Proofgold Address

Param nat_p^nat_p : ι → ο
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param add_nat^add_nat : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
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 Subq_ref^Subq_ref : ∀ x0 . x0 ⊆ x0
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Known Subq_tra^Subq_tra : ∀ x0 x1 x2 . x0 ⊆ x1 ⟶ x1 ⊆ x2 ⟶ x0 ⊆ x2
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Theorem 4d87b.. : ∀ x0 x1 . nat_p x1 ⟶ x0 ⊆ add_nat x0 x1 (proof)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param mul_nat^mul_nat : ι → ι → ι
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
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 mul_nat_p^mul_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (mul_nat x0 x1)
Theorem 2bbf0..^{mul_nat_0_or_Subq} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ or (x1 = 0) (x0 ⊆ mul_nat x0 x1) (proof)
Theorem nat_inv_impred^{nat_inv_impred} : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 . nat_p x1 ⟶ x0 (ordsucc x1)) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1 (proof)
Theorem 7be16.. : ∀ x0 : ι → ο . x0 0 ⟶ x0 1 ⟶ (∀ x1 . nat_p x1 ⟶ x0 (ordsucc (ordsucc x1))) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1 (proof)
Theorem 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 (proof)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
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
Theorem 3fa1a.. : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 x1 = 0 ⟶ and (x0 = 0) (x1 = 0) (proof)
Known ordsucc_inj^ordsucc_inj : ∀ x0 x1 . ordsucc x0 = ordsucc x1 ⟶ x0 = x1
Theorem c89ba.. : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 x1 = x1 ⟶ x0 = 0 (proof)
Known mul_nat_0R^mul_nat_0R : ∀ x0 . mul_nat x0 0 = 0
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
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)
Theorem 75048.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ mul_nat x1 x0 = x1 ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ x0 = 1 (proof)
Param omega^omega : ι
Definition not^not := λ x0 : ο . x0 ⟶ False
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)
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 dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Param ordinal^ordinal : ι → ο
Known ordinal_In_Or_Subq^{ordinal_In_Or_Subq} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ or (x0 ∈ x1) (x1 ⊆ x0)
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known mul_nat_com^mul_nat_com : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ mul_nat x0 x1 = mul_nat x1 x0
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known mul_nat_1R^mul_nat_1R : ∀ x0 . mul_nat x0 1 = x0
Known ordinal_ordsucc_In^{ordinal_ordsucc_In} : ∀ x0 . ordinal 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 ordinal_trichotomy_or_impred^{ordinal_trichotomy_or_impred} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ ∀ x2 : ο . (x0 ∈ x1 ⟶ x2) ⟶ (x0 = x1 ⟶ x2) ⟶ (x1 ∈ x0 ⟶ x2) ⟶ x2
Known ordinal_1^ordinal_1 : ordinal 1
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Known ordinal_Empty^{ordinal_Empty} : ordinal 0
Known Empty_Subq_eq^{Empty_Subq_eq} : ∀ x0 . x0 ⊆ 0 ⟶ x0 = 0
Known mul_nat_0L^mul_nat_0L : ∀ x0 . nat_p x0 ⟶ mul_nat 0 x0 = 0
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Theorem 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 (proof)
Param SNo^SNo : ι → ο
Param SNoLt^SNoLt : ι → ι → ο
Known ordinal_SNoLt_In^{ordinal_SNoLt_In} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNoLt x0 x1 ⟶ x0 ∈ x1
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
Param add_SNo^add_SNo : ι → ι → ι
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Param minus_SNo^minus_SNo : ι → ι
