theorem Elligator.Elligator1.x_ne_zero {F : Type u_1} [Field F] [Fintype F] (s : F) (s_h1 : s ≠ 0) (s_h2 : (s ^ 2 - 2) * (s ^ 2 + 2) ≠ 0) (q : ℕ) (field_cardinality : Fintype.card F = q) (q_prime_power : IsPrimePow q) (q_mod_4_congruent_3 : q % 4 = 3) (t : { n : F // n ≠ 1 ∧ n ≠ -1 }) : have x_of_t := x t s s_h1 s_h2 q field_cardinality q_prime_power q_mod_4_congruent_3; x_of_t ≠ 0

theorem Elligator.Elligator1.x_comparison {F : Type u_1} [Field F] [Fintype F] (t : { t : F // t ≠ 1 ∧ t ≠ -1 }) (s : F) (s_h1 : s ≠ 0) (s_h2 : (s ^ 2 - 2) * (s ^ 2 + 2) ≠ 0) (q : ℕ) (field_cardinality : Fintype.card F = q) (q_prime_power : IsPrimePow q) (q_mod_4_congruent_3 : q % 4 = 3) : let t1 := ↑t; let t2 := -t1; have h2_2 := ⋯; have x1 := x t s s_h1 s_h2 q field_cardinality q_prime_power q_mod_4_congruent_3; have x2 := x ⟨t2, h2_2⟩ s s_h1 s_h2 q field_cardinality q_prime_power q_mod_4_congruent_3; x2 = x1

theorem Elligator.Elligator1.x_y_eq_zero_sign_one {F : Type u_1} [Field F] [Fintype F] (s : F) (s_h1 : s ≠ 0) (s_h2 : (s ^ 2 - 2) * (s ^ 2 + 2) ≠ 0) (q : ℕ) (field_cardinality : Fintype.card F = q) (q_prime_power : IsPrimePow q) (q_mod_4_congruent_3 : q % 4 = 3) (point : { p : F × F // p ∈ E_over_F s s_h1 s_h2 q field_cardinality q_prime_power q_mod_4_congruent_3 }) (x_eq_zero : (↑point).1 = 0) : ↑point = (0, 1) ∨ ↑point = (0, -1)