diff --git a/PolyType.lean b/PolyType.lean
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+import Mathlib
+import Mathlib.Algebra.MonoidAlgebra.Basic
+import Mathlib.Data.Finset.Sort
+import Mathlib.Order.Height
+import Mathlib.Order.KrullDimension
+import Mathlib.Order.JordanHolder
+import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
+import Mathlib.Order.Height
+import Mathlib.RingTheory.Ideal.Basic
+import Mathlib.RingTheory.Ideal.Operations
+import Mathlib.LinearAlgebra.Finsupp
+import Mathlib.RingTheory.GradedAlgebra.Basic
+import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
+import Mathlib.Algebra.Module.GradedModule
+import Mathlib.RingTheory.Ideal.AssociatedPrime
+import Mathlib.RingTheory.Noetherian
+import Mathlib.RingTheory.Artinian
+import Mathlib.Algebra.Module.GradedModule
+import Mathlib.RingTheory.Noetherian
+import Mathlib.RingTheory.Finiteness
+import Mathlib.RingTheory.Ideal.Operations
+
+-- Setting for "library_search"
+set_option maxHeartbeats 0
+macro "ls" : tactic => `(tactic|library_search)
+
+-- New tactic "obviously"
+macro "obviously" : tactic =>
+  `(tactic| (
+      first
+        | dsimp; simp; done; dbg_trace "it was dsimp simp"
+        | simp; done; dbg_trace "it was simp"
+        | tauto; done; dbg_trace "it was tauto"
+        | simp; tauto; done; dbg_trace "it was simp tauto"
+        | rfl; done; dbg_trace "it was rfl"
+        | norm_num; done; dbg_trace "it was norm_num"
+        | /-change (@Eq ℝ _ _);-/ linarith; done; dbg_trace "it was linarith"
+        -- | gcongr; done
+        | ring; done; dbg_trace "it was ring"
+        | trivial; done; dbg_trace "it was trivial"
+        -- | nlinarith; done
+        | fail "No, this is not obvious."))
+
+
+-- Testing of Polynomial
+section Polynomial
+variable [Semiring ℕ]
+variable [Semiring ℤ]
+variable [Semiring ℚ]
+noncomputable section
+#check Polynomial 
+#check Polynomial (ℚ)
+#check Polynomial.eval
+
+
+example (f : Polynomial ℚ) (hf : f = Polynomial.C (1 : ℚ)) : Polynomial.eval 2 f = 1 := by
+  have : ∀ (q : ℚ), Polynomial.eval q f = 1 := by
+    sorry
+  obviously
+
+-- example (f : ℤ → ℤ) (hf : ∀ x, f x = x ^ 2) : Polynomial.eval 2 f = 4 := by
+--   sorry
+
+-- degree of a constant function is ⊥ (is this same as -1 ???)
+#print Polynomial.degree_zero
+
+def F : Polynomial ℚ := Polynomial.C (2 : ℚ)
+#print F
+#check F
+#check Polynomial.degree F
+#check Polynomial.degree 0
+#check WithBot ℕ
+-- #eval Polynomial.degree F
+#check Polynomial.eval 1 F
+example : Polynomial.eval (100 : ℚ) F = (2 : ℚ) := by
+  refine Iff.mpr (Rat.ext_iff (Polynomial.eval 100 F) 2) ?_
+  simp only [Rat.ofNat_num, Rat.ofNat_den]
+  rw [F]
+  simp
+
+-- Treat polynomial f ∈ ℚ[X] as a function f : ℚ → ℚ
+#check CoeFun
+
+
+
+
+end section
+
+-- @[BH, 4.1.2]
+-- All the polynomials are in ℚ[X], all the functions are considered as ℤ → ℤ
+noncomputable section
+-- Polynomial type of degree d
+@[simp]
