| import OAI.Combinatorics.LowSpaceSubsetSum.Counting.CountedReplay |
| import OAI.Combinatorics.LowSpaceSubsetSum.Streams.JoinOne |
|
|
| namespace OAI |
|
|
| /-! # Counted buffered and suffix replay for one matching occurrence |
|
|
| A suffix restore invokes the supplied concrete cloning procedure. Its cost |
| is charged only on the long-group branch. Query and report counters continue |
| to denote the exact stopped execution. |
| -/ |
|
|
| namespace LowSpaceSubsetSum.CountedJoinOne |
|
|
| universe u v w |
| variable {α : Type u} {σ : Type v} {τ : Type w} |
|
|
| def run (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) |
| (report : α → τ → (τ × Bool) × ℕ) (clone : σ → σ × ℕ) |
| (buffer : Array α) (saved : σ) (long : Bool) (fuel : ℕ) (a : τ) : |
| TieReplay.Result σ τ × ℕ := |
| let first := CountedReplay.run (fun i => (BufferReplay.read buffer i, 4)) key K report buffer.size 0 a |
| if first.1.halted then |
| (⟨saved, first.1.state, first.1.queries, first.1.reports, false, true⟩, first.2 + 4) |
| else if long then |
| let copied := clone saved |
| let rest := CountedReplay.run read key K report fuel copied.1 first.1.state |
| (⟨rest.1.cursor, rest.1.state, first.1.queries + rest.1.queries, |
| first.1.reports + rest.1.reports, rest.1.complete, rest.1.halted⟩, |
| first.2 + copied.2 + rest.2 + 5) |
| else (⟨saved, first.1.state, first.1.queries, first.1.reports, true, false⟩, first.2 + 4) |
|
|
| theorem run_value (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) |
| (report : α → τ → (τ × Bool) × ℕ) (clone : σ → σ × ℕ) |
| (hclone : ∀ s, (clone s).1 = s) |
| (buffer : Array α) (saved : σ) (long : Bool) (fuel : ℕ) (a : τ) : |
| (run read key K report clone buffer saved long fuel a).1 = |
| JoinOne.run (fun s => (read s).1) (fun x => (key x).1) K |
| (fun x a => (report x a).1) buffer saved long fuel a := by |
| simp only [run, CountedReplay.run_value, hclone, JoinOne.run, BufferReplay.run] |
| split |
| · simp_all only [↓reduceIte] |
| · simp_all only |
| split <;> rfl |
|
|
| theorem run_work (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) |
| (report : α → τ → (τ × Bool) × ℕ) (clone : σ → σ × ℕ) |
| (R L C B : ℕ) (hr : ∀ s, (read s).2 ≤ R) (hk : ∀ x, (key x).2 ≤ L) |
| (hc : ∀ x a, (report x a).2 ≤ C) (hcopy : ∀ s, (clone s).2 ≤ B) |
| (buffer : Array α) (saved : σ) (long : Bool) (fuel : ℕ) (a : τ) : |
| (run read key K report clone buffer saved long fuel a).2 ≤ |
| (max R 4 + L + 8) * (run read key K report clone buffer saved long fuel a).1.queries + |
| C * (run read key K report clone buffer saved long fuel a).1.reports + |
| (if long then B else 0) + 7 := by |
| let first := CountedReplay.run (fun i => (BufferReplay.read buffer i, 4)) key K report buffer.size 0 a |
| have hf0 := CountedReplay.run_work (fun i => (BufferReplay.read buffer i, 4)) key K report |
| 4 L C (fun _ => le_rfl) hk hc buffer.size 0 a |
| have hf : first.2 ≤ (max R 4 + L + 8) * first.1.queries + C * first.1.reports + 1 := by |
| change first.2 ≤ (4 + L + 8) * first.1.queries + C * first.1.reports + 1 at hf0 |
| have hmul := Nat.mul_le_mul_right first.1.queries |
| (show 4 + L + 8 ≤ max R 4 + L + 8 by omega) |
| exact hf0.trans (by omega) |
| have hfirst : CountedReplay.run (fun i => (BufferReplay.read buffer i, 4)) key K report |
| buffer.size 0 a = first := rfl |
| simp only [run, hfirst] |
| by_cases hh : first.1.halted = true |
| · simp only [hh, ↓reduceIte] |
| change first.2 + 4 ≤ (max R 4 + L + 8) * first.1.queries + C * first.1.reports + _ + 7 |
| omega |
| · have hfalse : first.1.halted = false := Bool.eq_false_iff.mpr hh |
| by_cases hl : long = true |
| · let rest := CountedReplay.run read key K report fuel (clone saved).1 first.1.state |
| have hr0 := CountedReplay.run_work read key K report R L C hr hk hc fuel (clone saved).1 first.1.state |
| have hrest : rest.2 ≤ (max R 4 + L + 8) * rest.1.queries + C * rest.1.reports + 1 := by |
| change rest.2 ≤ (R + L + 8) * rest.1.queries + C * rest.1.reports + 1 at hr0 |
| have hmul := Nat.mul_le_mul_right rest.1.queries |
| (show R + L + 8 ≤ max R 4 + L + 8 by omega) |
| exact hr0.trans (by omega) |
| have hcp := hcopy saved |
| simp only [hfalse, Bool.false_eq_true, hl, ↓reduceIte] |
| change first.2 + (clone saved).2 + rest.2 + 5 ≤ |
| (max R 4 + L + 8) * (first.1.queries + rest.1.queries) + |
| C * (first.1.reports + rest.1.reports) + B + 7 |
| simp only [Nat.mul_add] |
| omega |
| · have hlfalse : long = false := Bool.eq_false_iff.mpr hl |
| simp only [hfalse, hlfalse, Bool.false_eq_true, ↓reduceIte] |
| change first.2 + 4 ≤ (max R 4 + L + 8) * first.1.queries + C * first.1.reports + 0 + 7 |
| omega |
|
|
| end LowSpaceSubsetSum.CountedJoinOne |
|
|
| end OAI |
|
|
