Proofgold Address

Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param omega^omega : ι
Param SNoS_^SNoS_ : ι → ι
Param ordinal^ordinal : ι → ο
Param SNo_^SNo_ : ι → ι → ο
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known omega_ordinal^{omega_ordinal} : ordinal omega
Param SNoLev^SNoLev : ι → ι
Known ordinal_SNoLev^{ordinal_SNoLev} : ∀ x0 . ordinal x0 ⟶ SNoLev x0 = x0
Param nat_p^nat_p : ι → ο
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
Param SNo^SNo : ι → ο
Known SNoLev_^SNoLev_ : ∀ x0 . SNo x0 ⟶ SNo_ (SNoLev x0) x0
Known omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Theorem omega_SNoS_omega^{omega_SNoS_omega} : omega ⊆ SNoS_ omega (proof)
Param add_SNo^add_SNo : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
Param add_nat^add_nat : 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
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_1^nat_1 : nat_p 1
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Known nat_0^nat_0 : nat_p 0
Known add_nat_0R^add_nat_0R : ∀ x0 . add_nat x0 0 = x0
Theorem add_SNo_1_ordsucc^{add_SNo_1_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ add_SNo x0 1 = ordsucc x0 (proof)
Param SNoLt^SNoLt : ι → ι → ο
Param abs_SNo^abs_SNo : ι → ι
Param minus_SNo^minus_SNo : ι → ι
Param eps_^eps_ : ι → ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known SNoS_E2^SNoS_E2 : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ SNoS_ x0 ⟶ ∀ x2 : ο . (SNoLev x1 ∈ x0 ⟶ ordinal (SNoLev x1) ⟶ SNo x1 ⟶ SNo_ (SNoLev x1) x1 ⟶ x2) ⟶ x2
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param SNoLe^SNoLe : ι → ι → ο
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known SNo_0^SNo_0 : SNo 0
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known abs_SNo_minus^{abs_SNo_minus} : ∀ x0 . SNo x0 ⟶ abs_SNo (minus_SNo x0) = abs_SNo x0
Known pos_abs_SNo^pos_abs_SNo : ∀ x0 . SNoLt 0 x0 ⟶ abs_SNo x0 = x0
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Known SNo_eps_^SNo_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo (eps_ x0)
Known SNo_eps_SNoS_omega^{SNo_eps_SNoS_omega} : ∀ x0 . x0 ∈ omega ⟶ eps_ x0 ∈ SNoS_ omega
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Known SNo_eps_pos^SNo_eps_pos : ∀ x0 . x0 ∈ omega ⟶ SNoLt 0 (eps_ 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 SNo_eps_decr^SNo_eps_decr : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt (eps_ x0) (eps_ x1)
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known add_SNo_Lt1^add_SNo_Lt1 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x1)
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Theorem SNo_prereal_incr_lower_pos^{SNo_prereal_incr_lower_pos} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ (∀ x1 . x1 ∈ SNoS_ omega ⟶ (∀ x2 . x2 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x1 (minus_SNo x0))) (eps_ x2)) ⟶ x1 = x0) ⟶ (∀ x1 . x1 ∈ omega ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ SNoS_ omega) (and (SNoLt x3 x0) (SNoLt x0 (add_SNo x3 (eps_ x1)))) ⟶ x2) ⟶ x2) ⟶ ∀ x1 . x1 ∈ omega ⟶ ∀ x2 : ο . (∀ x3 . x3 ∈ SNoS_ omega ⟶ SNoLt 0 x3 ⟶ SNoLt x3 x0 ⟶ SNoLt x0 (add_SNo x3 (eps_ x1)) ⟶ x2) ⟶ x2 (proof)
Param mul_SNo^mul_SNo : ι → ι → ι
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo 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)
Theorem pos_mul_SNo_Lt'^{pos_mul_SNo_Lt} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt 0 x2 ⟶ SNoLt x0 x1 ⟶ SNoLt (mul_SNo x0 x2) (mul_SNo x1 x2) (proof)
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Theorem pos_mul_SNo_Lt2^{pos_mul_SNo_Lt2} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ SNoLt x0 x2 ⟶ SNoLt x1 x3 ⟶ SNoLt (mul_SNo x0 x1) (mul_SNo x2 x3) (proof)
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 nonneg_mul_SNo_Le'^{nonneg_mul_SNo_Le} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe 0 x2 ⟶ SNoLe x0 x1 ⟶ SNoLe (mul_SNo x0 x2) (mul_SNo x1 x2) (proof)
Known SNoLe_tra^SNoLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLe x0 x2
