Proofgold Signed Transaction

Param omega^omega : ι
Param mul_nat^mul_nat : ι → ι → ι
Param mul_SNo^mul_SNo : ι → ι → ι
Param nat_p^nat_p : ι → ο
Param ordinal^ordinal : ι → ο
Param SNo^SNo : ι → ο
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
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 mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known mul_nat_0R^mul_nat_0R : ∀ x0 . mul_nat x0 0 = 0
Param add_nat^add_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)
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
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known mul_nat_p^mul_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (mul_nat x0 x1)
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known ordinal_SNo^ordinal_SNo : ∀ x0 . ordinal x0 ⟶ SNo x0
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known add_SNo_ordinal_SL^{add_SNo_ordinal_SL} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ add_SNo (ordsucc x0) x1 = ordsucc (add_SNo x0 x1)
Known ordinal_Empty^{ordinal_Empty} : ordinal 0
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_1^SNo_1 : SNo 1
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Theorem mul_nat_mul_SNo^{mul_nat_mul_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat x0 x1 = mul_SNo x0 x1 (proof)
Param minus_SNo^minus_SNo : ι → ι
Param binunion^binunion : ι → ι → ι
Param minus_CSNo^minus_CSNo : ι → ι
Definition int_alt1^int := binunion omega (prim5 omega minus_CSNo)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known ReplE_impred^ReplE_impred : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ prim5 x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3
Known minus_SNo_minus_CSNo^{minus_SNo_minus_CSNo} : ∀ x0 . SNo x0 ⟶ minus_SNo x0 = minus_CSNo x0
Theorem a3283..^{int_SNo_cases} : ∀ x0 : ι → ο . (∀ x1 . x1 ∈ omega ⟶ x0 x1) ⟶ (∀ x1 . x1 ∈ omega ⟶ x0 (minus_SNo x1)) ⟶ ∀ x1 . x1 ∈ int_alt1 ⟶ x0 x1 (proof)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Theorem c3d99..^{Subq_omega_int} : omega ⊆ int_alt1 (proof)
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Theorem 1b4d5..^{int_minus_SNo_omega} : ∀ x0 . x0 ∈ omega ⟶ minus_SNo x0 ∈ int_alt1 (proof)
Theorem ordinal_ordsucc_SNo_eq^{ordinal_ordsucc_SNo_eq} : ∀ x0 . ordinal x0 ⟶ ordsucc x0 = add_SNo 1 x0 (proof)
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
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)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
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 minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Known nat_1^nat_1 : nat_p 1
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)
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 omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Theorem 6fbb0..^{int_add_SNo_lem} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . nat_p x1 ⟶ add_SNo (minus_SNo x0) x1 ∈ int_alt1 (proof)
Known add_SNo_In_omega^{add_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_SNo x0 x1 ∈ omega
