Reaches

formal source

  21lemma reach_le_in_ball {K : Kinematics α} {x y : α} {k n : Nat}
  22  (hk : k ≤ n) (h : ReachN K k x y) : inBall K x n y := ⟨k, hk, h⟩
  23
  24def Reaches (K : Kinematics α) (x y : α) : Prop := ∃ n, ReachN K n x y
  25
  26lemma reaches_of_reachN {K : Kinematics α} {x y : α} {n : Nat}
  27  (h : ReachN K n x y) : Reaches K x y := ⟨n, h⟩
  28
  29lemma inBall_mono {K : Kinematics α} {x y : α} {n m : Nat}
  30  (hnm : n ≤ m) : inBall K x n y → inBall K x m y := by
  31  intro ⟨k, hk, hkreach⟩
  32  exact ⟨k, le_trans hk hnm, hkreach⟩
  33
  34def ballP (K : Kinematics α) (x : α) : Nat → α → Prop
  35| 0, y => y = x
  36| Nat.succ n, y => ballP K x n y ∨ ∃ z, ballP K x n z ∧ K.step z y
  37
  38lemma ballP_mono {K : Kinematics α} {x : α} {n m : Nat}
  39  (hnm : n ≤ m) : {y | ballP K x n y} ⊆ {y | ballP K x m y} := by
  40  induction hnm with
  41  | refl => intro y hy; simpa using hy
  42  | @step m hm ih =>
  43      intro y hy
  44      exact Or.inl (ih hy)
  45
  46lemma reach_mem_ballP {K : Kinematics α} {x y : α} :
  47  ∀ {n}, ReachN K n x y → ballP K x n y := by
  48  intro n h; induction h with
  49  | zero => simp [ballP]
  50  | @succ n x y z hxy hyz ih =>
  51      exact Or.inr ⟨y, ih, hyz⟩
  52
  53lemma inBall_subset_ballP {K : Kinematics α} {x y : α} {n : Nat} :
  54  inBall K x n y → ballP K x n y := by