golfed delta_

CommAlg/final_poly_type.lean

@@ -53,8 +53,6 @@ noncomputable section
 def PolyType (f : ℤ → ℤ) (d : ℕ) := ∃ Poly : Polynomial ℚ, ∃ (N : ℤ), (∀ (n : ℤ), N ≤ n → f n = Polynomial.eval (n : ℚ) Poly) ∧ d = Polynomial.degree Poly
 section
 
-#check PolyType
-
 example (f : ℤ → ℤ) (hf : ∀ x, f x = x ^ 2) : PolyType f 2 := by
   unfold PolyType
   sorry
@@ -132,8 +130,8 @@ lemma PolyType_0 (f : ℤ → ℤ) : (PolyType f 0) ↔ (∃ (c : ℤ), ∃ (N :
 
 -- Δ of 0 times preserves the function
 lemma Δ_0 (f : ℤ → ℤ) : (Δ f 0) = f := by rfl
-  --simp only [Δ]
--- Δ of 1 times decreaes the polynomial type by one
+
+-- Δ of 1 times decreaes the polynomial type by one --can be golfed
 lemma Δ_1 (f : ℤ → ℤ) (d : ℕ) : PolyType f (d + 1) → PolyType (Δ f 1) d := by
   intro h
   simp only [PolyType, Δ, Int.cast_sub, exists_and_right]
@@ -186,20 +184,15 @@ lemma Δ_d_PolyType_d_to_PolyType_0 (f : ℤ → ℤ) (d : ℕ): PolyType f d
 
 -- The "reverse" of Δ of 1 times increases the polynomial type by one
 lemma Δ_1_ (f : ℤ → ℤ) (d : ℕ) : PolyType (Δ f 1) d → PolyType f (d + 1) := by
-  intro h
+  rintro ⟨P, N, ⟨h1, h2⟩⟩ 
   simp only [PolyType, Nat.cast_add, Nat.cast_one, exists_and_right]
-  rcases h with ⟨P, N, h⟩
-  rcases h with ⟨h1, h2⟩
   let G := fun (q : ℤ) => f (N)
   sorry
 
-
 lemma foo (d : ℕ) : (f : ℤ → ℤ) → (∃ (c : ℤ), ∃ (N : ℤ), (∀ (n : ℤ), N ≤ n → 
     (Δ f d) (n) = c) ∧ c ≠ 0) → (PolyType f d)  := by
   induction' d with d hd
-  -- Base case
   · rintro f ⟨c, N, hh⟩; rw [PolyType_0 f]; exact ⟨c, N, hh⟩
-  -- Induction step
   · exact fun f ⟨c, N, ⟨H, c0⟩⟩ =>
       Δ_1_ f d (hd (Δ f 1) ⟨c, N, fun n h => by rw [← H n h, Δ_1_s_equiv_Δ_s_1], c0⟩)