import Erdos1135.Tao.Renewal.HoldPoint

/-!
# Basic Renewal Path Recursion

This low module owns the total finite renewal path and its list-locality
identities.  Source-hit, growth, triangle, and stopping consequences remain in
their higher proof modules.
-/

namespace Erdos1135
namespace Tao

/-- Point reached after consuming the first `n` renewal increments from `holds`. -/
def taoSection7RenewalPathPoint
    (p : TaoSection7RenewalPoint) : List TaoSection7RenewalPoint → ℕ →
      TaoSection7RenewalPoint
  | _holds, 0 => p
  | [], _n + 1 => p
  | h :: hs, n + 1 => taoSection7RenewalPathPoint (p + h) hs n

@[simp] theorem taoSection7RenewalPathPoint_zero
    (p : TaoSection7RenewalPoint) (holds : List TaoSection7RenewalPoint) :
    taoSection7RenewalPathPoint p holds 0 = p := by
  cases holds <;> rfl

@[simp] theorem taoSection7RenewalPathPoint_nil
    (p : TaoSection7RenewalPoint) (n : ℕ) :
    taoSection7RenewalPathPoint p [] n = p := by
  cases n <;> rfl

@[simp] theorem taoSection7RenewalPathPoint_cons_succ
    (p h : TaoSection7RenewalPoint) (hs : List TaoSection7RenewalPoint)
    (n : ℕ) :
    taoSection7RenewalPathPoint p (h :: hs) (n + 1) =
      taoSection7RenewalPathPoint (p + h) hs n := by
  rfl

/-- Advancing after dropping `q` increments agrees with advancing `q+n` at once. -/
theorem taoSection7RenewalPathPoint_drop_add
    (p : TaoSection7RenewalPoint) :
    ∀ (holds : List TaoSection7RenewalPoint) (q n : ℕ),
      taoSection7RenewalPathPoint
          (taoSection7RenewalPathPoint p holds q)
          (holds.drop q) n =
        taoSection7RenewalPathPoint p holds (q + n) := by
  intro holds q
  induction q generalizing p holds with
  | zero =>
      intro n
      simp
  | succ q ih =>
      intro n
      cases holds with
      | nil =>
          simp
      | cons h hs =>
          rw [show q + 1 + n = q + n + 1 by omega]
          simpa [taoSection7RenewalPathPoint] using
            ih (p + h) hs n

theorem taoSection7RenewalPathPoint_drop_candidate_sub
    {p : TaoSection7RenewalPoint}
    {holds : List TaoSection7RenewalPoint} {q candidate : ℕ}
    (hq : q ≤ candidate) :
    taoSection7RenewalPathPoint
        (taoSection7RenewalPathPoint p holds q)
        (holds.drop q) (candidate - q) =
      taoSection7RenewalPathPoint p holds candidate := by
  have h :=
    taoSection7RenewalPathPoint_drop_add p holds q (candidate - q)
  have hsum : q + (candidate - q) = candidate := Nat.add_sub_of_le hq
  simpa [hsum] using h

/-- Taking at least `n` increments does not change the path point at time `n`. -/
theorem taoSection7RenewalPathPoint_take_eq_of_le
    (start : TaoSection7RenewalPoint) :
    ∀ (holds : List TaoSection7RenewalPoint) (n m : ℕ),
      n ≤ m →
        taoSection7RenewalPathPoint start (holds.take m) n =
          taoSection7RenewalPathPoint start holds n
  | holds, 0, m, _hm => by simp
  | [], n + 1, m, _hm => by simp
  | h :: hs, n + 1, 0, hm => by omega
  | h :: hs, n + 1, m + 1, hm => by
      have hn : n ≤ m := Nat.succ_le_succ_iff.mp hm
      simpa [taoSection7RenewalPathPoint] using
        taoSection7RenewalPathPoint_take_eq_of_le (start + h) hs n m hn

end Tao
end Erdos1135