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Theorem a0aa5..^int_add_SNo : ∀ x0 . x0 ∈ int_alt1 ⟶ ∀ x1 . x1 ∈ int_alt1 ⟶ add_SNo x0 x1 ∈ int_alt1 (proof)
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Theorem daaad..^{int_minus_SNo} : ∀ x0 . x0 ∈ int_alt1 ⟶ minus_SNo x0 ∈ int_alt1 (proof)
Theorem mul_SNo_In_omega^{mul_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_SNo x0 x1 ∈ omega (proof)
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_distrL^{mul_SNo_minus_distrL} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo (minus_SNo x0) x1 = minus_SNo (mul_SNo x0 x1)
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Theorem eefdf..^int_mul_SNo : ∀ x0 . x0 ∈ int_alt1 ⟶ ∀ x1 . x1 ∈ int_alt1 ⟶ mul_SNo x0 x1 ∈ int_alt1 (proof)
Param SNo_pair^SNo_pair : ι → ι → ι
Param CSNo_Re^CSNo_Re : ι → ι
Param CSNo_Im^CSNo_Im : ι → ι
Definition mul_CSNo^mul_CSNo := λ x0 x1 . SNo_pair (add_SNo (mul_SNo (CSNo_Re x0) (CSNo_Re x1)) (minus_SNo (mul_SNo (CSNo_Im x0) (CSNo_Im x1)))) (add_SNo (mul_SNo (CSNo_Re x0) (CSNo_Im x1)) (mul_SNo (CSNo_Im x0) (CSNo_Re x1)))
Known SNo_Re^SNo_Re : ∀ x0 . SNo x0 ⟶ CSNo_Re x0 = x0
Known SNo_pair_0^SNo_pair_0 : ∀ x0 . SNo_pair x0 0 = x0
Known SNo_Im^SNo_Im : ∀ x0 . SNo x0 ⟶ CSNo_Im x0 = 0
Known SNo_0^SNo_0 : SNo 0
Theorem 15de6..^{mul_SNo_mul_CSNo} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_CSNo x0 x1 (proof)
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 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))
Theorem add_SNo_rotate_5_1^{add_SNo_rotate_5_1} : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ add_SNo x0 (add_SNo x1 (add_SNo x2 (add_SNo x3 x4))) = add_SNo x4 (add_SNo x0 (add_SNo x1 (add_SNo x2 x3))) (proof)
Theorem add_SNo_rotate_5_2^{add_SNo_rotate_5_2} : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ add_SNo x0 (add_SNo x1 (add_SNo x2 (add_SNo x3 x4))) = add_SNo x3 (add_SNo x4 (add_SNo x0 (add_SNo x1 x2))) (proof)
Param SNoL^SNoL : ι → ι
Param SNoLe^SNoLe : ι → ι → ο
Param SNoR^SNoR : ι → ι
Param SNoLt^SNoLt : ι → ι → ο
Param SNoS_^SNoS_ : ι → ι
Param SNoLev^SNoLev : ι → ι
Known SNoL_E^SNoL_E : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoL x0 ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x2) ⟶ x2
Known SNoL_SNoS^SNoL_SNoS : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoL x0 ⟶ x1 ∈ SNoS_ (SNoLev x0)
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known SNoR_E^SNoR_E : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoR x0 ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x2) ⟶ x2
Known SNoR_SNoS^SNoR_SNoS : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoR x0 ⟶ x1 ∈ SNoS_ (SNoLev x0)
Param SNo_^SNo_ : ι → ι → ο
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
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Known add_SNo_minus_Lt1b3^{add_SNo_minus_Lt1b3} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x3 x2) ⟶ SNoLt (add_SNo x0 (add_SNo x1 (minus_SNo x2))) x3