+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
+-- structure PolyType (f : ℤ → ℤ) where
+--   Poly : Polynomial ℤ
+--   d : 
+--   N : ℤ
+--   Poly_equal : ∀ n ∈ ℤ → f n = Polynomial.eval n : ℤ Poly
+
+#check PolyType
+
+example (f : ℤ → ℤ) (hf : ∀ x, f x = x ^ 2) : PolyType f 2 := by
+  unfold PolyType
+  sorry
+  -- use Polynomial.monomial (2 : ℤ) (1 : ℤ)
+  -- have' := hf 0; ring_nf at this
+  -- exact this
+
+end section
+
+-- Δ operator (of d times)
+@[simp]
+def Δ : (ℤ → ℤ) → ℕ → (ℤ → ℤ)
+  | f, 0 => f
+  | f, d + 1 => fun (n : ℤ) ↦ (Δ f d) (n + 1) - (Δ f d) (n)  
+section
+-- def Δ (f : ℤ → ℤ) (d : ℕ) := fun (n : ℤ) ↦ f (n + 1) - f n
+-- def add' : ℕ → ℕ → ℕ
+--   | 0, m => m 
+--   | n+1, m => (add' n m) + 1
+-- #eval add' 5 10
+#check Δ
+def f (n : ℤ) := n
+#eval (Δ f 1) 100
+-- #check (by (show_term unfold Δ) : Δ f 0=0)
+end section
+
+
+-- Constant polynomial function = constant function
+lemma Poly_constant (F : Polynomial ℚ) (c : ℚ) : 
+  (F = Polynomial.C c) ↔ (∀ r : ℚ, (Polynomial.eval r F) = c) := by
+  constructor
+  · intro h
+    rintro r
+    refine Iff.mpr (Rat.ext_iff (Polynomial.eval r F) c) ?_
+    simp only [Rat.ofNat_num, Rat.ofNat_den]
+    rw [h]
+    simp
+  · sorry
+
+-- PolyType 0 = constant function
+lemma PolyType_0 (f : ℤ → ℤ) : (PolyType f 0) ↔ (∃ (c : ℤ), ∃ (N : ℤ), ∀ (n : ℤ), (N ≤ n → f n = c) ∧ c ≠ 0) := by
+  constructor
+  · intro h
+    rcases h with ⟨Poly, hN⟩
+    rcases hN with ⟨N, hh⟩
+    have H1 := λ n hn => (hh n hn).left
+    have H2 := λ n hn => (hh n hn).right
+    clear hh
+    specialize H2 (N + 1)
+    have this1 : Polynomial.degree Poly = 0 := by
+      have : N ≤ N + 1 := by
+        dsimp
+        simp
+      tauto
+    have this2 : ∃ (c : ℤ), Poly = Polynomial.C (c : ℚ) := by
+      have HH : ∃ (c : ℚ), Poly = Polynomial.C (c : ℚ) := by
+        use Poly.coeff 0
+        apply Polynomial.eq_C_of_degree_eq_zero
+        exact this1
+      cases' HH with c HHH
+      have HHHH : ∃ (d : ℤ), d = c := by
+        sorry
+      cases' HHHH with d H5
+      use d
+      rw [H5]
+      exact HHH
+    clear this1
+    rcases this2 with ⟨c, hthis2⟩
+    use c
+    use N
+    intro n
+    specialize H1 n
+    constructor
+    · intro HH1
+      have this3 : f n = Polynomial.eval (n : ℚ) Poly := by
+        tauto
+      have this4 : Polynomial.eval (n : ℚ) Poly = c := by
+        rw [hthis2]
+        dsimp
+        simp
+      have this5 : f n = (c : ℚ) := by
+        rw [←this4, this3]
+      exact Iff.mp (Rat.coe_int_inj (f n) c) this5
+    · sorry
+
+
+  · intro h
+    rcases h with ⟨c, N, aaa⟩
+    let (Poly : Polynomial ℚ) := Polynomial.C (c : ℚ)
+    use Poly
+    use N
+    intro n Nn
+    specialize aaa n
+    have this1 : c ≠ 0 →  f n = c := by
+      tauto
+    constructor
+    · sorry
+    · sorry
+      -- apply Polynomial.degree_C c
+
+-- Δ of d times maps polynomial of degree d to polynomial of degree 0
+lemma Δ_PolyType_d_to_PolyType_0 (f : ℤ → ℤ) (d : ℕ): PolyType f d → PolyType (Δ f d) 0 := by
