openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Search /BudgetedJoinExecution.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Search.BudgetedStoppedLoop | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Search.BudgetedStreamPotential | |
| namespace OAI | |
| namespace LowSpaceSubsetSum.BudgetedJoinExecution | |
| universe u v s t w | |
| variable {α : Type u} {β : Type v} {σ₁ : Type s} {σ₂ : Type t} {τ : Type w} | |
| abbrev step (base front fill outer inner copy : ℕ) | |
| (read₁ : σ₁ → Option (α × σ₁) × ℕ) (key₁ : α → ℤ × ℕ) | |
| (read₂ : σ₂ → Option (β × σ₂) × ℕ) (key₂ : β → ℤ × ℕ) | |
| (cells : β → ℕ) (report : α → β → τ → (τ × Bool) × ℕ) | |
| (clone : σ₂ → σ₂ × ℕ) (B fuel₁ fuel₂ : ℕ) := | |
| BudgetedStreamStep.execute base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ | |
| abbrev run (base front fill outer inner copy : ℕ) | |
| (read₁ : σ₁ → Option (α × σ₁) × ℕ) (key₁ : α → ℤ × ℕ) | |
| (read₂ : σ₂ → Option (β × σ₂) × ℕ) (key₂ : β → ℤ × ℕ) | |
| (cells : β → ℕ) (report : α → β → τ → (τ × Bool) × ℕ) | |
| (clone : σ₂ → σ₂ × ℕ) (B fuel₁ fuel₂ : ℕ) := | |
| BudgetedStoppedLoop.run | |
| (step base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂) | |
| theorem reference_eq (read₁ : σ₁ → Option (α × σ₁) × ℕ) (key₁ : α → ℤ × ℕ) | |
| (read₂ : σ₂ → Option (β × σ₂) × ℕ) (key₂ : β → ℤ × ℕ) | |
| (cells : β → ℕ) (report : α → β → τ → (τ × Bool) × ℕ) | |
| (clone : σ₂ → σ₂ × ℕ) (B fuel₁ fuel₂ fuel : ℕ) (st : StreamJoin.State σ₁ σ₂ τ) : | |
| BudgetedStoppedJoin.reference | |
| (fun s => (CountedStreamJoin.step read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ s).1) fuel st = | |
| (CountedStreamJoin.runAcc read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ fuel st).1 := by | |
| induction fuel generalizing st with | |
| | zero => rfl | |
| | succ fuel ih => | |
| simp only [BudgetedStoppedJoin.reference, CountedStreamJoin.runAcc] | |
| split <;> simp only [ih] | |
| theorem run_value (base front fill outer inner copy : ℕ) | |
| (read₁ : σ₁ → Option (α × σ₁) × ℕ) (key₁ : α → ℤ × ℕ) | |
| (read₂ : σ₂ → Option (β × σ₂) × ℕ) (key₂ : β → ℤ × ℕ) | |
| (cells : β → ℕ) (report : α → β → τ → (τ × Bool) × ℕ) | |
| (clone : σ₂ → σ₂ × ℕ) (B fuel₁ fuel₂ fuel : ℕ) | |
| (st : StreamJoin.State σ₁ σ₂ τ) (budget : ℕ) (out : StreamJoin.StoppedResult σ₁ σ₂ τ) (left : ℕ) | |
| (h : (run base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone | |
| B fuel₁ fuel₂ fuel st budget).1 = some (out, left)) : | |
| out = (CountedStreamJoin.runAcc read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ fuel st).1 ∧ | |
| left ≤ budget := by | |
| simp only [run, BudgetedStoppedLoop.run_eq] at h | |
| have hv := BudgetedStoppedJoin.execute_value | |
| (step base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂) | |
| (fun s => (CountedStreamJoin.step read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ s).1) | |
| (BudgetedStreamStep.execute_value base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂) | |
| fuel st budget out left h | |
| rw [reference_eq] at hv | |
| exact hv | |
| theorem run_work | |
| (read₁ : σ₁ → Option (α × σ₁) × ℕ) (key₁ : α → ℤ × ℕ) | |
| (read₂ : σ₂ → Option (β × σ₂) × ℕ) (key₂ : β → ℤ × ℕ) | |
| (cells : β → ℕ) (report : α → β → τ → (τ × Bool) × ℕ) (clone : σ₂ → σ₂ × ℕ) | |
| (R₁ L₁ R₂ L₂ S C base front fill outer inner copy : ℕ) | |
| (hr₁ : ∀ s, (read₁ s).2 ≤ R₁) (hk₁ : ∀ x, (key₁ x).2 ≤ L₁) | |
| (hr₂ : ∀ s, (read₂ s).2 ≤ R₂) (hk₂ : ∀ y, (key₂ y).2 ≤ L₂) | |
| (hs : ∀ y, cells y ≤ S) (hc : ∀ x y a, (report x y a).2 ≤ C) | |
| (hbase : R₁ + L₁ + R₂ + L₂ + 16 ≤ base) | |
| (hfront : R₁ + L₁ + R₂ + L₂ + 16 ≤ front) | |
| (hfill : R₂ + L₂ + S + 8 ≤ fill) (houter : R₁ + L₁ + 8 ≤ outer) | |
| (hinner : max R₂ 4 + L₂ + C + 8 ≤ inner) (hcopy : ∀ s, (clone s).2 ≤ copy) | |
| (B fuel₁ fuel₂ fuel : ℕ) (st : StreamJoin.State σ₁ σ₂ τ) (budget : ℕ) : | |
| (run base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone | |
| B fuel₁ fuel₂ fuel st budget).2 ≤ budget + 1 := by | |
| simp only [run, BudgetedStoppedLoop.run_eq] | |
| apply BudgetedStoppedJoin.execute_work | |
| · exact BudgetedStreamStep.execute_work read₁ key₁ read₂ key₂ cells report clone | |
| R₁ L₁ R₂ L₂ S C base front fill outer inner copy hr₁ hk₁ hr₂ hk₂ hs hc hbase hfront hfill houter hinner hcopy | |
| B fuel₁ fuel₂ | |
| · intro s b out left h | |
