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fixed merge conflics

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Andre 2023-06-16 00:44:03 -04:00
parent f225a9e262
commit 0089e927e1
1 changed files with 0 additions and 27 deletions
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CommAlg/final_poly_type.lean
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@ -139,11 +139,6 @@ lemma Poly_shifting (f : ℤ → ℤ) (g : ℤ → ℤ) (hf : PolyType f d) (s :
rcases hh with ⟨N,ss⟩ rcases hh with ⟨N,ss⟩
sorry sorry
-- set_option pp.all true in
-- PolyType 0 = constant function -- PolyType 0 = constant function
lemma PolyType_0 (f : ℤ → ℤ) : (PolyType f 0) ↔ (∃ (c : ℤ), ∃ (N : ℤ), (∀ (n : ℤ), lemma PolyType_0 (f : ℤ → ℤ) : (PolyType f 0) ↔ (∃ (c : ℤ), ∃ (N : ℤ), (∀ (n : ℤ),
(N ≤ n → f n = c)) ∧ c ≠ 0) := by (N ≤ n → f n = c)) ∧ c ≠ 0) := by
@ -179,11 +174,6 @@ lemma Δ_0 (f : ℤ → ℤ) : (Δ f 0) = f := by tauto
lemma Δ_1 (f : ℤ → ℤ) (d : ℕ): d > 0 → PolyType f d → PolyType (Δ f 1) (d - 1) := by lemma Δ_1 (f : ℤ → ℤ) (d : ℕ): d > 0 → PolyType f d → PolyType (Δ f 1) (d - 1) := by
sorry sorry
-- Δ of d times maps polynomial of degree d to polynomial of degree 0 -- 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 lemma Δ_1_s_equiv_Δ_s_1 (f : ℤ → ℤ) (s : ℕ) : Δ (Δ f 1) s = (Δ f (s + 1)) := by
sorry sorry
@ -208,23 +198,6 @@ lemma foofoo (d : ℕ) : (f : ℤ → ℤ) → (PolyType f d) → (PolyType (Δ
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 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 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
lemma Δ_d_PolyType_d_to_PolyType_0 (f : ℤ → ℤ) (d : ℕ): PolyType f d → PolyType (Δ f d) 0 := by
intro h
have this : ∀ (d : ℕ), ∀ (f :ℤ → ℤ), (PolyType f d) → (PolyType (Δ f d) 0) := by
exact foofoo
specialize this d f
tauto
lemma foofoofoo (d : ℕ) : (f : ℤ → ℤ) → (∃ (c : ℤ), ∃ (N : ℤ), (∀ (n : ℤ), N ≤ n → (Δ f d) (n) = c) ∧ c ≠ 0) → (PolyType f d) := by lemma foofoofoo (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 induction' d with d hd