openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Search /BudgetedStreamContracts.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Search.BudgetedStreamAllowance | |
| 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} | |
| theorem execute_remaining (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) (out : StreamJoin.State σ₁ σ₂ τ) (left : ℕ) | |
| (h : (execute base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone | |
| B fuel₁ fuel₂ st budget).1 = some (out, left)) : | |
| budget ≤ left + allowance base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone | |
| B fuel₁ fuel₂ st := by | |
| rw [execute] at h | |
| split at h | |
| next hb => | |
| by_cases hd : st.done = true | |
| · simp only [hd, ↓reduceIte] at h | |
| cases h | |
| simp only [allowance, hd, ↓reduceIte] | |
| omega | |
| · simp only [hd, Bool.false_eq_true, ↓reduceIte] at h | |
| cases hx : (read₁ st.left).1 with | |
| | none => | |
| simp only [hx] at h | |
| cases h | |
| simp only [allowance, hd, Bool.false_eq_true, ↓reduceIte, hx] | |
| omega | |
| | some p => | |
| rcases p with ⟨x, x'⟩ | |
| simp only [hx] at h | |
| cases hy : (read₂ st.right).1 with | |
| | none => | |
| simp only [hy] at h | |
| cases h | |
| simp only [allowance, hd, Bool.false_eq_true, ↓reduceIte, hx, hy] | |
| omega | |
| | some p => | |
| rcases p with ⟨y, y'⟩ | |
| simp only [hy] at h | |
| by_cases hk : (key₁ x).1 < (key₂ y).1 | |
| · simp only [hk, ↓reduceIte] at h | |
| cases h | |
| simp only [allowance, hd, Bool.false_eq_true, ↓reduceIte, hx, hy, hk] | |
| omega | |
| · simp only [hk, ↓reduceIte] at h | |
| by_cases hk' : (key₂ y).1 < (key₁ x).1 | |
| · simp only [hk', ↓reduceIte] at h | |
| cases h | |
| simp only [allowance, hd, Bool.false_eq_true, ↓reduceIte, hx, hy, hk, hk'] | |
| omega | |
| · simp only [hk', ↓reduceIte] at h | |
| obtain ⟨pair, hg, he⟩ := Option.map_eq_some_iff.mp h | |
| rcases pair with ⟨group, remaining⟩ | |
| have hr := BudgetedTieGroup.execute_remaining 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) group remaining hg | |
| cases he | |
| simp only [allowance, hd, Bool.false_eq_true, ↓reduceIte, hx, hy, hk, hk'] | |
| omega | |
| next _ => contradiction | |
| theorem execute_spent | |
| (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 : ℕ) (st : StreamJoin.State σ₁ σ₂ τ) (budget : ℕ) | |
| (out : StreamJoin.State σ₁ σ₂ τ) (left : ℕ) | |
| (h : (execute base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone | |
| B fuel₁ fuel₂ st budget).1 = some (out, left)) : | |
| (execute base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone | |
| B fuel₁ fuel₂ st budget).2 ≤ budget - left := by | |
| rw [execute] at h ⊢ | |
| split at h | |
| next hb => | |
| simp only [hb, ↓reduceIte] at h ⊢ | |
| by_cases hd : st.done = true | |
| · simp only [hd, ↓reduceIte] at h ⊢ | |
| cases h | |
| omega | |
| · simp only [hd, Bool.false_eq_true, ↓reduceIte] at h ⊢ | |
| have hxw := hr₁ st.left | |
| cases hx : (read₁ st.left).1 with | |
| | none => | |
| simp only [hx] at h ⊢ | |
| cases h | |
| omega | |
| | some p => | |
| rcases p with ⟨x, x'⟩ | |
| simp only [hx] at h ⊢ | |
| have hyw := hr₂ st.right | |
| cases hy : (read₂ st.right).1 with | |
| | none => | |
| simp only [hy] at h ⊢ | |
| cases h | |
| omega | |
| | some p => | |
| rcases p with ⟨y, y'⟩ | |
| have hxk := hk₁ x | |
| have hyk := hk₂ y | |
| simp only [hy] at h ⊢ | |
| by_cases hk : (key₁ x).1 < (key₂ y).1 | |
| · simp only [hk, ↓reduceIte] at h ⊢ | |
| cases h | |
| omega | |
| · simp only [hk, ↓reduceIte] at h ⊢ | |
| by_cases hk' : (key₂ y).1 < (key₁ x).1 | |
| · simp only [hk', ↓reduceIte] at h ⊢ | |
| cases h | |
| omega | |
| · simp only [hk', ↓reduceIte] at h ⊢ | |
| obtain ⟨pair, hg, he⟩ := Option.map_eq_some_iff.mp h | |
| rcases pair with ⟨group, remaining⟩ | |
| have hgv := (BudgetedTieGroup.execute_value 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) group remaining hg).2 | |
| have hgs := BudgetedTieGroup.execute_spent read₁ key₁ read₂ key₂ cells (key₁ x).1 report clone | |
| R₁ L₁ R₂ L₂ S C front fill outer inner copy hr₁ hk₁ hr₂ hk₂ hs hc hfront hfill houter hinner hcopy | |
| B fuel₁ fuel₂ st.left st.right st.user (budget - base) group remaining hg | |
| cases he | |
| omega | |
| next _ => contradiction | |
| end LowSpaceSubsetSum.BudgetedStreamStep | |
| end OAI | |