x0) (SNoLev x1) (λ x2 . x2 ∈ x1)
Known SNoLe_tra^SNoLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLe x0 x2
Known nonneg_mul_SNo_Le^{nonneg_mul_SNo_Le} : ∀ x0 x1 x2 . SNo x0 ⟶ SNoLe 0 x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x1 x2 ⟶ SNoLe (mul_SNo x0 x1) (mul_SNo x0 x2)
Theorem 55103.. : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNoLe 0 x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLe 0 x3 ⟶ SNoLe x0 x1 ⟶ SNoLe x2 x3 ⟶ SNoLe (mul_SNo x0 x2) (mul_SNo x1 x3) (proof)
Known add_SNo_minus_SNo_linv^{add_SNo_minus_SNo_linv} : ∀ x0 . SNo x0 ⟶ add_SNo (minus_SNo x0) x0 = 0
Theorem bd5ca.. : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ add_SNo (minus_SNo (ordsucc x0)) (ordsucc x1) = add_SNo (minus_SNo x0) x1 (proof)
Known int_SNo_cases^{int_SNo_cases} : ∀ x0 : ι → ο . (∀ x1 . x1 ∈ omega ⟶ x0 x1) ⟶ (∀ x1 . x1 ∈ omega ⟶ x0 (minus_SNo x1)) ⟶ ∀ x1 . x1 ∈ int ⟶ x0 x1
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Known omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Known divides_int_minus_SNo^{divides_int_minus_SNo} : ∀ x0 x1 . divides_int x0 x1 ⟶ divides_int x0 (minus_SNo x1)
Known mul_SNo_minus_distrL^{mul_SNo_minus_distrL} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo (minus_SNo x0) x1 = minus_SNo (mul_SNo x0 x1)
Known mul_SNo_minus_minus^{mul_SNo_minus_minus} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo (minus_SNo x0) (minus_SNo x1) = mul_SNo x0 x1
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 ordinal_SNoLt_In^{ordinal_SNoLt_In} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNoLt x0 x1 ⟶ x0 ∈ x1
Known aa7e8..^{nonneg_int_nat_p} : ∀ x0 . x0 ∈ int ⟶ SNoLe 0 x0 ⟶ nat_p x0
Known int_add_SNo^int_add_SNo : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ add_SNo x0 x1 ∈ int
Known int_minus_SNo^{int_minus_SNo} : ∀ x0 . x0 ∈ int ⟶ minus_SNo x0 ∈ int
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known add_SNo_minus_Lt2b^{add_SNo_minus_Lt2b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x2 x1) x0 ⟶ SNoLt x2 (add_SNo x0 (minus_SNo x1))
Known add_SNo_Lt2^add_SNo_Lt2 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x1 x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x0 x2)
Known minus_SNo_Lt_contra1^{minus_SNo_Lt_contra1} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt (minus_SNo x0) x1 ⟶ SNoLt (minus_SNo x1) x0
Known divides_int_1^{divides_int_1} : ∀ x0 . x0 ∈ int ⟶ divides_int 1 x0
Known SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 x2
Known 84495.. : ∀ x0 . nat_p x0 ⟶ SNoLe 0 x0
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known pos_mul_SNo_Lt^{pos_mul_SNo_Lt} : ∀ x0 x1 x2 . SNo x0 ⟶ SNoLt 0 x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x1 x2 ⟶ SNoLt (mul_SNo x0 x1) (mul_SNo x0 x2)
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known mul_SNo_nonneg_nonneg^{mul_SNo_nonneg_nonneg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe 0 x0 ⟶ SNoLe 0 x1 ⟶ SNoLe 0 (mul_SNo x0 x1)
Known mul_SNo_nonzero_cancel^{mul_SNo_nonzero_cancel_L} : ∀ x0 x1 x2 . SNo x0 ⟶ (x0 = 0 ⟶ ∀ x3 : ο . x3) ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 x1 = mul_SNo x0 x2 ⟶ x1 = x2
