| import OAI.Combinatorics.LowSpaceSubsetSum.Streams.BufferReplay |
|
|
| namespace OAI |
|
|
| /-! |
| # Reporting one item's matches against a buffered tie group |
|
|
| The immutable prefix buffer is reused. A generator suffix is restored only |
| when the group was longer than the buffer. The callback may stop in either |
| part; its work and stream queries are bounded on every outcome. |
| -/ |
|
|
| namespace LowSpaceSubsetSum |
| namespace JoinOne |
|
|
| universe u v w |
|
|
| variable {α : Type u} {σ : Type v} {τ : Type w} |
|
|
| def run (read : σ → Option (α × σ)) (key : α → ℤ) (K : ℤ) |
| (report : α → τ → τ × Bool) (buffer : Array α) (saved : σ) |
| (long : Bool) (fuel : ℕ) (a : τ) : TieReplay.Result σ τ := |
| let first := BufferReplay.run key K report buffer a |
| if first.halted then |
| ⟨saved, first.state, first.queries, first.reports, false, true⟩ |
| else if long then |
| let rest := TieReplay.run read key K report fuel saved first.state |
| ⟨rest.cursor, rest.state, first.queries + rest.queries, |
| first.reports + rest.reports, rest.complete, rest.halted⟩ |
| else ⟨saved, first.state, first.queries, first.reports, true, false⟩ |
|
|
| theorem reports_le (read : σ → Option (α × σ)) (key : α → ℤ) (K : ℤ) |
| (report : α → τ → τ × Bool) (view : σ → List α) (hread : RowMerge.IsReader read view) |
| (buffer : Array α) (saved : σ) (long : Bool) (fuel : ℕ) (a : τ) : |
| (run read key K report buffer saved long fuel a).reports ≤ |
| buffer.size + (TieBuffer.groupView key K view saved).length := by |
| have hb := BufferReplay.reports_le key K report buffer a |
| dsimp only [run] |
| split |
| · exact hb.trans (Nat.le_add_right _ _) |
| · split |
| · have hr := TieReplay.run_reports_le_potential read key K report view hread |
| fuel saved (BufferReplay.run key K report buffer a).state |
| exact Nat.add_le_add hb hr |
| · exact hb.trans (Nat.le_add_right _ _) |
|
|
| theorem queries_le (read : σ → Option (α × σ)) (key : α → ℤ) (K : ℤ) |
| (report : α → τ → τ × Bool) (view : σ → List α) (hread : RowMerge.IsReader read view) |
| (buffer : Array α) (saved : σ) (long : Bool) (fuel : ℕ) (a : τ) : |
| (run read key K report buffer saved long fuel a).queries ≤ |
| buffer.size + (TieBuffer.groupView key K view saved).length + 1 := by |
| have hb := BufferReplay.queries_le key K report buffer a |
| dsimp only [run] |
| split |
| · dsimp only |
| omega |
| · split |
| · have hr := TieReplay.run_queries_le_potential read key K report view hread |
| fuel saved (BufferReplay.run key K report buffer a).state |
| dsimp only |
| omega |
| · dsimp only |
| omega |
|
|
| theorem halted_false (read : σ → Option (α × σ)) (key : α → ℤ) (K : ℤ) |
| (report : α → τ → τ × Bool) (hreport : ∀ x a, (report x a).2 = false) |
| (buffer : Array α) (saved : σ) (long : Bool) (fuel : ℕ) (a : τ) : |
| (run read key K report buffer saved long fuel a).halted = false := by |
| have hb : (BufferReplay.run key K report buffer a).halted = false := |
| TieReplay.run_halted_false _ _ _ _ hreport _ _ _ |
| simp only [run, hb, Bool.false_eq_true, ite_false] |
| split |
| · exact TieReplay.run_halted_false _ _ _ _ hreport _ _ _ |
| · rfl |
|
|
| /-- Completed reporting is precisely prefix reporting followed by suffix |
| reporting; a short group has no suffix to restore. -/ |
| theorem state_complete (read : σ → Option (α × σ)) (key : α → ℤ) (K : ℤ) |
| (report : α → τ → τ × Bool) (hreport : ∀ x a, (report x a).2 = false) |
| (view : σ → List α) (hread : RowMerge.IsReader read view) |
| (buffer : Array α) (hkeys : ∀ x ∈ buffer.toList, key x = K) |
| (saved : σ) (fuel : ℕ) (a : τ) |
| (hfuel : (TieBuffer.groupView key K view saved).length ≤ fuel) : |
| (run read key K report buffer saved (TieBuffer.hasKey read key K saved) fuel a).state = |
