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06/16 version
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chelseaandmadrid
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b3332d77d7
1 changed files with 81 additions and 36 deletions
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@ -44,24 +44,27 @@ example (f : Polynomial ℚ) (hf : f = Polynomial.C (1 : ℚ)) : Polynomial.eval
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-- degree of a constant function is ⊥ (is this same as -1 ???)
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#print Polynomial.degree_zero
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def F : Polynomial ℚ := Polynomial.C (2 : ℚ)
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#print F
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#check F
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#check Polynomial.degree F
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def FF : Polynomial ℚ := Polynomial.C (2 : ℚ)
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#print FF
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#check FF
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#check Polynomial.degree FF
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#check Polynomial.degree 0
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#check WithBot ℕ
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-- #eval Polynomial.degree F
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#check Polynomial.eval 1 F
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example : Polynomial.eval (100 : ℚ) F = (2 : ℚ) := by
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refine Iff.mpr (Rat.ext_iff (Polynomial.eval 100 F) 2) ?_
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-- #eval Polynomial.degree FF
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#check Polynomial.eval 1 FF
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example : Polynomial.eval (100 : ℚ) FF = (2 : ℚ) := by
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refine Iff.mpr (Rat.ext_iff (Polynomial.eval 100 FF) 2) ?_
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simp only [Rat.ofNat_num, Rat.ofNat_den]
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rw [F]
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rw [FF]
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simp
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sorry
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-- Treat polynomial f ∈ ℚ[X] as a function f : ℚ → ℚ
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#check CoeFun
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#check Polynomial.eval₂
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#check Polynomial.comp
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#check Polynomial.eval₂.comp
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#check Polynomial.card_roots
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end section
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@ -98,7 +101,7 @@ end section
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@[simp]
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def Δ : (ℤ → ℤ) → ℕ → (ℤ → ℤ)
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| f, 0 => f
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| f, d + 1 => fun (n : ℤ) ↦ (Δ f d) (n + 1) - (Δ f d) (n)
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| f, d + 1 => fun (n : ℤ) ↦ (Δ f d) (n + 1) - (Δ f d) (n)
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section
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-- def Δ (f : ℤ → ℤ) (d : ℕ) := fun (n : ℤ) ↦ f (n + 1) - f n
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-- def add' : ℕ → ℕ → ℕ
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@ -106,8 +109,8 @@ section
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-- | n+1, m => (add' n m) + 1
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-- #eval add' 5 10
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#check Δ
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def f (n : ℤ) := n
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#eval (Δ f 1) 100
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def fff (n : ℤ) := n
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#eval (Δ fff 1) 100
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-- #check (by (show_term unfold Δ) : Δ f 0=0)
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end section
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@ -125,12 +128,30 @@ lemma Poly_constant (F : Polynomial ℚ) (c : ℚ) :
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simp
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· sorry
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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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-- 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 : ℤ) (hfg : ∀ (n : ℤ), f (n + s) = g (n)) : PolyType g d := by
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simp only [PolyType]
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rcases hf with ⟨F, hh⟩
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rcases hh with ⟨N,ss⟩
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sorry
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rcases hh with ⟨N,s1, s2⟩
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have this : ∃ (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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exact Polynomial_shifting F s
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rcases this with ⟨Poly, h1, h2⟩
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use Poly
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use N
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constructor
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· intro n
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specialize s1 (n + s)
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intro hN
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have this1 : f (n + s) = Polynomial.eval (n + s : ℚ) F := by
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sorry
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sorry
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· rw [h2, s2]
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-- PolyType 0 = constant function
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lemma PolyType_0 (f : ℤ → ℤ) : (PolyType f 0) ↔ (∃ (c : ℤ), ∃ (N : ℤ), (∀ (n : ℤ),
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@ -142,22 +163,22 @@ lemma PolyType_0 (f : ℤ → ℤ) : (PolyType f 0) ↔ (∃ (c : ℤ), ∃ (N :
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have HH : ∃ (c : ℚ), Poly = Polynomial.C (c : ℚ) :=
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⟨Poly.coeff 0, Polynomial.eq_C_of_degree_eq_zero (by rw[← H2]; rfl)⟩
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cases' HH with c HHH
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have HHHH : ∃ (d : ℤ), d = c :=
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have HHHH : ∃ (d : ℤ), d = c :=
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⟨f N, by simp [(Poly_constant Poly c).mp HHH N, H1 N (le_refl N)]⟩
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cases' HHHH with d H5; exact ⟨d, by rw[← H5] at HHH; exact HHH⟩
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rcases this2 with ⟨c, hthis2⟩
