| import OAI.Geometry.PeriodicTiling.ConfigurationDifferences |
| import OAI.Geometry.PeriodicTiling.ForwardDifference |
| import Mathlib.Algebra.Order.Archimedean.Real.Basic |
| import Mathlib.Tactic.Linarith |
|
|
| namespace OAI |
|
|
| universe uG uA |
|
|
| namespace PeriodicTilingThree |
|
|
| variable {G : Type uG} [AddCommGroup G] |
|
|
| theorem configDiff_bounded (v : G) (f : G → ℝ) |
| (hf : ∃ B : ℝ, ∀ x, |f x| ≤ B) : |
| ∃ B : ℝ, ∀ x, |configDiff v f x| ≤ B := by |
| obtain ⟨B, hB⟩ := hf |
| refine ⟨B + B, fun x => ?_⟩ |
| calc |
| |configDiff v f x| = |f (x + v) + -f x| := rfl |
| _ ≤ |f (x + v)| + |-f x| := abs_add_le _ _ |
| _ = |f (x + v)| + |f x| := by rw [abs_neg] |
| _ ≤ B + B := add_le_add (hB _) (hB _) |
|
|
| theorem configDiff_eq_zero_of_second (v : G) (f : G → ℝ) |
| (hf : ∃ B : ℝ, ∀ x, |f x| ≤ B) |
| (hsecond : configDiff v (configDiff v f) = 0) : configDiff v f = 0 := by |
| obtain ⟨B, hB⟩ := hf |
| let g := configDiff v f |
| have hg : Function.Periodic g v := (configDiff_eq_zero_iff v g).mp hsecond |
| ext x |
| have hg_nat : ∀ n : ℕ, g (x + n • v) = g x := by |
| intro n |
| induction n with |
| | zero => simp |
| | succ n ih => |
| simpa only [succ_nsmul, ← add_assoc] using (hg (x + n • v)).trans ih |
| have hf_nat : ∀ n : ℕ, f (x + n • v) = f x + (n : ℝ) * g x := by |
| intro n |
| induction n with |
| | zero => simp |
| | succ n ih => |
| have hd := hg_nat n |
| change f (x + n • v + v) - f (x + n • v) = g x at hd |
| simp only [succ_nsmul, ← add_assoc, Nat.cast_add, Nat.cast_one] |
| linarith |
| have hx := abs_le.mp (hB x) |
| have hz : g x = 0 := by |
| rcases lt_trichotomy (g x) 0 with hneg | hzero | hpos |
| · obtain ⟨n, hn⟩ := exists_nat_gt (2 * B / (-g x)) |
| have hn' : 2 * B < (n : ℝ) * (-g x) := |
| (div_lt_iff₀ (neg_pos.mpr hneg)).mp hn |
| have hy := abs_le.mp (hB (x + n • v)) |
| have he := hf_nat n |
| nlinarith |
| · exact hzero |
| · obtain ⟨n, hn⟩ := exists_nat_gt (2 * B / g x) |
| have hn' : 2 * B < (n : ℝ) * g x := (div_lt_iff₀ hpos).mp hn |
| have hy := abs_le.mp (hB (x + n • v)) |
| have he := hf_nat n |
| nlinarith |
| exact hz |
|
|
| theorem configDiff_eq_zero_of_iterate (v : G) (n : ℕ) (f : G → ℝ) |
| (hf : ∃ B : ℝ, ∀ x, |f x| ≤ B) |
| (hn : 0 < n) (hiter : (configDiff v)^[n] f = 0) : |
| configDiff v f = 0 := by |
| induction n generalizing f with |
| | zero => exact (Nat.not_lt_zero _ hn).elim |
| | succ n ih => |
| cases n with |
| | zero => |
| change configDiff v f = 0 at hiter |
| exact hiter |
| | succ n => |
| have hnext : (configDiff v)^[n + 1] (configDiff v f) = 0 := by |
| simpa only [Function.iterate_succ_apply] using hiter |
| have hsecond : configDiff v (configDiff v f) = 0 := |
| ih (configDiff v f) (configDiff_bounded v f hf) (Nat.succ_pos _) hnext |
| exact configDiff_eq_zero_of_second v f hf hsecond |
|
|
| theorem periodic_of_bounded_configDiff_iterate (v : G) (n : ℕ) (f : G → ℝ) |
| (hf : ∃ B : ℝ, ∀ x, |f x| ≤ B) |
| (hn : 0 < n) (hiter : (configDiff v)^[n] f = 0) : |
| Function.Periodic f v := |
| (configDiff_eq_zero_iff v f).mp (configDiff_eq_zero_of_iterate v n f hf hn hiter) |
|
|
| theorem forwardDiff_eq_configDiff {A : Type uA} [AddCommGroup A] : |
| (forwardDiff : (ℤ → A) → ℤ → A) = configDiff (1 : ℤ) := rfl |
|
|
| theorem forwardDiff_eq_zero_of_bounded_iterate (n : ℕ) (f : ℤ → ℝ) |
| (hf : ∃ B : ℝ, ∀ x, |f x| ≤ B) |
| (hn : 0 < n) (hiter : (forwardDiff^[n]) f = 0) : forwardDiff f = 0 := by |
| rw [forwardDiff_eq_configDiff] at hiter ⊢ |
| exact configDiff_eq_zero_of_iterate 1 n f hf hn hiter |
|
|
| end PeriodicTilingThree |
|
|
| end OAI |
|
|
