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)
Param nat_p^nat_p : ι → ο
Definition iff^iff := λ x0 x1 : ο . and (x0 ⟶ x1) (x1 ⟶ x0)
Param add_nat^add_nat : ι → ι → ι
Definition odd_nat^odd_nat := λ x0 . and (x0 ∈ omega) (∀ x1 . x1 ∈ omega ⟶ x0 = mul_nat 2 x1 ⟶ ∀ x2 : ο . 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
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Known even_nat_0^even_nat_0 : even_nat 0
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known even_nat_not_odd_nat^{even_nat_not_odd_nat} : ∀ x0 . even_nat x0 ⟶ not (odd_nat x0)
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Known odd_nat_even_nat_S^{odd_nat_even_nat_S} : ∀ x0 . odd_nat x0 ⟶ even_nat (ordsucc x0)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known even_nat_or_odd_nat^{even_nat_or_odd_nat} : ∀ x0 . nat_p x0 ⟶ or (even_nat x0) (odd_nat x0)
Known even_nat_odd_nat_S^{even_nat_odd_nat_S} : ∀ x0 . even_nat x0 ⟶ odd_nat (ordsucc x0)
Known add_nat_p^add_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (add_nat x0 x1)
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Theorem d465d.. : ∀ x0 . even_nat x0 ⟶ ∀ x1 . nat_p x1 ⟶ and (iff (even_nat x1) (even_nat (add_nat x0 x1))) (iff (odd_nat x1) (odd_nat (add_nat x0 x1))) (proof)
Theorem de99a.. : ∀ x0 . odd_nat x0 ⟶ ∀ x1 . nat_p x1 ⟶ and (iff (even_nat x1) (odd_nat (add_nat x0 x1))) (iff (odd_nat x1) (even_nat (add_nat x0 x1))) (proof)
Theorem a723f.. : ∀ x0 x1 . even_nat x0 ⟶ even_nat x1 ⟶ even_nat (add_nat x0 x1) (proof)
Theorem 96034.. : ∀ x0 x1 . even_nat x0 ⟶ odd_nat x1 ⟶ odd_nat (add_nat x0 x1) (proof)
Theorem c68f4.. : ∀ x0 x1 . odd_nat x0 ⟶ odd_nat x1 ⟶ even_nat (add_nat x0 x1) (proof)
Param add_SNo^add_SNo : ι → ι → ι
Known add_nat_add_SNo^{add_nat_add_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_nat x0 x1 = add_SNo x0 x1
Theorem fcda7.. : ∀ x0 x1 . even_nat x0 ⟶ even_nat x1 ⟶ even_nat (add_SNo x0 x1) (proof)
Theorem 04348.. : ∀ x0 x1 . even_nat x0 ⟶ odd_nat x1 ⟶ odd_nat (add_SNo x0 x1) (proof)
Known add_nat_com^add_nat_com : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ add_nat x0 x1 = add_nat x1 x0
Theorem 2f1dc.. : ∀ x0 x1 . odd_nat x0 ⟶ even_nat x1 ⟶ odd_nat (add_SNo x0 x1) (proof)
Theorem 80563.. : ∀ x0 x1 . odd_nat x0 ⟶ odd_nat x1 ⟶ even_nat (add_SNo x0 x1) (proof)
Known mul_nat_0R^mul_nat_0R : ∀ x0 . mul_nat x0 0 = 0
Known mul_nat_SR^mul_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ mul_nat x0 (ordsucc x1) = add_nat x0 (mul_nat x0 x1)
Theorem fc7ba.. : ∀ x0 . even_nat x0 ⟶ ∀ x1 . nat_p x1 ⟶ even_nat (mul_nat x0 x1) (proof)
Known iff_refl^iff_refl : ∀ x0 : ο . iff x0 x0
Known mul_nat_p^mul_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (mul_nat x0 x1)
Known mul_nat_com^mul_nat_com : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ mul_nat x0 x1 = mul_nat x1 x0
Theorem odd_nat_iff_odd_mul_nat^{odd_nat_iff_odd_mul_nat} : ∀ x0 . odd_nat x0 ⟶ ∀ x1 . nat_p x1 ⟶ iff (odd_nat x1) (odd_nat (mul_nat x0 x1)) (proof)
Theorem odd_nat_mul_nat^{odd_nat_mul_nat} : ∀ x0 x1 . odd_nat x0 ⟶ odd_nat x1 ⟶ odd_nat (mul_nat x0 x1) (proof)
Param mul_SNo^mul_SNo : ι → ι → 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
Theorem 7dfbb.. : ∀ x0 x1 . even_nat x0 ⟶ nat_p x1 ⟶ even_nat (mul_SNo x0 x1) (proof)
Theorem 5144a.. : ∀ x0 x1 . nat_p x0 ⟶ even_nat x1 ⟶ even_nat (mul_SNo x0 x1) (proof)