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use c; use N; intro n
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constructor
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· have this4 : Polynomial.eval (n : ℚ) Poly = c := by
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use c; use N; constructor
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· intro n
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have this4 : Polynomial.eval (n : ℚ) Poly = c := by
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rw [hthis2]; simp only [map_intCast, Polynomial.eval_int_cast]
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exact fun HH1 => Iff.mp (Rat.coe_int_inj (f n) c) (by rw [←this4, H1 n HH1])
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· intro c0
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simp only [hthis2, c0, Int.cast_zero, map_zero, Polynomial.degree_zero]
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at this1
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· rintro ⟨c, N, hh⟩
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have H2 : (c : ℚ) ≠ 0 := by simp only [ne_eq, Int.cast_eq_zero]; exact (hh 0).2
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have H2 : (c : ℚ) ≠ 0 := by simp only [ne_eq, Int.cast_eq_zero, hh]
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exact ⟨Polynomial.C (c : ℚ), N, fun n Nn
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=> by rw [(hh n).1 Nn]; exact (((Poly_constant (Polynomial.C (c : ℚ))
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=> by rw [hh.1 n Nn]; exact (((Poly_constant (Polynomial.C (c : ℚ))
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(c : ℚ)).mp rfl) n).symm, by rw [Polynomial.degree_C H2]; rfl⟩
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-- Δ of 0 times preserves the function
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@ -167,13 +188,46 @@ lemma Δ_0 (f : ℤ → ℤ) : (Δ f 0) = f := by tauto
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lemma Δ_1 (f : ℤ → ℤ) (d : ℕ) : PolyType f (d + 1) → PolyType (Δ f 1) d := by
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intro h
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simp only [PolyType, Δ, Int.cast_sub, exists_and_right]
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rcases h with ⟨Poly, N, h⟩
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sorry
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rcases h with ⟨F, N, h⟩
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rcases h with ⟨h1, h2⟩
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have this : ∃ (G : Polynomial ℚ), (∀ (x : ℚ), Polynomial.eval x G = Polynomial.eval (x + 1) F) ∧ (Polynomial.degree G = Polynomial.degree F) := by
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exact Polynomial_shifting F 1
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rcases this with ⟨G, hG, hGG⟩
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let Poly := G - F
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use Poly
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constructor
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· use N
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intro n hn
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specialize hG n
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norm_num
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rw [hG]
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let h3 := h1
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specialize h3 n
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have this1 : f n = Polynomial.eval (n : ℚ) F := by tauto
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have this2 : f (n + 1) = Polynomial.eval ((n + 1) : ℚ) F := by
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specialize h1 (n + 1)
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have this3 : N ≤ n + 1 := by linarith
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aesop
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rw [←this1, ←this2]
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· have this1 : Polynomial.degree Poly = d := by
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have this2 : Polynomial.degree Poly ≤ d := by
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sorry
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have this3 : Polynomial.degree Poly ≥ d := by
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sorry
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sorry
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tauto
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-- The "reverse" of Δ of 1 times increases the polynomial type by one
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lemma Δ_1_ (f : ℤ → ℤ) (d : ℕ) : PolyType (Δ f 1) d → PolyType f (d + 1) := by
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intro h
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simp only [PolyType, Nat.cast_add, Nat.cast_one, exists_and_right]
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rcases h with ⟨P, N, h⟩
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rcases h with ⟨h1, h2⟩
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let G := fun (q : ℤ) => f (N)
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sorry
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-- Δ of d times maps polynomial of degree d to polynomial of degree 0
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lemma Δ_1_s_equiv_Δ_s_1 (f : ℤ → ℤ) (s : ℕ) : Δ (Δ f 1) s = (Δ f (s + 1)) := by
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induction' s with s hs
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@ -183,19 +237,10 @@ lemma foofoo (d : ℕ) : (f : ℤ → ℤ) → (PolyType f d) → (PolyType (Δ
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induction' d with d hd
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· intro f h
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rw [Δ_0]
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tauto
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exact h
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· intro f hf
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have this1 : PolyType f (d + 1) := by tauto
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have this2 : PolyType (Δ f (d + 1)) 0 := by
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have this3 : PolyType (Δ f 1) d := by
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have this5 : PolyType f (d + 1) → PolyType (Δ f 1) d := Δ_1 f d
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exact this5 this1
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clear hf
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specialize hd (Δ f 1)
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have this4 : PolyType (Δ (Δ f 1) d) 0 := by tauto
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rw [Δ_1_s_equiv_Δ_s_1] at this4
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tauto
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tauto
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have this4 := hd (Δ f 1) $ (Δ_1 f d) hf
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rwa [Δ_1_s_equiv_Δ_s_1] at this4
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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
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