Known add_SNo_minus_Lt2^{add_SNo_minus_Lt2} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x2 (add_SNo x0 (minus_SNo x1)) ⟶ SNoLt (add_SNo x2 x1) x0
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known SNo_0^SNo_0 : SNo 0
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = 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)
Known SNo_1^SNo_1 : SNo 1
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_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
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)
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo 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 ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Known ordinal_SNo^ordinal_SNo : ∀ x0 . ordinal x0 ⟶ SNo x0
Theorem 58a75.. : ∀ x0 x1 . nat_p x0 ⟶ nat_p x1 ⟶ 0 ∈ x0 ⟶ 1 ∈ x1 ⟶ x0 ∈ mul_nat x0 x1 (proof)
Known nat_trans^nat_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known In_0_1^In_0_1 : 0 ∈ 1
Theorem 7d1f8.. : ∀ x0 x1 . nat_p x0 ⟶ nat_p x1 ⟶ 1 ∈ x0 ⟶ 1 ∈ x1 ⟶ and (x0 ∈ mul_nat x0 x1) (x1 ∈ mul_nat x0 x1) (proof)
Known 25618.. : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι . (∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x0 (x1 x2 x3)) ⟶ ∀ x2 x3 x4 x5 . x0 x2 ⟶ x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x0 (x1 x2 (x1 x3 (x1 x4 x5)))
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))))))))
Theorem 41130.. : ∀ 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 . x0 x5 ⟶ x0 x6 ⟶ ∀ x7 x8 x9 x10 . x0 x7 ⟶ x0 x8 ⟶ x0 x9 ⟶ x0 x10 ⟶ x5 = x1 (x3 x7) (x1 (x3 x8) (x1 (x3 x9) (x3 x10))) ⟶ ∀ x11 x12 x13 x14 . x0 x11 ⟶ x0 x12 ⟶ x0 x13 ⟶ x0 x14 ⟶ x6 = x1 (x3 x11) (x1 (x3 x12) (x1 (x3 x13) (x3 x14))) ⟶ ∀ x15 : ο . (∀ x16 x17 x18 x19 . x0 x16 ⟶ x0 x17 ⟶ x0 x18 ⟶ x0 x19 ⟶ x2 x5 x6 = x1 (x3 x16) (x1 (x3 x17) (x1 (x3 x18) (x3 x19))) ⟶ x15) ⟶ x15 (proof)
Theorem 277a1.. : ∀ 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 : ι → ο . (∀ x6 . x0 x6 ⟶ ∀ x7 : ο . (∀ x8 . and (x5 x8) (x3 x8 = x3 x6) ⟶ x7) ⟶ x7) ⟶ ∀ x6 x7 . x0 x6 ⟶ x0 x7 ⟶ ∀ x8 x9 x10 x11 . x0 x8 ⟶ x0 x9 ⟶ x0 x10 ⟶ x0 x11 ⟶ x6 = x1 (x3 x8) (x1 (x3 x9) (x1 (x3 x10) (x3 x11))) ⟶ ∀ x12 x13 x14 x15 . x0 x12 ⟶ x0 x13 ⟶ x0 x14 ⟶ x0 x15 ⟶ x7 = x1 (x3 x12) (x1 (x3 x13) (x1 (x3 x14) (x3 x15))) ⟶ ∀ x16 : ο . (∀ x17 x18 x19 x20 . x5 x17 ⟶ x5 x18 ⟶ x5 x19 ⟶ x5 x20 ⟶ x2 x6 x7 = x1 (x3 x17) (x1 (x3 x18) (x1 (x3 x19) (x3 x20))) ⟶ x16) ⟶ x16 (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 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 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_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
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 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
Theorem 634fb.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 x3 x4 x5 . SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNo x5 ⟶ x0 = add_SNo (mul_SNo x2 x2) (add_SNo (mul_SNo x3 x3) (add_SNo (mul_SNo x4 x4) (mul_SNo x5 x5))) ⟶ ∀ x6 x7 x8 x9 . SNo x6 ⟶ SNo x7 ⟶ SNo x8 ⟶ SNo x9 ⟶ 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 x12 x13 x14 . SNo x11 ⟶ SNo x12 ⟶ SNo x13 ⟶ SNo x14 ⟶ 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 (proof)
Param int^int : ι
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 int_minus_SNo^{int_minus_SNo} : ∀ x0 . x0 ∈ int ⟶ minus_SNo x0 ∈ int
Known int_SNo^int_SNo : ∀ x0 . x0 ∈ int ⟶ SNo x0
Theorem d6e7f.. : ∀ 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 ∈ int ⟶ ∀ x12 . x12 ∈ int ⟶ ∀ x13 . x13 ∈ int ⟶ ∀ x14 . x14 ∈ int ⟶ 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 (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 omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Theorem 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 (proof)