Theorem 41e14.. : ∀ x0 x1 . odd_nat x0 ⟶ odd_nat x1 ⟶ odd_nat (mul_SNo x0 x1) (proof)
Known nat_2^nat_2 : nat_p 2
Known even_nat_double^{even_nat_double} : ∀ x0 . nat_p x0 ⟶ even_nat (mul_nat 2 x0)
Theorem even_nat_2x : ∀ x0 . x0 ∈ omega ⟶ even_nat (mul_SNo 2 x0) (proof)
Theorem 02cd1.. : ∀ x0 . even_nat x0 ⟶ even_nat (mul_SNo x0 x0) (proof)
Theorem 222d4.. : ∀ x0 . odd_nat x0 ⟶ odd_nat (mul_SNo x0 x0) (proof)
Param int^int : ι
Param divides_int^divides_int : ι → ι → ο
Definition 79148.. := λ x0 . and (x0 ∈ int) (divides_int 2 x0)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known Subq_omega_int^{Subq_omega_int} : omega ⊆ int
Param divides_nat^divides_nat : ι → ι → ο
Known divides_nat_divides_int^{divides_nat_divides_int} : ∀ x0 x1 . divides_nat x0 x1 ⟶ divides_int x0 x1
Known b2900.. : ∀ x0 . even_nat x0 ⟶ divides_nat 2 x0
Theorem b06fc.. : ∀ x0 . even_nat x0 ⟶ 79148.. x0 (proof)
Param SNoLe^SNoLe : ι → ι → ο
Param minus_SNo^minus_SNo : ι → ι
Param SNo^SNo : ι → ο
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
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 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 nat_p_int^nat_p_int : ∀ x0 . nat_p x0 ⟶ x0 ∈ int
Known int_minus_SNo^{int_minus_SNo} : ∀ x0 . x0 ∈ int ⟶ minus_SNo x0 ∈ int
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 SNo_0^SNo_0 : SNo 0
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Theorem 48090.. : ∀ x0 x1 . even_nat x0 ⟶ even_nat x1 ⟶ SNoLe x0 x1 ⟶ even_nat (add_SNo x1 (minus_SNo x0)) (proof)
Theorem a28e8.. : ∀ x0 x1 . odd_nat x0 ⟶ odd_nat x1 ⟶ SNoLe x0 x1 ⟶ even_nat (add_SNo x1 (minus_SNo x0)) (proof)
Known SNo_foil^SNo_foil : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo (add_SNo x0 x1) (add_SNo x2 x3) = add_SNo (mul_SNo x0 x2) (add_SNo (mul_SNo x0 x3) (add_SNo (mul_SNo x1 x2) (mul_SNo x1 x3)))
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
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 SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known d67ed.. : ∀ x0 . SNo x0 ⟶ mul_SNo 2 x0 = add_SNo x0 x0
Theorem 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)) (proof)
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 SNo_2^SNo_2 : SNo 2
Known bc1cf.. : ∀ x0 . SNo x0 ⟶ mul_SNo (minus_SNo x0) (minus_SNo x0) = mul_SNo x0 x0
Theorem 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)) (proof)
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 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 add_SNo_minus_SNo_rinv^{add_SNo_minus_SNo_rinv} : ∀ x0 . SNo x0 ⟶ add_SNo x0 (minus_SNo x0) = 0
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Theorem e33a0.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (mul_SNo (add_SNo x0 x1) (add_SNo x0 x1)) (mul_SNo (add_SNo x0 (minus_SNo x1)) (add_SNo x0 (minus_SNo x1))) = mul_SNo 2 (add_SNo (mul_SNo x0 x0) (mul_SNo x1 x1)) (proof)
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 48da5.. : SNo 4
Known neq_4_0^neq_4_0 : 4 = 0 ⟶ ∀ x0 : ο . x0
Known nat_p_SNo^nat_p_SNo : ∀ x0 . nat_p x0 ⟶ SNo 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 55f68.. : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo x3 (add_SNo x0 (add_SNo x1 x2)) = add_SNo (mul_SNo x3 x0) (add_SNo (mul_SNo x3 x1) (mul_SNo x3 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 3c0dd.. : ∀ x0 . SNo x0 ⟶ mul_SNo 4 (mul_SNo x0 x0) = mul_SNo (mul_SNo 2 x0) (mul_SNo 2 x0)