| (buffer.toList ++ TieBuffer.groupView key K view saved).foldl |
| (fun a x => (report x a).1) a := by |
| have hb : (BufferReplay.run key K report buffer a).halted = false := |
| TieReplay.run_halted_false _ _ _ _ hreport _ _ _ |
| have hs := BufferReplay.state_complete key K report hreport buffer a hkeys |
| simp only [run, hb, Bool.false_eq_true, ite_false] |
| by_cases hl : TieBuffer.hasKey read key K saved = true |
| · simp only [hl, ite_true] |
| rw [TieReplay.run_state_complete read key K report hreport view hread fuel saved _ hfuel, |
| hs, List.foldl_append] |
| · have he : TieBuffer.groupView key K view saved = [] := by |
| by_contra he |
| exact hl ((TieBuffer.hasKey_iff read key K view hread saved).mpr he) |
| have hfalse : TieBuffer.hasKey read key K saved = false := Bool.eq_false_iff.mpr hl |
| simp only [hfalse, Bool.false_eq_true, ite_false, he, List.append_nil] |
| exact hs |
|
|
| theorem suffix_empty (read : σ → Option (α × σ)) (key : α → ℤ) (K : ℤ) |
| (report : α → τ → τ × Bool) (hreport : ∀ x a, (report x a).2 = false) |
| (view : σ → List α) (hread : RowMerge.IsReader read view) |
| (buffer : Array α) (saved : σ) (fuel : ℕ) (a : τ) |
| (hfuel : (TieBuffer.groupView key K view saved).length ≤ fuel) : |
| TieBuffer.groupView key K view |
| (run read key K report buffer saved (TieBuffer.hasKey read key K saved) fuel a).cursor = [] := by |
| have hb : (BufferReplay.run key K report buffer a).halted = false := |
| TieReplay.run_halted_false _ _ _ _ hreport _ _ _ |
| simp only [run, hb, Bool.false_eq_true, ite_false] |
| by_cases hl : TieBuffer.hasKey read key K saved = true |
| · simp only [hl, ite_true] |
| have h := (TieReplay.runAcc_nonhalting_spec read key K report hreport view hread |
| fuel saved (BufferReplay.run key K report buffer a).state 0 0).2.2.1 |
| exact h.trans (List.drop_eq_nil_of_le hfuel) |
| · simp only [Bool.eq_false_iff.mpr hl, Bool.false_eq_true, ite_false] |
| by_contra he |
| exact hl ((TieBuffer.hasKey_iff read key K view hread saved).mpr he) |
|
|
| theorem view_complete (read : σ → Option (α × σ)) (key : α → ℤ) (K : ℤ) |
| (report : α → τ → τ × Bool) (hreport : ∀ x a, (report x a).2 = false) |
| (view : σ → List α) (hread : RowMerge.IsReader read view) |
| (buffer : Array α) (saved : σ) (fuel : ℕ) (a : τ) |
| (hfuel : (TieBuffer.groupView key K view saved).length ≤ fuel) : |
| view saved = TieBuffer.groupView key K view saved ++ |
| view (run read key K report buffer saved (TieBuffer.hasKey read key K saved) fuel a).cursor := by |
| have hb : (BufferReplay.run key K report buffer a).halted = false := |
| TieReplay.run_halted_false _ _ _ _ hreport _ _ _ |
| simp only [run, hb, Bool.false_eq_true, ite_false] |
| by_cases hl : TieBuffer.hasKey read key K saved = true |
| · simp only [hl, ite_true] |
| have h := (TieReplay.runAcc_nonhalting_spec read key K report hreport view hread |
| fuel saved (BufferReplay.run key K report buffer a).state 0 0).2.2.2 |
| simpa only [TieReplay.run, List.take_of_length_le hfuel] using h |
| · have he : TieBuffer.groupView key K view saved = [] := by |
| by_contra he |
| exact hl ((TieBuffer.hasKey_iff read key K view hread saved).mpr he) |
| simp only [Bool.eq_false_iff.mpr hl, Bool.false_eq_true, ite_false, he, List.nil_append] |
|
|
| def runFilled (read : σ → Option (α × σ)) (key : α → ℤ) (K : ℤ) |
| (report : α → τ → τ × Bool) (B fuel : ℕ) (s : σ) (a : τ) : TieReplay.Result σ τ := |
| let filled := TieBuffer.fill read key K B s |
| run read key K report filled.buffer filled.cursor |
| (TieBuffer.hasKey read key K filled.cursor) fuel a |
|
|
| theorem filled_reports_le (read : σ → Option (α × σ)) (key : α → ℤ) (K : ℤ) |
