finish a => b except some other lemmas

FinalPolyType.lean

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+import Mathlib.Order.Height
+import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
+
+-- 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"
+        | aesop; done; dbg_trace "it was aesop"
+        -- | nlinarith; done
+        | fail "No, this is not obvious."))
+
+
+-- Testing of Polynomial
+section Polynomial
+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
+
+
+
+-- (NO need to prove another direction) Constant polynomial function = constant function
+lemma Poly_constant (F : Polynomial ℚ) (c : ℚ) : 
+  (F = Polynomial.C (c : ℚ)) ↔ (∀ r : ℚ, (Polynomial.eval r F) = (c : ℚ)) := 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
+
+-- Shifting doesn't change the polynomial type
+lemma Poly_shifting (f : ℤ → ℤ) (g : ℤ → ℤ) (hf : PolyType f d) (s : ℤ) (hfg : ∀ (n : ℤ), f (n + s) = g (n)) : PolyType g d := by
+  simp only [PolyType]
+  rcases hf with ⟨F, hh⟩
+  rcases hh with ⟨N,ss⟩
+  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
+  · rintro ⟨Poly, ⟨N, ⟨H1, H2⟩⟩⟩ 
+    have this1 : Polynomial.degree Poly = 0 := by rw [← H2]; rfl
+    have this2 : ∃ (c : ℤ), Poly = Polynomial.C (c : ℚ) := by
+      have HH : ∃ (c : ℚ), Poly = Polynomial.C (c : ℚ) := 
+        ⟨Poly.coeff 0, Polynomial.eq_C_of_degree_eq_zero (by rw[← H2]; rfl)⟩
+      cases' HH with c HHH
+      have HHHH : ∃ (d : ℤ), d = c := 
+        ⟨f N, by simp [(Poly_constant Poly c).mp HHH N, H1 N (le_refl N)]⟩
+      cases' HHHH with d H5; exact ⟨d, by rw[← H5] at HHH; exact HHH⟩
+    rcases this2 with ⟨c, hthis2⟩ 
+    use c; use N; intro n
+    constructor
+    · have this4 : Polynomial.eval (n : ℚ) Poly = c := by
+        rw [hthis2]; simp only [map_intCast, Polynomial.eval_int_cast]
+      exact fun HH1 => Iff.mp (Rat.coe_int_inj (f n) c) (by rw [←this4, H1 n HH1])
+    · intro c0
+      simp only [hthis2, c0, Int.cast_zero, map_zero, Polynomial.degree_zero] 
+        at this1
+  · rintro ⟨c, N, hh⟩
+    have H2 : (c : ℚ) ≠ 0 := by simp only [ne_eq, Int.cast_eq_zero]; exact (hh 0).2 
+    exact ⟨Polynomial.C (c : ℚ), N, fun n Nn 
+      => by rw [(hh n).1 Nn]; exact (((Poly_constant (Polynomial.C (c : ℚ)) 
+      (c : ℚ)).mp rfl) n).symm, by rw [Polynomial.degree_C H2]; rfl⟩
+
+-- Δ of 0 times preserves the function
+lemma Δ_0 (f : ℤ → ℤ) : (Δ f 0) = f := by tauto
+
+-- Δ of 1 times decreaes the polynomial type by one
+lemma Δ_1 (f : ℤ → ℤ) (d : ℕ) : PolyType f (d + 1) → PolyType (Δ f 1) d := by
+  sorry
+
+-- 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
+  sorry
+
+-- Δ of d times maps polynomial of degree d to polynomial of degree 0
+lemma Δ_1_s_equiv_Δ_s_1 (f : ℤ → ℤ) (s : ℕ) : Δ (Δ f 1) s = (Δ f (s + 1)) := by
+  sorry
+lemma foofoo (d : ℕ) : (f : ℤ → ℤ) → (PolyType f d) → (PolyType (Δ f d) 0):= by
+  induction' d with d hd
+  · intro f h
+    rw [Δ_0]
+    tauto
+  · intro f hf
+    have this1 : PolyType f (d + 1) := by tauto
+    have this2 : PolyType (Δ f (d + 1)) 0 := by
+      have this3 : PolyType (Δ f 1) d := by
+        have this5 : PolyType f (d + 1) → PolyType (Δ f 1) d := Δ_1 f d
+        exact this5 this1
+      clear hf
+      specialize hd (Δ f 1)
+      have this4 : PolyType (Δ (Δ f 1) d) 0 := by tauto
+      rw [Δ_1_s_equiv_Δ_s_1] at this4
+      tauto
+    tauto
+lemma Δ_d_PolyType_d_to_PolyType_0 (f : ℤ → ℤ) (d : ℕ): PolyType f d → PolyType (Δ f d) 0 := fun h => (foofoo d f) h
+
+
+-- [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 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
+  · intro f
+    intro h
+    rcases h with ⟨c, N, hh⟩
+    rw [PolyType_0]
+    use c
+    use N
+    tauto
+
+  -- Induction step
+  · intro f
+    intro h
+    rcases h with ⟨c, N, h⟩
+    have this : PolyType f (d + 1) := by
+      rcases h with ⟨H,c0⟩
+      let g := (Δ f 1)
+      -- let g := fun (x : ℤ) => (f (x + 1) - f (x))
+      have this1 : (∃ (c : ℤ), ∃ (N : ℤ), (∀ (n : ℤ), N ≤ n → (Δ g d) (n) = c) ∧ c ≠ 0) := by
+        use c; use N
+        constructor
+        · intro n
+          specialize H n
+          intro h
+          have this : Δ f (d + 1) n = c := by tauto
+          rw [←this]
+          rw [Δ_1_s_equiv_Δ_s_1] 
+        · tauto 
+      have this2 : PolyType g d := by
+        apply hd
+        tauto
+      exact Δ_1_ f d this2
+    tauto
+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 := fun h => (foo d f) h
+
+  
+
+
+
+-- [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_d_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
+-- Given a SES 0 → A → B → C → 0, then length (A) - length (B) + length (C) = 0 
+section
+open LinearMap
+
+-- 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
+
+-- 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 = ⊤
+
+-- 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