theorem Elligator.Elligator1.X_pow_two_add_one_over_c_pow_two_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; have c_of_s := c s s_h1 s_h2 q field_cardinality q_prime_power q_mod_4_congruent_3; X_of_t ^ 2 + 1 / c_of_s ^ 2 ≠ 0

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 : { n : F // n ≠ 1 ∧ n ≠ -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 = 1 / X1

theorem Elligator.Elligator1.X_of_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) : have h1 := ⋯; have X_of_t := X ⟨0, h1⟩ s s_h1 s_h2 q field_cardinality q_prime_power q_mod_4_congruent_3; X_of_t = 1