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known mul_SNo_com_3_0_1^{mul_SNo_com_3_0_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (mul_SNo x1 x2) = mul_SNo x1 (mul_SNo x0 x2)
Known int_SNo^int_SNo : ∀ x0 . x0 ∈ int ⟶ SNo x0
Known divides_int_mul_SNo_L^{divides_int_mul_SNo_L} : ∀ x0 x1 x2 . x2 ∈ int ⟶ divides_int x0 x1 ⟶ divides_int x0 (mul_SNo x1 x2)
Known ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Known nat_trans^nat_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known divides_int_divides_nat^{divides_int_divides_nat} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ divides_int x0 x1 ⟶ divides_nat x0 x1
Theorem Euclid_lemma^Euclid_lemma : ∀ x0 . prime_nat x0 ⟶ ∀ x1 . x1 ∈ int ⟶ ∀ x2 . x2 ∈ int ⟶ divides_int x0 (mul_SNo x1 x2) ⟶ or (divides_int x0 x1) (divides_int x0 x2) (proof)
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Known add_SNo_minus_R2^{add_SNo_minus_R2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (add_SNo x0 x1) (minus_SNo x1) = x0
Known add_SNo_Lt3^add_SNo_Lt3 : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt x0 x2 ⟶ SNoLt x1 x3 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x3)
Known mul_nat_mul_SNo^{mul_nat_mul_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat x0 x1 = mul_SNo x0 x1
Known add_nat_add_SNo^{add_nat_add_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_nat x0 x1 = add_SNo x0 x1
Known add_SNo_Le2^add_SNo_Le2 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x1 x2 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x0 x2)
Known minus_SNo_Le_contra^{minus_SNo_Le_contra} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ SNoLe (minus_SNo x1) (minus_SNo x0)
Known add_SNo_Le1^add_SNo_Le1 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x2 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x2 x1)
Theorem d3d9f.. : ∀ x0 . prime_nat x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ divides_int x0 (add_SNo (mul_SNo x1 x1) (minus_SNo (mul_SNo x2 x2))) ⟶ or (x1 = x2) (x1 = add_SNo x0 (minus_SNo x2)) (proof)
Definition odd_nat^odd_nat := λ x0 . and (x0 ∈ omega) (∀ x1 . x1 ∈ omega ⟶ x0 = mul_nat 2 x1 ⟶ ∀ x2 : ο . x2)
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Definition even_nat^even_nat := λ x0 . and (x0 ∈ omega) (∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (x0 = mul_nat 2 x2) ⟶ x1) ⟶ x1)
Known odd_nat_even_nat_S^{odd_nat_even_nat_S} : ∀ x0 . odd_nat x0 ⟶ even_nat (ordsucc x0)
Known nat_2^nat_2 : nat_p 2
Known d67ed.. : ∀ x0 . SNo x0 ⟶ mul_SNo 2 x0 = add_SNo x0 x0
Definition inj^inj := λ x0 x1 . λ x2 : ι → ι . and (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4)
Definition atleastp^atleastp := λ x0 x1 . ∀ x2 : ο . (∀ x3 : ι → ι . inj x0 x1 x3 ⟶ x2) ⟶ x2
Known 4fb58..^{Pigeonhole_not_atleastp_ordsucc} : ∀ x0 . nat_p x0 ⟶ not (atleastp (ordsucc x0) x0)
Param setsum^setsum : ι → ι → ι
Known atleastp_tra^atleastp_tra : ∀ x0 x1 x2 . atleastp x0 x1 ⟶ atleastp x1 x2 ⟶ atleastp x0 x2
Param equip^equip : ι → ι → ο
Known equip_atleastp^{equip_atleastp} : ∀ x0 x1 . equip x0 x1 ⟶ atleastp x0 x1
Known c88e0.. : ∀ x0 x1 x2 x3 . nat_p x0 ⟶ nat_p x1 ⟶ equip x0 x2 ⟶ equip x1 x3 ⟶ equip (add_nat x0 x1) (setsum x2 x3)
Known equip_ref^equip_ref : ∀ x0 . equip x0 x0
Param combine_funcs^{combine_funcs} : ι → ι → (ι → ι) → (ι → ι) → ι → ι
