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add something in Polynomial_shifting
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@ -80,7 +80,33 @@ lemma Poly_constant (F : Polynomial ℚ) (c : ℚ) :
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-- Get the polynomial G (X) = F (X + s) from the polynomial F(X)
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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
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sorry
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let Shift := (Polynomial.monomial 1 (1 : ℚ)) + (Polynomial.C s)
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let G := Polynomial.comp Shift F
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use G
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constructor
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· intro t
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have this : Polynomial.eval t G = Polynomial.eval (Polynomial.eval t Shift) F:= by
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sorry
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have this1 : Polynomial.eval t Shift = t + s := by
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dsimp; simp
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rw [this1] at this
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exact this
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· have this : Polynomial.degree G = (Polynomial.degree F) * (Polynomial.degree Shift) := by
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sorry
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have Shift_degree : Polynomial.degree Shift = 1 := by
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have this2 : Polynomial.degree (Polynomial.monomial 1 (1 : ℚ)) = 1 := by
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have this3 : (1 : ℚ) ≠ 0 := by norm_num
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exact Polynomial.degree_monomial 1 this3
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have this1 : Polynomial.degree Shift = (Polynomial.degree (Polynomial.monomial 1 (1 : ℚ))) + (Polynomial.degree (Polynomial.C s)) := by
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sorry
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have this3 : Polynomial.degree (Polynomial.C s) = 0 := by
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sorry
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rw [this1, this2, this3]
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trivial
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rw [Shift_degree] at this
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rw [this]
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norm_num
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-- Shifting doesn't change the polynomial type
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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
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