+  intro h
+  rcases h with ⟨Poly, hN⟩
+  rcases hN with ⟨N, hh⟩
+  have H1 := λ n hn => (hh n hn).left
+  have H2 := λ n hn => (hh n hn).right
+  clear hh
+  have HH2 : d = Polynomial.degree Poly := by
+    sorry
+  induction' d with d hd
+  · rw [PolyType_0]
+    sorry
+  · sorry
+
+-- [BH, 4.1.2] (a) => (b)
+-- Δ^d f (n) = c for some nonzero integer c for n >> 0 → f is of polynomial type d
+lemma a_to_b (f : ℤ → ℤ) (d : ℕ) : (∃ (c : ℤ), ∃ (N : ℤ), ∀ (n : ℤ), ((N ≤ n → (Δ f d) (n) = c) ∧ c ≠ 0)) → PolyType f d := by
+  intro h
+  rcases h with ⟨c, N, hh⟩
+  have H1 := λ n => (hh n).left
+  have H2 := λ n => (hh n).right
+  clear hh
+  have H2 : c ≠ 0 := by
+    tauto
+  induction' d with d hd
+  · rw [PolyType_0]
+    use c
+    use N
+    tauto
+  · sorry
+
+-- [BH, 4.1.2] (a) <= (b)
+-- f is of polynomial type d → Δ^d f (n) = c for some nonzero integer c for n >> 0
+lemma b_to_a (f : ℤ → ℤ) (d : ℕ) : PolyType f d → (∃ (c : ℤ), ∃ (N : ℤ), ∀ (n : ℤ), ((N ≤ n → (Δ f d) (n) = c) ∧ c ≠ 0)) := by
+  intro h
+  have : PolyType (Δ f d) 0 := by
+    apply Δ_PolyType_d_to_PolyType_0
+    exact h
+  have this1 : (∃ (c : ℤ), ∃ (N : ℤ), ∀ (n : ℤ), ((N ≤ n → (Δ f d) n = c) ∧ c ≠ 0)) := by
+    rw [←PolyType_0]
+    exact this
+  exact this1
+end
+
+-- @Additive lemma of length for a SES
+section
+-- variable {R M N : Type _} [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N]
+--   (f : M →[R] N)
+open LinearMap
+-- variable {R M : Type _} [CommRing R] [AddCommGroup M] [Module R M]
+-- noncomputable def length := Set.chainHeight {M' : Submodule R M | M' < ⊤}
+
+
+-- Definitiion of the length of a module
+noncomputable def length (R M : Type _) [CommRing R] [AddCommGroup M] [Module R M] := Set.chainHeight {M' : Submodule R M | M' < ⊤}
+#check length ℤ ℤ
+-- #eval length ℤ ℤ
+
+
+-- @[ext]
+-- structure SES (R : Type _) [CommRing R] where
+--   A : Type _
+--   B : Type _
+--   C : Type _
+--   f : A →ₗ[R] B
+--   g : B →ₗ[R] C
+-- left_exact : LinearMap.ker f = 0
+-- middle_exact : LinearMap.range f = LinearMap.ker g
+-- right_exact : LinearMap.range g = C
+
+
+
+-- Definition of a SES (Short Exact Sequence)
+-- @[ext]
+structure SES {R A B C : Type _} [CommRing R] [AddCommGroup A] [AddCommGroup B]
+  [AddCommGroup C] [Module R A] [Module R B] [Module R C] 
+  (f : A →ₗ[R] B) (g : B →ₗ[R] C)
+  where
+  left_exact : LinearMap.ker f = ⊥
+  middle_exact : LinearMap.range f = LinearMap.ker g
+  right_exact : LinearMap.range g = ⊤
+
+#check SES.right_exact
+#check SES
+
+
+-- Additive lemma
+lemma length_Additive (R A B C : Type _) [CommRing R] [AddCommGroup A] [AddCommGroup B] [AddCommGroup C] [Module R A] [Module R B] [Module R C] 
+  (f : A →ₗ[R] B) (g : B →ₗ[R] C)
+  : (SES f g) → ((length R A) + (length R C) = (length R B)) := by
+  intro h
+  rcases h with ⟨left_exact, middle_exact, right_exact⟩
+  sorry
+
+end section
+
+
+
+
+
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