Param Inj0^Inj0 : ι → ι
Param Inj1^Inj1 : ι → ι
Known setsum_Inj_inv^{setsum_Inj_inv} : ∀ x0 x1 x2 . x2 ∈ setsum x0 x1 ⟶ or (∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (x2 = Inj0 x4) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ x1) (x2 = Inj1 x4) ⟶ x3) ⟶ x3)
Known combine_funcs_eq1^{combine_funcs_eq1} : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 . combine_funcs x0 x1 x2 x3 (Inj0 x4) = x2 x4
Known combine_funcs_eq2^{combine_funcs_eq2} : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 . combine_funcs x0 x1 x2 x3 (Inj1 x4) = x3 x4
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known SNo_2^SNo_2 : SNo 2
Known ordinal_Empty^{ordinal_Empty} : ordinal 0
Known add_SNo_Le3^add_SNo_Le3 : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLe x0 x2 ⟶ SNoLe x1 x3 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x2 x3)
Known add_SNo_minus_Lt1b^{add_SNo_minus_Lt1b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 (add_SNo x2 x1) ⟶ SNoLt (add_SNo x0 (minus_SNo x1)) x2
Known SNoLt_0_1^SNoLt_0_1 : SNoLt 0 1
Known neq_1_0^neq_1_0 : 1 = 0 ⟶ ∀ x0 : ο . x0
Known SNo_zero_or_sqr_pos^{SNo_zero_or_sqr_pos} : ∀ x0 . SNo x0 ⟶ or (x0 = 0) (SNoLt 0 (mul_SNo x0 x0))
Known SNo_add_SNo_4^{SNo_add_SNo_4} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo (add_SNo x0 (add_SNo x1 (add_SNo x2 x3)))
Known SNo_add_SNo_3^{SNo_add_SNo_3} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo (add_SNo x0 (add_SNo x1 x2))
Known SNo_sqr_nonneg^{SNo_sqr_nonneg} : ∀ x0 . SNo x0 ⟶ SNoLe 0 (mul_SNo x0 x0)
Known SNoLe_antisym^{SNoLe_antisym} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x0 ⟶ x0 = x1
Known 48da5.. : SNo 4
Known 23c65.. : ∀ x0 . SNo x0 ⟶ mul_SNo 4 x0 = add_SNo x0 (add_SNo x0 (add_SNo x0 x0))
Known 7b468.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNo x5 ⟶ SNo x6 ⟶ SNo x7 ⟶ SNoLe x0 x4 ⟶ SNoLe x1 x5 ⟶ SNoLe x2 x6 ⟶ SNoLe x3 x7 ⟶ SNoLe (add_SNo x0 (add_SNo x1 (add_SNo x2 x3))) (add_SNo x4 (add_SNo x5 (add_SNo x6 x7)))
Known b5021.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo (add_SNo x0 (minus_SNo x1)) (add_SNo x0 (minus_SNo x1)) = add_SNo (mul_SNo x0 x0) (add_SNo (minus_SNo (mul_SNo 2 (mul_SNo x0 x1))) (mul_SNo x1 x1))
Known mul_SNo_com_4_inner_mid^{mul_SNo_com_4_inner_mid} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo (mul_SNo x0 x1) (mul_SNo x2 x3) = mul_SNo (mul_SNo x0 x2) (mul_SNo x1 x3)
Known ecc46.. : mul_SNo 2 2 = 4
Known 8b4bf.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo (add_SNo x0 x1) (add_SNo x0 x1) = add_SNo (mul_SNo x0 x0) (add_SNo (mul_SNo 2 (mul_SNo x0 x1)) (mul_SNo x1 x1))
Known add_SNo_assoc_4^{add_SNo_assoc_4} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo x0 (add_SNo x1 (add_SNo x2 x3)) = add_SNo (add_SNo x0 (add_SNo x1 x2)) x3
Known add_SNo_rotate_3_1^{add_SNo_rotate_3_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo x2 (add_SNo x0 x1)
Known nat_3^nat_3 : nat_p 3
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known SNoLt_1_2^SNoLt_1_2 : SNoLt 1 2
Known divides_int_mul_SNo_R^{divides_int_mul_SNo_R} : ∀ x0 x1 x2 . x2 ∈ int ⟶ divides_int x0 x1 ⟶ divides_int x0 (mul_SNo x2 x1)
Known int_mul_SNo^int_mul_SNo : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ mul_SNo x0 x1 ∈ int
