openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Search /BudgetedStreamAllowance.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Search.BudgetedStreamStep | |
| namespace OAI | |
| namespace LowSpaceSubsetSum.BudgetedStreamStep | |
| universe u v s t w | |
| variable {α : Type u} {β : Type v} {σ₁ : Type s} {σ₂ : Type t} {τ : Type w} | |
| /-- Proof-only allowance for one comparison or one complete tie group. -/ | |
| def allowance (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) (st : StreamJoin.State σ₁ σ₂ τ) : ℕ := | |
| if st.done then base else | |
| match (read₁ st.left).1 with | |
| | none => base | |
| | some (x, _) => match (read₂ st.right).1 with | |
| | none => base | |
| | some (y, _) => | |
| if (key₁ x).1 < (key₂ y).1 then base | |
| else if (key₂ y).1 < (key₁ x).1 then base | |
| else base + BudgetedTieGroup.allowance front fill outer inner copy read₁ key₁ read₂ key₂ | |
| cells (key₁ x).1 report clone B fuel₁ fuel₂ st.left st.right st.user | |
| theorem execute_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 : ℕ) | |
| (st : StreamJoin.State σ₁ σ₂ τ) (budget : ℕ) | |
| (hb : allowance base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone | |
| B fuel₁ fuel₂ st ≤ budget) : | |
| ∃ left, (execute base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone | |
| B fuel₁ fuel₂ st budget).1 = | |
| some ((CountedStreamJoin.step read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ st).1, left) := by | |
| by_cases hd : st.done = true | |
| · simp only [allowance, hd, ↓reduceIte] at hb | |
| exact ⟨budget - base, by simp only [execute, hb, hd, ↓reduceIte, CountedStreamJoin.step]⟩ | |
| · simp only [allowance, hd, Bool.false_eq_true, ↓reduceIte] at hb | |
| cases hx : (read₁ st.left).1 with | |
| | none => | |
| simp only [hx] at hb | |
| exact ⟨budget - base, by simp only [execute, hb, hd, Bool.false_eq_true, ↓reduceIte, hx, | |
| CountedStreamJoin.step]⟩ | |
| | some p => | |
| rcases p with ⟨x, x'⟩ | |
| simp only [hx] at hb | |
| cases hy : (read₂ st.right).1 with | |
| | none => | |
| simp only [hy] at hb | |
| exact ⟨budget - base, by simp only [execute, hb, hd, Bool.false_eq_true, ↓reduceIte, hx, hy, | |
| CountedStreamJoin.step]⟩ | |
| | some p => | |
| rcases p with ⟨y, y'⟩ | |
| simp only [hy] at hb | |
| by_cases hk : (key₁ x).1 < (key₂ y).1 | |
| · simp only [hk, ↓reduceIte] at hb | |
| exact ⟨budget - base, by simp only [execute, hb, hd, Bool.false_eq_true, ↓reduceIte, hx, hy, hk, | |
| CountedStreamJoin.step]⟩ | |
| · simp only [hk, ↓reduceIte] at hb | |
| by_cases hk' : (key₂ y).1 < (key₁ x).1 | |
| · simp only [hk', ↓reduceIte] at hb | |
| exact ⟨budget - base, by simp only [execute, hb, hd, Bool.false_eq_true, ↓reduceIte, hx, hy, hk, hk', | |
| CountedStreamJoin.step]⟩ | |
| · simp only [hk', ↓reduceIte] at hb | |
| have hbase : base ≤ budget := by omega | |
| obtain ⟨left, hg⟩ := BudgetedTieGroup.execute_complete front fill outer inner copy | |
| read₁ key₁ read₂ key₂ cells (key₁ x).1 report clone B fuel₁ fuel₂ st.left st.right st.user | |
| (budget - base) (by omega) | |
| refine ⟨left, ?_⟩ | |
| simp only [execute, hbase, hd, Bool.false_eq_true, ↓reduceIte, hx, hy, hk, hk', hg, | |
| Option.map_some, CountedStreamJoin.step] | |
| rfl | |
| end LowSpaceSubsetSum.BudgetedStreamStep | |
| end OAI | |