Theorem nonneg_mul_SNo_Le2^{nonneg_mul_SNo_Le2} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLe 0 x0 ⟶ SNoLe 0 x1 ⟶ SNoLe x0 x2 ⟶ SNoLe x1 x3 ⟶ SNoLe (mul_SNo x0 x1) (mul_SNo x2 x3) (proof)
Param exp_SNo_nat^exp_SNo_nat : ι → 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 exp_SNo_nat_0^{exp_SNo_nat_0} : ∀ x0 . SNo x0 ⟶ exp_SNo_nat x0 0 = 1
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Known exp_SNo_nat_S^{exp_SNo_nat_S} : ∀ x0 . SNo x0 ⟶ ∀ x1 . nat_p x1 ⟶ exp_SNo_nat x0 (ordsucc x1) = mul_SNo x0 (exp_SNo_nat x0 x1)
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Known SNo_1^SNo_1 : SNo 1
Known SNo_exp_SNo_nat^{SNo_exp_SNo_nat} : ∀ x0 . SNo x0 ⟶ ∀ x1 . nat_p x1 ⟶ SNo (exp_SNo_nat x0 x1)
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known SNoLt_0_1^SNoLt_0_1 : SNoLt 0 1
Theorem exp_SNo_1_bd^exp_SNo_1_bd : ∀ x0 . SNo x0 ⟶ SNoLe 1 x0 ⟶ ∀ x1 . nat_p x1 ⟶ SNoLe 1 (exp_SNo_nat x0 x1) (proof)
Known SNo_2^SNo_2 : SNo 2
Known add_SNo_1_1_2^{add_SNo_1_1_2} : add_SNo 1 1 = 2
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 nat_p_SNo^nat_p_SNo : ∀ x0 . nat_p x0 ⟶ SNo x0
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 SNoLt_1_2^SNoLt_1_2 : SNoLt 1 2
Theorem exp_SNo_2_bd^exp_SNo_2_bd : ∀ x0 . nat_p x0 ⟶ SNoLt x0 (exp_SNo_nat 2 x0) (proof)
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Known mul_SNo_eps_power_2^{mul_SNo_eps_power_2} : ∀ x0 . nat_p x0 ⟶ mul_SNo (eps_ x0) (exp_SNo_nat 2 x0) = 1
Theorem eps_bd_1^eps_bd_1 : ∀ x0 . x0 ∈ omega ⟶ SNoLe (eps_ x0) 1 (proof)
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 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_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 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))) (proof)
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 minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Theorem 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 (proof)
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)
Theorem add_SNo_Lt4^add_SNo_Lt4 : ∀ x0 x1 x2 x3 x4 x5 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNo x5 ⟶ SNoLt x0 x3 ⟶ SNoLt x1 x4 ⟶ SNoLt x2 x5 ⟶ SNoLt (add_SNo x0 (add_SNo x1 x2)) (add_SNo x3 (add_SNo x4 x5)) (proof)
Theorem mul_SNo_Lt1_pos_Lt^{mul_SNo_Lt1_pos_Lt} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 1 ⟶ SNoLt 0 x1 ⟶ SNoLt (mul_SNo x0 x1) x1 (proof)
Theorem mul_SNo_Le1_nonneg_Le^{mul_SNo_Le1_nonneg_Le} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 1 ⟶ SNoLe 0 x1 ⟶ SNoLe (mul_SNo x0 x1) x1 (proof)
Theorem SNo_mul_SNo_3^{SNo_mul_SNo_3} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo (mul_SNo x0 (mul_SNo x1 x2)) (proof)
Theorem 82b86.. : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo (mul_SNo x0 (mul_SNo x1 (mul_SNo x2 x3))) (proof)
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 624bf.. : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo x0 (mul_SNo x1 (mul_SNo x2 x3)) = mul_SNo (mul_SNo x0 (mul_SNo x1 x2)) x3 (proof)
Theorem 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) (proof)
Theorem 85f24.. : ∀ x0 x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo x0 (mul_SNo x1 (mul_SNo x2 x3)) = mul_SNo x0 (mul_SNo x2 (mul_SNo x1 x3)) (proof)
Theorem mul_SNo_com_3b_1_2^{mul_SNo_com_3b_1_2} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo (mul_SNo x0 x1) x2 = mul_SNo (mul_SNo x0 x2) x1 (proof)
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)
Theorem mul_SNo_rotate_3_1^{mul_SNo_rotate_3_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (mul_SNo x1 x2) = mul_SNo x2 (mul_SNo x0 x1) (proof)
Theorem mul_SNo_rotate_4_1^{mul_SNo_rotate_4_1} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo x0 (mul_SNo x1 (mul_SNo x2 x3)) = mul_SNo x3 (mul_SNo x0 (mul_SNo x1 x2)) (proof)
Theorem b4aa3.. : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ mul_SNo x0 (mul_SNo x1 (mul_SNo x2 (mul_SNo x3 x4))) = mul_SNo x4 (mul_SNo x0 (mul_SNo x1 (mul_SNo x2 x3))) (proof)
Theorem 1b9b9.. : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ mul_SNo x0 (mul_SNo x1 (mul_SNo x2 (mul_SNo x3 x4))) = mul_SNo x3 (mul_SNo x4 (mul_SNo x0 (mul_SNo x1 x2))) (proof)