Known add_SNo_minus_Lt1^{add_SNo_minus_Lt1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x0 (minus_SNo x1)) x2 ⟶ SNoLt x0 (add_SNo x2 x1)
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known add_SNo_minus_Le2b^{add_SNo_minus_Le2b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe (add_SNo x2 x1) x0 ⟶ SNoLe x2 (add_SNo x0 (minus_SNo x1))
Known ordinal_Subq_SNoLe^{ordinal_Subq_SNoLe} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ x0 ⊆ x1 ⟶ SNoLe x0 x1
Known ordinal_ordsucc_In_Subq^{ordinal_ordsucc_In_Subq} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ⊆ x0
Known add_SNo_1_1_2^{add_SNo_1_1_2} : add_SNo 1 1 = 2
Theorem 12389.. : ∀ x0 . prime_nat x0 ⟶ odd_nat x0 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ setminus x0 (Sing 0)) (∀ x3 : ο . (∀ x4 . and (x4 ∈ omega) (∀ x5 : ο . (∀ x6 . and (x6 ∈ omega) (∀ x7 : ο . (∀ x8 . and (x8 ∈ omega) (∀ x9 : ο . (∀ x10 . and (x10 ∈ omega) (mul_SNo x2 x0 = add_SNo (mul_SNo x4 x4) (add_SNo (mul_SNo x6 x6) (add_SNo (mul_SNo x8 x8) (mul_SNo x10 x10)))) ⟶ x9) ⟶ x9) ⟶ x7) ⟶ x7) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1 (proof)
Known 69709.. : (∀ x0 . prime_nat x0 ⟶ odd_nat x0 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ setminus x0 (Sing 0)) (∀ x3 : ο . (∀ x4 . and (x4 ∈ omega) (∀ x5 : ο . (∀ x6 . and (x6 ∈ omega) (∀ x7 : ο . (∀ x8 . and (x8 ∈ omega) (∀ x9 : ο . (∀ x10 . and (x10 ∈ omega) (mul_SNo x2 x0 = add_SNo (mul_SNo x4 x4) (add_SNo (mul_SNo x6 x6) (add_SNo (mul_SNo x8 x8) (mul_SNo x10 x10)))) ⟶ x9) ⟶ x9) ⟶ x7) ⟶ x7) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1) ⟶ (∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ omega ⟶ ∀ x2 . x2 ∈ omega ⟶ ∀ x3 . x3 ∈ omega ⟶ ∀ x4 . x4 ∈ omega ⟶ mul_SNo 2 x0 = add_SNo (mul_SNo x1 x1) (add_SNo (mul_SNo x2 x2) (add_SNo (mul_SNo x3 x3) (mul_SNo x4 x4))) ⟶ ∀ x5 : ο . (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ omega ⟶ ∀ x8 . x8 ∈ omega ⟶ ∀ x9 . x9 ∈ omega ⟶ x0 = add_SNo (mul_SNo x6 x6) (add_SNo (mul_SNo x7 x7) (add_SNo (mul_SNo x8 x8) (mul_SNo x9 x9))) ⟶ x5) ⟶ x5) ⟶ ∀ 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
Known 36e2b.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ omega ⟶ ∀ x2 . x2 ∈ omega ⟶ ∀ x3 . x3 ∈ omega ⟶ ∀ x4 . x4 ∈ omega ⟶ mul_SNo 2 x0 = add_SNo (mul_SNo x1 x1) (add_SNo (mul_SNo x2 x2) (add_SNo (mul_SNo x3 x3) (mul_SNo x4 x4))) ⟶ ∀ x5 : ο . (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ omega ⟶ ∀ x8 . x8 ∈ omega ⟶ ∀ x9 . x9 ∈ omega ⟶ x0 = add_SNo (mul_SNo x6 x6) (add_SNo (mul_SNo x7 x7) (add_SNo (mul_SNo x8 x8) (mul_SNo x9 x9))) ⟶ x5) ⟶ x5
Theorem f673c.. : ∀ 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 (proof)
Known 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
Theorem 09f1b.. : ∀ 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)
Known mul_SNo_In_omega^{mul_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_SNo x0 x1 ∈ omega
Known add_SNo_In_omega^{add_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_SNo x0 x1 ∈ omega
Theorem 47d21.. : ∀ 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_nat (mul_nat x2 x2) (add_nat (mul_nat x4 x4) (add_nat (mul_nat x6 x6) (mul_nat x8 x8)))) ⟶ x7) ⟶ x7) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1 (proof)