| exact ⟨(BudgetedStreamStep.execute_value base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone | |
| B fuel₁ fuel₂ s b out left h).2, | |
| BudgetedStreamStep.execute_spent read₁ key₁ read₂ key₂ cells report clone | |
| R₁ L₁ R₂ L₂ S C base front fill outer inner copy hr₁ hk₁ hr₂ hk₂ hs hc hbase hfront hfill houter hinner hcopy | |
| B fuel₁ fuel₂ s b out left h⟩ | |
| theorem run_complete (base front fill outer inner copy : ℕ) | |
| (read₁ : σ₁ → Option (α × σ₁) × ℕ) (key₁ : α → ℤ × ℕ) | |
| (read₂ : σ₂ → Option (β × σ₂) × ℕ) (key₂ : β → ℤ × ℕ) | |
| (cells : β → ℕ) (report : α → β → τ → (τ × Bool) × ℕ) | |
| (clone : σ₂ → σ₂ × ℕ) (B fuel₁ fuel₂ fuel : ℕ) | |
| (st : StreamJoin.State σ₁ σ₂ τ) (budget : ℕ) | |
| (hb : let out := (CountedStreamJoin.runAcc read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ fuel st).1 | |
| (base + front + fill + outer + 2 * inner + 12) * out.state.queries + copy * out.state.copies + | |
| (base + 4) * out.steps + 4 ≤ budget) : | |
| ∃ left, (run base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone | |
| B fuel₁ fuel₂ fuel st budget).1 = | |
| some ((CountedStreamJoin.runAcc read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ fuel st).1, left) := by | |
| let pure := fun s => (CountedStreamJoin.step read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ s).1 | |
| let bill := BudgetedStreamStep.allowance base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ | |
| let M := base + front + fill + outer + 2 * inner + 12 | |
| have hp := BudgetedStoppedJoin.allowance_potential pure bill (CountedStreamJoin.potential M 0 copy) base | |
| (BudgetedStreamStep.allowance_potential base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂) | |
| fuel st | |
| rw [reference_eq] at hp | |
| simp only [CountedStreamJoin.potential, Nat.zero_mul, Nat.add_zero] at hp | |
| have ha : BudgetedStoppedJoin.allowance pure bill fuel st ≤ budget := by | |
| dsimp only [M] at hp | |
| omega | |
| have h := BudgetedStoppedJoin.execute_complete | |
| (step base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂) pure bill | |
| (BudgetedStreamStep.execute_complete base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂) | |
| (BudgetedStreamStep.execute_remaining base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂) | |
| fuel st budget ha | |
| simpa only [run, BudgetedStoppedLoop.run_eq, pure, reference_eq] using h | |
| /-- From fresh cursors, every control iteration performs at least one query. -/ | |
| theorem run_complete_queries (base front fill outer inner copy : ℕ) | |
| (read₁ : σ₁ → Option (α × σ₁) × ℕ) (key₁ : α → ℤ × ℕ) | |
| (read₂ : σ₂ → Option (β × σ₂) × ℕ) (key₂ : β → ℤ × ℕ) | |
| (cells : β → ℕ) (report : α → β → τ → (τ × Bool) × ℕ) | |
| (clone : σ₂ → σ₂ × ℕ) (hclone : ∀ s, (clone s).1 = s) | |
| (B fuel₁ fuel₂ fuel : ℕ) (s₁ : σ₁) (s₂ : σ₂) (a : τ) (budget : ℕ) | |
| (hb : let out := (CountedStreamJoin.runAcc read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ fuel | |
| (StreamJoin.start s₁ s₂ a)).1 | |
| (2 * base + front + fill + outer + 2 * inner + 16) * out.state.queries + | |
| copy * out.state.copies + 4 ≤ budget) : | |
| ∃ left, (run base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone | |
| B fuel₁ fuel₂ fuel (StreamJoin.start s₁ s₂ a) budget).1 = | |
| some ((CountedStreamJoin.runAcc read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ fuel | |
| (StreamJoin.start s₁ s₂ a)).1, left) := by | |
| have hv := CountedStreamJoin.runAcc_value read₁ key₁ read₂ key₂ cells report clone hclone | |
| B fuel₁ fuel₂ fuel (StreamJoin.start s₁ s₂ a) | |
| have hq := StreamJoin.stopped_steps_le_queries (fun s => (read₁ s).1) (fun x => (key₁ x).1) | |
| (fun s => (read₂ s).1) (fun x => (key₂ x).1) (fun x y a => (report x y a).1) | |
| B fuel₁ fuel₂ fuel s₁ s₂ a | |
| rw [← StreamJoin.stopped_eq] at hq | |
| change (StreamJoin.stoppedAcc _ _ _ _ _ _ _ _ _ _).steps ≤ | |
| (StreamJoin.stoppedAcc _ _ _ _ _ _ _ _ _ _).state.queries at hq | |
| rw [← hv] at hq | |
| have hm := Nat.mul_le_mul_left (base + 4) hq | |
| apply run_complete | |
| have he : 2 * base + front + fill + outer + 2 * inner + 16 = | |
| (base + front + fill + outer + 2 * inner + 12) + (base + 4) := by omega | |
| dsimp only at hb | |
| rw [he, Nat.add_mul] at hb | |
| omega | |
| end LowSpaceSubsetSum.BudgetedJoinExecution | |
| end OAI | |