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add something in Polynomial_shifting

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chelseaandmadrid 2023-06-16 14:34:28 -07:00
parent cb4e0ead26
commit 3588faa23c
1 changed files with 27 additions and 1 deletions
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28
CommAlg/final_poly_type.lean
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@ -80,7 +80,33 @@ lemma Poly_constant (F : Polynomial ℚ) (c : ℚ) :
-- Get the polynomial G (X) = F (X + s) from the polynomial F(X)
lemma Polynomial_shifting (F : Polynomial ℚ) (s : ℚ) : ∃ (G : Polynomial ℚ), (∀ (x : ℚ), Polynomial.eval x G = Polynomial.eval (x + s) F) ∧ (Polynomial.degree G = Polynomial.degree F) := by
sorry
let Shift := (Polynomial.monomial 1 (1 : ℚ)) + (Polynomial.C s)
let G := Polynomial.comp Shift F
use G
constructor
· intro t
have this : Polynomial.eval t G = Polynomial.eval (Polynomial.eval t Shift) F:= by
sorry
have this1 : Polynomial.eval t Shift = t + s := by
dsimp; simp
rw [this1] at this
exact this
· have this : Polynomial.degree G = (Polynomial.degree F) * (Polynomial.degree Shift) := by
sorry
have Shift_degree : Polynomial.degree Shift = 1 := by
have this2 : Polynomial.degree (Polynomial.monomial 1 (1 : ℚ)) = 1 := by
have this3 : (1 : ℚ) ≠ 0 := by norm_num
exact Polynomial.degree_monomial 1 this3
have this1 : Polynomial.degree Shift = (Polynomial.degree (Polynomial.monomial 1 (1 : ℚ))) + (Polynomial.degree (Polynomial.C s)) := by
sorry
have this3 : Polynomial.degree (Polynomial.C s) = 0 := by
sorry
rw [this1, this2, this3]
trivial
rw [Shift_degree] at this
rw [this]
norm_num
-- Shifting doesn't change the polynomial type
lemma Poly_shifting (f : ℤ → ℤ) (g : ℤ → ℤ) (hf : PolyType f d) (s : ℕ) (hfg : ∀ (n : ℤ), f (n + s) = g (n)) : PolyType g d := by