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version of 06/15

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chelseaandmadrid 2023-06-15 23:29:40 -07:00
parent 25b85edbea
commit ed83762764
1 changed files with 7 additions and 3 deletions
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10
FinalPolyType.lean
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@ -20,6 +20,7 @@ macro "obviously" : tactic =>
| ring; done; dbg_trace "it was ring" | ring; done; dbg_trace "it was ring"
| trivial; done; dbg_trace "it was trivial" | trivial; done; dbg_trace "it was trivial"
| aesop; done; dbg_trace "it was aesop" | aesop; done; dbg_trace "it was aesop"
| assumption; done; dbg_trace "it was assumption"
-- | nlinarith; done -- | nlinarith; done
| fail "No, this is not obvious.")) | fail "No, this is not obvious."))
@ -164,6 +165,9 @@ lemma Δ_0 (f : ℤ → ℤ) : (Δ f 0) = f := by tauto
-- Δ of 1 times decreaes the polynomial type by one -- 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 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]
rcases h with ⟨Poly, N, h⟩
sorry sorry
-- The "reverse" of Δ of 1 times increases the polynomial type by one -- The "reverse" of Δ of 1 times increases the polynomial type by one
@ -172,7 +176,9 @@ lemma Δ_1_ (f : ℤ → ℤ) (d : ℕ) : PolyType (Δ f 1) d → PolyType f (d
-- 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 induction' s with s hs
· norm_num
· aesop
lemma foofoo (d : ℕ) : (f : ℤ → ℤ) → (PolyType f d) → (PolyType (Δ f d) 0):= by lemma foofoo (d : ℕ) : (f : ℤ → ℤ) → (PolyType f d) → (PolyType (Δ f d) 0):= by
induction' d with d hd induction' d with d hd
· intro f h · intro f h
@ -234,8 +240,6 @@ lemma a_to_b (f : ℤ → ℤ) (d : ℕ) : (∃ (c : ℤ), ∃ (N : ℤ), (∀ (
-- [BH, 4.1.2] (a) <= (b) -- [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 -- 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 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