Proofgold Signed Transaction

Param c7ce4.. : ι → ο
Param SNo^SNo : ι → ο
Param add_SNo^add_SNo : ι → ι → ι
Param mul_SNo^mul_SNo : ι → ι → ι
Param 28f8d.. : ι → ι
Param minus_SNo^minus_SNo : ι → ι
Param d634d.. : ι → ι
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known 85533.. : ∀ x0 . c7ce4.. x0 ⟶ SNo (28f8d.. x0)
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known eb0da.. : ∀ x0 . c7ce4.. x0 ⟶ SNo (d634d.. x0)
Theorem e197b.. : ∀ x0 x1 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ SNo (add_SNo (mul_SNo (28f8d.. x0) (28f8d.. x1)) (minus_SNo (mul_SNo (d634d.. x0) (d634d.. x1))))
Theorem 484b2.. : ∀ x0 x1 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ SNo (add_SNo (mul_SNo (28f8d.. x0) (d634d.. x1)) (mul_SNo (d634d.. x0) (28f8d.. x1)))
Param ad280.. : ι → ι → ι
Definition 22598.. := λ x0 x1 . ad280.. (add_SNo (mul_SNo (28f8d.. x0) (28f8d.. x1)) (minus_SNo (mul_SNo (d634d.. x0) (d634d.. x1)))) (add_SNo (mul_SNo (28f8d.. x0) (d634d.. x1)) (mul_SNo (d634d.. x0) (28f8d.. x1)))
Known 1c01b.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ 28f8d.. (ad280.. x0 x1) = x0
Theorem d5374.. : ∀ x0 x1 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ 28f8d.. (22598.. x0 x1) = add_SNo (mul_SNo (28f8d.. x0) (28f8d.. x1)) (minus_SNo (mul_SNo (d634d.. x0) (d634d.. x1)))
Known 5b520.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ d634d.. (ad280.. x0 x1) = x1
Theorem c5063.. : ∀ x0 x1 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ d634d.. (22598.. x0 x1) = add_SNo (mul_SNo (28f8d.. x0) (d634d.. x1)) (mul_SNo (d634d.. x0) (28f8d.. x1))
Known e4080.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ c7ce4.. (ad280.. x0 x1)
Theorem 03ef2.. : ∀ x0 x1 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ c7ce4.. (22598.. x0 x1)
Known 371c6.. : ∀ x0 x1 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ 28f8d.. x0 = 28f8d.. x1 ⟶ d634d.. x0 = d634d.. x1 ⟶ x0 = x1
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Theorem 8d8ef.. : ∀ x0 x1 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ 22598.. x0 x1 = 22598.. x1 x0
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_minus_distrR^{mul_SNo_minus_distrR} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 (minus_SNo x1) = minus_SNo (mul_SNo x0 x1)
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 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))
Known add_SNo_rotate_4_0312^{add_SNo_rotate_4_0312} : ∀ 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 x3) (add_SNo x1 x2)
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 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_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)
Theorem 76b9f.. : ∀ x0 x1 x2 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ c7ce4.. x2 ⟶ 22598.. x0 (22598.. x1 x2) = 22598.. (22598.. x0 x1) x2
Known 2c33a.. : ∀ x0 . SNo x0 ⟶ c7ce4.. x0
Known SNo_0^SNo_0 : SNo 0
Theorem feda5.. : c7ce4.. 0
Param ordsucc^ordsucc : ι → ι
Known SNo_1^SNo_1 : SNo 1
Theorem 187f6.. : c7ce4.. 1
Known 43bcd.. : ∀ x0 . ad280.. x0 0 = x0
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Known minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Known mul_SNo_zeroL^{mul_SNo_zeroL} : ∀ x0 . SNo x0 ⟶ mul_SNo 0 x0 = 0
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Theorem 7a36e.. : ∀ x0 . c7ce4.. x0 ⟶ 22598.. 1 x0 = x0
Theorem 9dbc9.. : ∀ x0 . c7ce4.. x0 ⟶ 22598.. x0 1 = x0
Param a0628.. : ι → ι → ι
Known 33aee.. : ∀ x0 x1 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ c7ce4.. (a0628.. x0 x1)
Known 5576f.. : ∀ x0 x1 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ 28f8d.. (a0628.. x0 x1) = add_SNo (28f8d.. x0) (28f8d.. x1)
Known 9325f.. : ∀ x0 x1 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ d634d.. (a0628.. x0 x1) = add_SNo (d634d.. x0) (d634d.. x1)
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)
Theorem 610b0.. : ∀ x0 x1 x2 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ c7ce4.. x2 ⟶ 22598.. x0 (a0628.. x1 x2) = a0628.. (22598.. x0 x1) (22598.. x0 x2)
Known 416cf.. : ∀ x0 . SNo x0 ⟶ 28f8d.. x0 = x0
Known 0de29.. : ∀ x0 . SNo x0 ⟶ d634d.. x0 = 0
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Theorem a2ab6.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = 22598.. x0 x1
Param 8d0f8.. : ι
Param b1848.. : ι → ι
Known 3e3aa.. : c7ce4.. 8d0f8..
