S_sq_add_S_sq — Mathlib

English

For every integer n, the equality (S_R(n)).comp(2 X) = U_R(n) holds.

Русский

Для каждого целого n выполняется (S_R(n)).comp(2 X) = U_R(n).

LaTeX

$$$ (S_R(n)).\circ(2 X) = U_R(n) $$$

Lean4

theorem S_sq_add_S_sq (n : ℤ) : S R n ^ 2 + S R (n + 1) ^ 2 - X * S R n * S R (n + 1) = 1 := by
  induction n with
  | zero => simp; ring
  | succ n ih =>
    have h₁ := S_add_two R n
    linear_combination (norm := ring_nf) (S R (2 + n) - S R n) * h₁ + ih
  | pred n ih =>
    have h₁ := S_sub_one R (-n)
    linear_combination (norm := ring_nf) (S R (-1 - n) - S R (1 - n)) * h₁ + ih

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