Known add_SNo_Lt1_cancel^{add_SNo_Lt1_cancel} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x1) ⟶ SNoLt 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 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_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 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 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)
Theorem mul_SNo_assoc_lem1^{mul_SNo_assoc_lem1} : ∀ x0 : ι → ι → ι . (∀ x1 x2 . SNo x1 ⟶ SNo x2 ⟶ SNo (x0 x1 x2)) ⟶ (∀ x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ x0 x1 (add_SNo x2 x3) = add_SNo (x0 x1 x2) (x0 x1 x3)) ⟶ (∀ x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ x0 (add_SNo x1 x2) x3 = add_SNo (x0 x1 x3) (x0 x2 x3)) ⟶ (∀ x1 x2 . SNo x1 ⟶ SNo x2 ⟶ ∀ x3 . x3 ∈ SNoL (x0 x1 x2) ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ SNoL x1 ⟶ ∀ x6 . x6 ∈ SNoL x2 ⟶ SNoLe (add_SNo x3 (x0 x5 x6)) (add_SNo (x0 x5 x2) (x0 x1 x6)) ⟶ x4) ⟶ (∀ x5 . x5 ∈ SNoR x1 ⟶ ∀ x6 . x6 ∈ SNoR x2 ⟶ SNoLe (add_SNo x3 (x0 x5 x6)) (add_SNo (x0 x5 x2) (x0 x1 x6)) ⟶ x4) ⟶ x4) ⟶ (∀ x1 x2 . SNo x1 ⟶ SNo x2 ⟶ ∀ x3 . x3 ∈ SNoR (x0 x1 x2) ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ SNoL x1 ⟶ ∀ x6 . x6 ∈ SNoR x2 ⟶ SNoLe (add_SNo (x0 x5 x2) (x0 x1 x6)) (add_SNo x3 (x0 x5 x6)) ⟶ x4) ⟶ (∀ x5 . x5 ∈ SNoR x1 ⟶ ∀ x6 . x6 ∈ SNoL x2 ⟶ SNoLe (add_SNo (x0 x5 x2) (x0 x1 x6)) (add_SNo x3 (x0 x5 x6)) ⟶ x4) ⟶ x4) ⟶ (∀ x1 x2 x3 x4 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNoLt x3 x1 ⟶ SNoLt x4 x2 ⟶ SNoLt (add_SNo (x0 x3 x2) (x0 x1 x4)) (add_SNo (x0 x1 x2) (x0 x3 x4))) ⟶ (∀ x1 x2 x3 x4 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNoLe x3 x1 ⟶ SNoLe x4 x2 ⟶ SNoLe (add_SNo (x0 x3 x2) (x0 x1 x4)) (add_SNo (x0 x1 x2) (x0 x3 x4))) ⟶ ∀ x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ x0 x4 (x0 x2 x3) = x0 (x0 x4 x2) x3) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x2) ⟶ x0 x1 (x0 x4 x3) = x0 (x0 x1 x4) x3) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x3) ⟶ x0 x1 (x0 x2 x4) = x0 (x0 x1 x2) x4) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x2) ⟶ x0 x4 (x0 x5 x3) = x0 (x0 x4 x5) x3) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x3) ⟶ x0 x4 (x0 x2 x5) = x0 (x0 x4 x2) x5) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x2) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x3) ⟶ x0 x1 (x0 x4 x5) = x0 (x0 x1 x4) x5) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x2) ⟶ ∀ x6 . x6 ∈ SNoS_ (SNoLev x3) ⟶ x0 x4 (x0 x5 x6) = x0 (x0 x4 x5) x6) ⟶ ∀ x4 . (∀ x5 . x5 ∈ x4 ⟶ ∀ x6 : ο . (∀ x7 . x7 ∈ SNoL x1 ⟶ ∀ x8 . x8 ∈ SNoL (x0 x2 x3) ⟶ x5 = add_SNo (x0 x7 (x0 x2 x3)) (add_SNo (x0 x1 x8) (minus_SNo (x0 x7 x8))) ⟶ x6) ⟶ (∀ x7 . x7 ∈ SNoR x1 ⟶ ∀ x8 . x8 ∈ SNoR (x0 x2 x3) ⟶ x5 = add_SNo (x0 x7 (x0 x2 x3)) (add_SNo (x0 x1 x8) (minus_SNo (x0 x7 x8))) ⟶ x6) ⟶ x6) ⟶ ∀ x5 . x5 ∈ x4 ⟶ SNoLt x5 (x0 (x0 x1 x2) x3) (proof)
Known add_SNo_minus_Lt2b3^{add_SNo_minus_Lt2b3} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt (add_SNo x3 x2) (add_SNo x0 x1) ⟶ SNoLt x3 (add_SNo x0 (add_SNo x1 (minus_SNo x2)))
Known SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 x2