Known 76017.. : ∀ x0 . c7ce4.. x0 ⟶ c7ce4.. (b1848.. x0)
Known d3d48.. : ∀ x0 . c7ce4.. x0 ⟶ 28f8d.. (b1848.. x0) = minus_SNo (28f8d.. x0)
Known f16c1.. : 28f8d.. 8d0f8.. = 0
Known b2ead.. : d634d.. 8d0f8.. = 1
Known 7d0dc.. : 28f8d.. 1 = 1
Known d1304.. : ∀ x0 . c7ce4.. x0 ⟶ d634d.. (b1848.. x0) = minus_SNo (d634d.. x0)
Known d3315.. : d634d.. 1 = 0
Theorem e377b.. : 22598.. 8d0f8.. 8d0f8.. = b1848.. 1
Param recip_SNo^recip_SNo : ι → ι
Definition div_SNo^div_SNo := λ x0 x1 . mul_SNo x0 (recip_SNo x1)
Param exp_SNo_nat^exp_SNo_nat : ι → ι → ι
Definition 41fb9.. := λ x0 . ad280.. (div_SNo (28f8d.. x0) (add_SNo (exp_SNo_nat (28f8d.. x0) 2) (exp_SNo_nat (d634d.. x0) 2))) (minus_SNo (div_SNo (d634d.. x0) (add_SNo (exp_SNo_nat (28f8d.. x0) 2) (exp_SNo_nat (d634d.. x0) 2))))
Known SNo_div_SNo^SNo_div_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (div_SNo x0 x1)
Param nat_p^nat_p : ι → ο
Known SNo_exp_SNo_nat^{SNo_exp_SNo_nat} : ∀ x0 . SNo x0 ⟶ ∀ x1 . nat_p x1 ⟶ SNo (exp_SNo_nat x0 x1)
Known nat_2^nat_2 : nat_p 2
Theorem b7add.. : ∀ x0 . c7ce4.. x0 ⟶ c7ce4.. (41fb9.. x0)
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Known 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)
Known add_SNo_minus_SNo_linv^{add_SNo_minus_SNo_linv} : ∀ x0 . SNo x0 ⟶ add_SNo (minus_SNo x0) x0 = 0
Theorem 9f4eb.. : ∀ x0 . c7ce4.. x0 ⟶ ∀ x1 . SNo x1 ⟶ mul_SNo (add_SNo (mul_SNo (28f8d.. x0) (28f8d.. x0)) (mul_SNo (d634d.. x0) (d634d.. x0))) x1 = 1 ⟶ 22598.. x0 (a0628.. (22598.. x1 (28f8d.. x0)) (b1848.. (22598.. 8d0f8.. (22598.. x1 (d634d.. x0))))) = 1
Known exp_SNo_nat_2^{exp_SNo_nat_2} : ∀ x0 . SNo x0 ⟶ exp_SNo_nat x0 2 = mul_SNo x0 x0
Theorem e1625.. : ∀ x0 . c7ce4.. x0 ⟶ ∀ x1 . SNo x1 ⟶ mul_SNo (add_SNo (exp_SNo_nat (28f8d.. x0) 2) (exp_SNo_nat (d634d.. x0) 2)) x1 = 1 ⟶ 22598.. x0 (a0628.. (22598.. x1 (28f8d.. x0)) (b1848.. (22598.. 8d0f8.. (22598.. x1 (d634d.. x0))))) = 1
Known 6c4fc.. : 28f8d.. 0 = 0
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param SNoLt^SNoLt : ι → ι → ο
Known SNo_zero_or_sqr_pos'^{SNo_zero_or_sqr_pos} : ∀ x0 . SNo x0 ⟶ or (x0 = 0) (SNoLt 0 (exp_SNo_nat x0 2))
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Param SNoLe^SNoLe : ι → ι → ο
Known SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 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_sqr_nonneg'^{SNo_sqr_nonneg} : ∀ x0 . SNo x0 ⟶ SNoLe 0 (exp_SNo_nat x0 2)
Known 1c92a.. : d634d.. 0 = 0
Known add_SNo_Le1^add_SNo_Le1 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x2 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x2 x1)
Theorem 64e45.. : ∀ x0 . c7ce4.. x0 ⟶ add_SNo (exp_SNo_nat (28f8d.. x0) 2) (exp_SNo_nat (d634d.. x0) 2) = 0 ⟶ x0 = 0
Known recip_SNo_invR^{recip_SNo_invR} : ∀ x0 . SNo x0 ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ mul_SNo x0 (recip_SNo x0) = 1
Known SNo_recip_SNo^{SNo_recip_SNo} : ∀ x0 . SNo x0 ⟶ SNo (recip_SNo x0)
Theorem a0e5c.. : ∀ x0 . c7ce4.. x0 ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ 22598.. x0 (41fb9.. x0) = 1
Theorem 607f7.. : ∀ x0 . c7ce4.. x0 ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ 22598.. (41fb9.. x0) x0 = 1
Definition 74066.. := λ x0 x1 . 22598.. x0 (41fb9.. x1)
Theorem e6598.. : ∀ x0 x1 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ c7ce4.. (74066.. x0 x1)
Theorem bd3b2.. : ∀ x0 x1 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ 22598.. (74066.. x0 x1) x1 = x0
Theorem 662fb.. : ∀ x0 x1 . c7ce4.. x0 ⟶ c7ce4.. x1 ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ 22598.. x1 (74066.. x0 x1) = x0