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92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | import Mathlib
namespace OAI
noncomputable section
namespace Problem340
def orbitClockLift {X : Type*} (f : X → X) (h : X → ℝ)
(x : X) (t₀ : ℝ) (n : ℕ) : ℝ :=
t₀ + ∑ i ∈ Finset.range n, h (f^[i] x)
@[simp] theorem orbitClockLift_zero {X : Type*} (f : X → X) (h : X → ℝ)
(x : X) (t₀ : ℝ) : orbitClockLift f h x t₀ 0 = t₀ := by
simp [orbitClockLift]
theorem orbitClockLift_succ {X : Type*} (f : X → X) (h : X → ℝ)
(x : X) (t₀ : ℝ) (n : ℕ) :
orbitClockLift f h x t₀ (n + 1) =
orbitClockLift f h x t₀ n + h (f^[n] x) := by
simp only [orbitClockLift, Finset.sum_range_succ, add_assoc]
theorem coe_orbitClockLift {X : Type*} (P : ℝ)
(f : X → X) (h : X → ℝ) (clock : X → AddCircle P)
(hclock : ∀ x, clock (f x) = clock x + (h x : AddCircle P))
(x : X) (t₀ : ℝ) (h₀ : (t₀ : AddCircle P) = clock x) (n : ℕ) :
(orbitClockLift f h x t₀ n : AddCircle P) = clock (f^[n] x) := by
induction n with
| zero => simpa using h₀
| succ n hn =>
rw [orbitClockLift_succ, Function.iterate_succ_apply', hclock, ← hn]
simp
theorem monotone_orbitClockLift {X : Type*} (f : X → X) (h : X → ℝ)
(hh : ∀ x, 0 ≤ h x) (x : X) (t₀ : ℝ) :
Monotone (orbitClockLift f h x t₀) := by
apply monotone_nat_of_le_succ
intro n
rw [orbitClockLift_succ]
exact le_add_of_nonneg_right (hh _)
theorem orbitClockLift_step_bounds {X : Type*} (f : X → X) (h : X → ℝ)
(hh : ∀ x, h x ∈ Set.Icc (0 : ℝ) 1) (x : X) (t₀ : ℝ) (n : ℕ) :
orbitClockLift f h x t₀ (n + 1) - orbitClockLift f h x t₀ n ∈
Set.Icc (0 : ℝ) 1 := by
rw [orbitClockLift_succ, add_sub_cancel_left]
exact hh _
theorem orbitClockLift_add_initial {X : Type*} (f : X → X) (h : X → ℝ)
(x : X) (t₀ c : ℝ) (n : ℕ) :
orbitClockLift f h x (t₀ + c) n = orbitClockLift f h x t₀ n + c := by
simp only [orbitClockLift]
ring
theorem orbitClockLift_restart {X : Type*} (f : X → X) (h : X → ℝ)
(x : X) (t₀ : ℝ) (m n : ℕ) :
orbitClockLift f h (f^[m] x) (orbitClockLift f h x t₀ m) n =
orbitClockLift f h x t₀ (m + n) := by
induction n with
| zero => simp
| succ n hn =>
rw [orbitClockLift_succ, hn, Nat.add_succ, orbitClockLift_succ]
congr 1
rw [← Function.iterate_add_apply, Nat.add_comm n m]
theorem exists_monotone_orbit_clock_lift {X : Type*} (P : ℝ)
(f : X → X) (h : X → ℝ) (clock : X → AddCircle P)
(hclock : ∀ x, clock (f x) = clock x + (h x : AddCircle P))
(hh : ∀ x, h x ∈ Set.Icc (0 : ℝ) 1) (x : X) :
∃ t : ℕ → ℝ,
(∀ n, (t n : AddCircle P) = clock (f^[n] x)) ∧
(∀ n, t (n + 1) = t n + h (f^[n] x)) ∧
Monotone t ∧ (∀ n, t (n + 1) - t n ∈ Set.Icc (0 : ℝ) 1) := by
obtain ⟨t₀, ht₀⟩ := QuotientAddGroup.mk_surjective (clock x)
refine ⟨orbitClockLift f h x t₀, ?_, ?_, ?_, ?_⟩
· exact coe_orbitClockLift P f h clock hclock x t₀ ht₀
· exact orbitClockLift_succ f h x t₀
· exact monotone_orbitClockLift f h (fun y => (hh y).1) x t₀
· exact orbitClockLift_step_bounds f h hh x t₀
end Problem340
end
end OAI
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