Known ecc46.. : mul_SNo 2 2 = 4
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
Theorem ff7fb.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 x2 x3 x4 . even_nat x1 ⟶ even_nat x2 ⟶ odd_nat x3 ⟶ odd_nat x4 ⟶ SNoLe x1 x2 ⟶ SNoLe x3 x4 ⟶ 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 (proof)
Param SNoLt^SNoLt : ι → ι → ο
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Theorem 568d4.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 x2 x3 x4 . even_nat x1 ⟶ even_nat x2 ⟶ odd_nat x3 ⟶ odd_nat x4 ⟶ 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 (proof)
Theorem 0a868.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 x2 x3 x4 . even_nat x1 ⟶ even_nat x2 ⟶ even_nat x3 ⟶ even_nat x4 ⟶ SNoLe x1 x2 ⟶ SNoLe x3 x4 ⟶ 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 (proof)
Theorem 6ca1c.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 x2 x3 x4 . even_nat x1 ⟶ even_nat x2 ⟶ even_nat x3 ⟶ even_nat x4 ⟶ 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 (proof)
Theorem fc389.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 x2 x3 x4 . odd_nat x1 ⟶ odd_nat x2 ⟶ odd_nat x3 ⟶ odd_nat x4 ⟶ SNoLe x1 x2 ⟶ SNoLe x3 x4 ⟶ 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 (proof)
Theorem ced26.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 x2 x3 x4 . odd_nat x1 ⟶ odd_nat x2 ⟶ odd_nat x3 ⟶ odd_nat x4 ⟶ 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 (proof)
Theorem 679ef.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 x2 x3 x4 . even_nat x1 ⟶ even_nat x2 ⟶ even_nat x3 ⟶ odd_nat x4 ⟶ 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 . x5 (proof)
Theorem ec8e6.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 x2 x3 x4 . even_nat x1 ⟶ odd_nat x2 ⟶ odd_nat x3 ⟶ odd_nat x4 ⟶ 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 . x5 (proof)
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 add_SNo_rotate_4_1^{add_SNo_rotate_4_1} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo x0 (add_SNo x1 (add_SNo x2 x3)) = add_SNo x3 (add_SNo x0 (add_SNo x1 x2))
Known 593c2.. : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι . (∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x0 (x1 x2 x3)) ⟶ (∀ x2 x3 x4 . x0 x2 ⟶ x0 x3 ⟶ x0 x4 ⟶ x1 x2 (x1 x3 x4) = x1 x3 (x1 x2 x4)) ⟶ (∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x1 x2 x3 = x1 x3 x2) ⟶ ∀ x2 x3 x4 x5 . x0 x2 ⟶ x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 x2 (x1 x3 (x1 x4 x5)) = x1 x5 (x1 x3 (x1 x4 x2))
Known 79d19.. : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι . (∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x0 (x1 x2 x3)) ⟶ (∀ x2 x3 x4 . x0 x2 ⟶ x0 x3 ⟶ x0 x4 ⟶ x1 x2 (x1 x3 x4) = x1 x3 (x1 x2 x4)) ⟶ (∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x1 x2 x3 = x1 x3 x2) ⟶ ∀ x2 x3 x4 . x0 x2 ⟶ x0 x3 ⟶ x0 x4 ⟶ x1 x2 (x1 x3 x4) = x1 x4 (x1 x3 x2)
Known 45f87.. : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι . (∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x0 (x1 x2 x3)) ⟶ (∀ x2 x3 x4 . x0 x2 ⟶ x0 x3 ⟶ x0 x4 ⟶ x1 x2 (x1 x3 x4) = x1 x3 (x1 x2 x4)) ⟶ ∀ x2 x3 x4 x5 . x0 x2 ⟶ x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 x2 (x1 x3 (x1 x4 x5)) = x1 x3 (x1 x4 (x1 x2 x5))
Theorem 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 (proof)