| (report : α → τ → τ × Bool) (view : σ → List α) (hread : RowMerge.IsReader read view) |
| (B fuel : ℕ) (s : σ) (a : τ) : |
| (runFilled read key K report B fuel s a).reports ≤ |
| (TieBuffer.groupView key K view s).length := by |
| have h := reports_le read key K report view hread (TieBuffer.fill read key K B s).buffer |
| (TieBuffer.fill read key K B s).cursor |
| (TieBuffer.hasKey read key K (TieBuffer.fill read key K B s).cursor) fuel a |
| have hb := congrArg List.length (TieBuffer.fill_buffer read key K view hread B s) |
| rw [TieBuffer.fill_suffix read key K view hread B s] at h |
| simp only [Array.length_toList, List.length_take] at hb |
| simp only [List.length_drop] at h |
| dsimp only [runFilled] |
| omega |
|
|
| theorem filled_queries_le (read : σ → Option (α × σ)) (key : α → ℤ) (K : ℤ) |
| (report : α → τ → τ × Bool) (view : σ → List α) (hread : RowMerge.IsReader read view) |
| (B fuel : ℕ) (s : σ) (a : τ) : |
| (runFilled read key K report B fuel s a).queries ≤ |
| (TieBuffer.groupView key K view s).length + 1 := by |
| have h := queries_le read key K report view hread (TieBuffer.fill read key K B s).buffer |
| (TieBuffer.fill read key K B s).cursor |
| (TieBuffer.hasKey read key K (TieBuffer.fill read key K B s).cursor) fuel a |
| have hb := congrArg List.length (TieBuffer.fill_buffer read key K view hread B s) |
| rw [TieBuffer.fill_suffix read key K view hread B s] at h |
| simp only [Array.length_toList, List.length_take] at hb |
| simp only [List.length_drop] at h |
| dsimp only [runFilled] |
| omega |
|
|
| theorem filled_state_complete (read : σ → Option (α × σ)) (key : α → ℤ) (K : ℤ) |
| (report : α → τ → τ × Bool) (hreport : ∀ x a, (report x a).2 = false) |
| (view : σ → List α) (hread : RowMerge.IsReader read view) |
| (B fuel : ℕ) (s : σ) (a : τ) |
| (hfuel : (TieBuffer.groupView key K view s).length ≤ fuel) : |
| (runFilled read key K report B fuel s a).state = |
| (TieBuffer.groupView key K view s).foldl (fun a x => (report x a).1) a := by |
| have hf : (TieBuffer.groupView key K view (TieBuffer.fill read key K B s).cursor).length ≤ |
| fuel := by |
| rw [TieBuffer.fill_suffix read key K view hread B s, List.length_drop] |
| omega |
| have h := state_complete read key K report hreport view hread |
| (TieBuffer.fill read key K B s).buffer |
| (BufferReplay.filled_buffer_keys read key K view hread B s) |
| (TieBuffer.fill read key K B s).cursor fuel a hf |
| rw [TieBuffer.fill_buffer read key K view hread B s, |
| TieBuffer.fill_suffix read key K view hread B s, List.take_append_drop] at h |
| exact h |
|
|
| theorem filled_view_complete (read : σ → Option (α × σ)) (key : α → ℤ) (K : ℤ) |
| (report : α → τ → τ × Bool) (hreport : ∀ x a, (report x a).2 = false) |
| (view : σ → List α) (hread : RowMerge.IsReader read view) |
| (B fuel : ℕ) (s : σ) (a : τ) |
| (hfuel : (TieBuffer.groupView key K view s).length ≤ fuel) : |
| view s = TieBuffer.groupView key K view s ++ |
| view (runFilled read key K report B fuel s a).cursor := by |
| have hf : (TieBuffer.groupView key K view (TieBuffer.fill read key K B s).cursor).length ≤ |
| fuel := by |
| rw [TieBuffer.fill_suffix read key K view hread B s, List.length_drop] |
| omega |
| have hs := view_complete read key K report hreport view hread |
| (TieBuffer.fill read key K B s).buffer (TieBuffer.fill read key K B s).cursor fuel a hf |
| have h := (TieBuffer.fillAcc_spec read key K view hread B s #[] 0).2.1 |
| change view s = (TieBuffer.groupView key K view s).take B ++ |
| view (TieBuffer.fill read key K B s).cursor at h |
| rw [hs, TieBuffer.fill_suffix read key K view hread B s, ← List.append_assoc, |
| List.take_append_drop] at h |
| exact h |
|
|
| end JoinOne |
| end LowSpaceSubsetSum |
|
|
| end OAI |
|
|