Theorem mul_SNo_assoc_lem2^{mul_SNo_assoc_lem2} : ∀ x0 : ι → ι → ι . (∀ x1 x2 . SNo x1 ⟶ SNo x2 ⟶ SNo (x0 x1 x2)) ⟶ (∀ x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ x0 x1 (add_SNo x2 x3) = add_SNo (x0 x1 x2) (x0 x1 x3)) ⟶ (∀ x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ x0 (add_SNo x1 x2) x3 = add_SNo (x0 x1 x3) (x0 x2 x3)) ⟶ (∀ x1 x2 . SNo x1 ⟶ SNo x2 ⟶ ∀ x3 . x3 ∈ SNoL (x0 x1 x2) ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ SNoL x1 ⟶ ∀ x6 . x6 ∈ SNoL x2 ⟶ SNoLe (add_SNo x3 (x0 x5 x6)) (add_SNo (x0 x5 x2) (x0 x1 x6)) ⟶ x4) ⟶ (∀ x5 . x5 ∈ SNoR x1 ⟶ ∀ x6 . x6 ∈ SNoR x2 ⟶ SNoLe (add_SNo x3 (x0 x5 x6)) (add_SNo (x0 x5 x2) (x0 x1 x6)) ⟶ x4) ⟶ x4) ⟶ (∀ x1 x2 . SNo x1 ⟶ SNo x2 ⟶ ∀ x3 . x3 ∈ SNoR (x0 x1 x2) ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ SNoL x1 ⟶ ∀ x6 . x6 ∈ SNoR x2 ⟶ SNoLe (add_SNo (x0 x5 x2) (x0 x1 x6)) (add_SNo x3 (x0 x5 x6)) ⟶ x4) ⟶ (∀ x5 . x5 ∈ SNoR x1 ⟶ ∀ x6 . x6 ∈ SNoL x2 ⟶ SNoLe (add_SNo (x0 x5 x2) (x0 x1 x6)) (add_SNo x3 (x0 x5 x6)) ⟶ x4) ⟶ x4) ⟶ (∀ x1 x2 x3 x4 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNoLt x3 x1 ⟶ SNoLt x4 x2 ⟶ SNoLt (add_SNo (x0 x3 x2) (x0 x1 x4)) (add_SNo (x0 x1 x2) (x0 x3 x4))) ⟶ (∀ x1 x2 x3 x4 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNoLe x3 x1 ⟶ SNoLe x4 x2 ⟶ SNoLe (add_SNo (x0 x3 x2) (x0 x1 x4)) (add_SNo (x0 x1 x2) (x0 x3 x4))) ⟶ ∀ x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ x0 x4 (x0 x2 x3) = x0 (x0 x4 x2) x3) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x2) ⟶ x0 x1 (x0 x4 x3) = x0 (x0 x1 x4) x3) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x3) ⟶ x0 x1 (x0 x2 x4) = x0 (x0 x1 x2) x4) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x2) ⟶ x0 x4 (x0 x5 x3) = x0 (x0 x4 x5) x3) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x3) ⟶ x0 x4 (x0 x2 x5) = x0 (x0 x4 x2) x5) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x2) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x3) ⟶ x0 x1 (x0 x4 x5) = x0 (x0 x1 x4) x5) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x2) ⟶ ∀ x6 . x6 ∈ SNoS_ (SNoLev x3) ⟶ x0 x4 (x0 x5 x6) = x0 (x0 x4 x5) x6) ⟶ ∀ x4 . (∀ x5 . x5 ∈ x4 ⟶ ∀ x6 : ο . (∀ x7 . x7 ∈ SNoL x1 ⟶ ∀ x8 . x8 ∈ SNoR (x0 x2 x3) ⟶ x5 = add_SNo (x0 x7 (x0 x2 x3)) (add_SNo (x0 x1 x8) (minus_SNo (x0 x7 x8))) ⟶ x6) ⟶ (∀ x7 . x7 ∈ SNoR x1 ⟶ ∀ x8 . x8 ∈ SNoL (x0 x2 x3) ⟶ x5 = add_SNo (x0 x7 (x0 x2 x3)) (add_SNo (x0 x1 x8) (minus_SNo (x0 x7 x8))) ⟶ x6) ⟶ x6) ⟶ ∀ x5 . x5 ∈ x4 ⟶ SNoLt (x0 (x0 x1 x2) x3) x5 (proof)
Known SNoLev_ind3^SNoLev_ind3 : ∀ x0 : ι → ι → ι → ο . (∀ x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ x0 x4 x2 x3) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x2) ⟶ x0 x1 x4 x3) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x3) ⟶ x0 x1 x2 x4) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x2) ⟶ x0 x4 x5 x3) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x3) ⟶ x0 x4 x2 x5) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x2) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x3) ⟶ x0 x1 x4 x5) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x2) ⟶ ∀ x6 . x6 ∈ SNoS_ (SNoLev x3) ⟶ x0 x4 x5 x6) ⟶ x0 x1 x2 x3) ⟶ ∀ x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ x0 x1 x2 x3
