Proofgold Address

Param SNo^SNo : ι → ο
Param ordsucc^ordsucc : ι → ι
Param nat_p^nat_p : ι → ο
Known nat_p_SNo^nat_p_SNo : ∀ x0 . nat_p x0 ⟶ SNo x0
Known nat_4^nat_4 : nat_p 4
Theorem 48da5.. : SNo 4 (proof)
Param SNoLt^SNoLt : ι → ι → ο
Param ordinal^ordinal : ι → ο
Known ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known In_0_4^In_0_4 : 0 ∈ 4
Theorem 32437.. : SNoLt 0 4 (proof)
Param mul_SNo^mul_SNo : ι → ι → ι
Known mul_SNo_assoc^{mul_SNo_assoc} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (mul_SNo x1 x2) = mul_SNo (mul_SNo x0 x1) 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
Theorem 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) (proof)
Param int^int : ι
Param add_SNo^add_SNo : ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
Known 88d93.. : ∀ x0 : ι → ο . ∀ x1 x2 : ι → ι → ι . (∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x1 x3 x4)) ⟶ (∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x2 x3 x4)) ⟶ (∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 x3 (x1 x4 x5) = x1 x4 (x1 x3 x5)) ⟶ (∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 (x1 x3 x4) x5 = x1 x3 (x1 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 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5)) ⟶ ∀ x3 : ι → ι . (∀ x4 . x0 x4 ⟶ x3 x4 = x2 x4 x4) ⟶ ∀ x4 : ι → ι . (∀ x5 . x0 x5 ⟶ x0 (x4 x5)) ⟶ (∀ x5 . x0 x5 ⟶ x4 (x4 x5) = x5) ⟶ (∀ x5 x6 . x0 x5 ⟶ x0 x6 ⟶ x1 (x4 x5) (x1 x5 x6) = x6) ⟶ (∀ x5 x6 . x0 x5 ⟶ x0 x6 ⟶ x1 x5 (x1 (x4 x5) x6) = x6) ⟶ (∀ x5 x6 . x0 x5 ⟶ x0 x6 ⟶ x2 (x4 x5) x6 = x4 (x2 x5 x6)) ⟶ (∀ x5 x6 . x0 x5 ⟶ x0 x6 ⟶ x2 x5 (x4 x6) = x4 (x2 x5 x6)) ⟶ (∀ x5 x6 . x0 x5 ⟶ x0 x6 ⟶ x2 x5 x6 = x2 x6 x5) ⟶ (∀ x5 x6 x7 x8 . x0 x5 ⟶ x0 x6 ⟶ x0 x7 ⟶ x0 x8 ⟶ x2 (x2 x5 x6) (x2 x7 x8) = x2 (x2 x5 x7) (x2 x6 x8)) ⟶ ∀ x5 x6 x7 x8 x9 x10 x11 x12 . x0 x5 ⟶ x0 x6 ⟶ x0 x7 ⟶ x0 x8 ⟶ x0 x9 ⟶ x0 x10 ⟶ x0 x11 ⟶ x0 x12 ⟶ x2 (x1 (x3 x5) (x1 (x3 x6) (x1 (x3 x7) (x3 x8)))) (x1 (x3 x9) (x1 (x3 x10) (x1 (x3 x11) (x3 x12)))) = x1 (x3 (x1 (x2 x5 x10) (x1 (x2 x6 x9) (x1 (x2 x7 x12) (x4 (x2 x8 x11)))))) (x1 (x3 (x1 (x2 x5 x11) (x1 (x4 (x2 x6 x12)) (x1 (x2 x7 x9) (x2 x8 x10))))) (x1 (x3 (x1 (x2 x5 x12) (x1 (x2 x6 x11) (x1 (x4 (x2 x7 x10)) (x2 x8 x9))))) (x3 (x1 (x2 x5 x9) (x1 (x4 (x2 x6 x10)) (x1 (x4 (x2 x7 x11)) (x4 (x2 x8 x12))))))))
Known int_add_SNo^int_add_SNo : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ add_SNo x0 x1 ∈ int
Known int_mul_SNo^int_mul_SNo : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ mul_SNo x0 x1 ∈ int
Known add_SNo_com_3_0_1^{add_SNo_com_3_0_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo x1 (add_SNo x0 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 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 int_minus_SNo^{int_minus_SNo} : ∀ x0 . x0 ∈ int ⟶ minus_SNo x0 ∈ int
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
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
Known add_SNo_minus_SNo_prop2^{add_SNo_minus_SNo_prop2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 (add_SNo (minus_SNo x0) x1) = 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_distrR^{mul_minus_SNo_distrR} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 (minus_SNo x1) = minus_SNo (mul_SNo x0 x1)
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 int_SNo^int_SNo : ∀ x0 . x0 ∈ int ⟶ SNo x0
Theorem a3d6a.. : ∀ 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))) ⟶ mul_SNo x0 x1 = add_SNo (mul_SNo (add_SNo (mul_SNo x2 x7) (add_SNo (mul_SNo x3 x6) (add_SNo (mul_SNo x4 x9) (minus_SNo (mul_SNo x5 x8))))) (add_SNo (mul_SNo x2 x7) (add_SNo (mul_SNo x3 x6) (add_SNo (mul_SNo x4 x9) (minus_SNo (mul_SNo x5 x8)))))) (add_SNo (mul_SNo (add_SNo (mul_SNo x2 x8) (add_SNo (minus_SNo (mul_SNo x3 x9)) (add_SNo (mul_SNo x4 x6) (mul_SNo x5 x7)))) (add_SNo (mul_SNo x2 x8) (add_SNo (minus_SNo (mul_SNo x3 x9)) (add_SNo (mul_SNo x4 x6) (mul_SNo x5 x7))))) (add_SNo (mul_SNo (add_SNo (mul_SNo x2 x9) (add_SNo (mul_SNo x3 x8) (add_SNo (minus_SNo (mul_SNo x4 x7)) (mul_SNo x5 x6)))) (add_SNo (mul_SNo x2 x9) (add_SNo (mul_SNo x3 x8) (add_SNo (minus_SNo (mul_SNo x4 x7)) (mul_SNo x5 x6))))) (mul_SNo (add_SNo (mul_SNo x2 x6) (add_SNo (minus_SNo (mul_SNo x3 x7)) (add_SNo (minus_SNo (mul_SNo x4 x8)) (minus_SNo (mul_SNo x5 x9))))) (add_SNo (mul_SNo x2 x6) (add_SNo (minus_SNo (mul_SNo x3 x7)) (add_SNo (minus_SNo (mul_SNo x4 x8)) (minus_SNo (mul_SNo x5 x9)))))))) (proof)
Param mul_nat^mul_nat : ι → ι → ι
Known c3da7.. : mul_nat 2 2 = 4
Param omega^omega : ι
Known mul_nat_mul_SNo^{mul_nat_mul_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat x0 x1 = mul_SNo x0 x1
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_2^nat_2 : nat_p 2
Theorem ecc46.. : mul_SNo 2 2 = 4 (proof)