Param SNoCutP^SNoCutP : ι → ι → ο
Param SNoCut^SNoCut : ι → ι → ι
Known mul_SNo_eq_3^mul_SNo_eq_3 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 : ο . (∀ x3 x4 . SNoCutP x3 x4 ⟶ (∀ x5 . x5 ∈ x3 ⟶ ∀ x6 : ο . (∀ x7 . x7 ∈ SNoL x0 ⟶ ∀ x8 . x8 ∈ SNoL x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ (∀ x7 . x7 ∈ SNoR x0 ⟶ ∀ x8 . x8 ∈ SNoR x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ SNoL x0 ⟶ ∀ x6 . x6 ∈ SNoL x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x3) ⟶ (∀ x5 . x5 ∈ SNoR x0 ⟶ ∀ x6 . x6 ∈ SNoR x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x3) ⟶ (∀ x5 . x5 ∈ x4 ⟶ ∀ x6 : ο . (∀ x7 . x7 ∈ SNoL x0 ⟶ ∀ x8 . x8 ∈ SNoR x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ (∀ x7 . x7 ∈ SNoR x0 ⟶ ∀ x8 . x8 ∈ SNoL x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ SNoL x0 ⟶ ∀ x6 . x6 ∈ SNoR x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x4) ⟶ (∀ x5 . x5 ∈ SNoR x0 ⟶ ∀ x6 . x6 ∈ SNoL x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x4) ⟶ mul_SNo x0 x1 = SNoCut x3 x4 ⟶ x2) ⟶ x2
Known SNoCut_ext^SNoCut_ext : ∀ x0 x1 x2 x3 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ SNoLt x4 (SNoCut x2 x3)) ⟶ (∀ x4 . x4 ∈ x1 ⟶ SNoLt (SNoCut x2 x3) x4) ⟶ (∀ x4 . x4 ∈ x2 ⟶ SNoLt x4 (SNoCut x0 x1)) ⟶ (∀ x4 . x4 ∈ x3 ⟶ SNoLt (SNoCut x0 x1) x4) ⟶ SNoCut x0 x1 = SNoCut x2 x3
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_SNoL_interpolate_impred^{mul_SNo_SNoL_interpolate_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoL (mul_SNo x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ SNoL x0 ⟶ ∀ x5 . x5 ∈ SNoL x1 ⟶ SNoLe (add_SNo x2 (mul_SNo x4 x5)) (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) ⟶ x3) ⟶ (∀ x4 . x4 ∈ SNoR x0 ⟶ ∀ x5 . x5 ∈ SNoR x1 ⟶ SNoLe (add_SNo x2 (mul_SNo x4 x5)) (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) ⟶ x3) ⟶ x3
Known mul_SNo_SNoR_interpolate_impred^{mul_SNo_SNoR_interpolate_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoR (mul_SNo x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ SNoL x0 ⟶ ∀ x5 . x5 ∈ SNoR x1 ⟶ SNoLe (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) (add_SNo x2 (mul_SNo x4 x5)) ⟶ x3) ⟶ (∀ x4 . x4 ∈ SNoR x0 ⟶ ∀ x5 . x5 ∈ SNoL x1 ⟶ SNoLe (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) (add_SNo x2 (mul_SNo x4 x5)) ⟶ x3) ⟶ x3
Known mul_SNo_Lt^mul_SNo_Lt : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt x2 x0 ⟶ SNoLt x3 x1 ⟶ SNoLt (add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3)) (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3))
Known mul_SNo_Le^mul_SNo_Le : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLe x2 x0 ⟶ SNoLe x3 x1 ⟶ SNoLe (add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3)) (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3))
Theorem 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 (proof)