openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Separators /BudgetedSeparatorFlow.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Separators.BudgetedSeparatorCall | |
| namespace OAI | |
| /-! # Sequencing the already charged checker operations | |
| The enclosing node charge includes branch bookkeeping. These combinators | |
| add only the work counters of their constituent charged operations. | |
| -/ | |
| namespace LowSpaceSubsetSum.BudgetedSeparatorFlow | |
| open BudgetedSeparatorCall | |
| variable {α β : Type*} | |
| def pure (a : α) (budget : ℕ) : Outcome α := (some (a, budget), 0) | |
| def bind (first : Outcome α) (next : α → ℕ → Outcome β) : Outcome β := | |
| match first.1 with | |
| | none => (none, first.2) | |
| | some (a, left) => | |
| let out := next a left | |
| (out.1, first.2 + out.2) | |
| def Account (budget : ℕ) (out : Outcome α) : Prop := | |
| out.2 ≤ budget + 1 ∧ | |
| ∀ a left, out.1 = some (a, left) → left ≤ budget ∧ out.2 ≤ budget - left | |
| theorem pure_account (a : α) (budget : ℕ) : Account budget (pure a budget) := by | |
| constructor | |
| · simp [pure] | |
| · intro b left h | |
| simp only [pure, Option.some.injEq, Prod.mk.injEq] at h | |
| rcases h with ⟨rfl, rfl⟩ | |
| simp [pure] | |
| theorem bind_account (first : Outcome α) (next : α → ℕ → Outcome β) | |
| (budget : ℕ) (hfirst : Account budget first) | |
| (hnext : ∀ a left, first.1 = some (a, left) → Account left (next a left)) : | |
| Account budget (bind first next) := by | |
| cases hf : first.1 with | |
| | none => | |
| constructor | |
| · simpa only [bind, hf] using hfirst.1 | |
| · intro a left h | |
| simp only [bind, hf] at h | |
| contradiction | |
| | some p => | |
| rcases p with ⟨a, left⟩ | |
| obtain ⟨hmleft, hmwork⟩ := hfirst.2 a left hf | |
| obtain ⟨hnwork, hnspent⟩ := hnext a left hf | |
| unfold Account | |
| simp only [bind, hf] | |
| constructor | |
| · omega | |
| · intro b rest h | |
| change (next a left).1 = some (b, rest) at h | |
| have hr := hnspent b rest h | |
| change rest ≤ budget ∧ first.2 + (next a left).2 ≤ budget - rest | |
| omega | |
| theorem bind_eq_some (first : Outcome α) (next : α → ℕ → Outcome β) | |
| (b : β) (rest : ℕ) : | |
| (bind first next).1 = some (b, rest) ↔ | |
| ∃ a left, first.1 = some (a, left) ∧ (next a left).1 = some (b, rest) := by | |
| cases hf : first.1 with | |
| | none => simp only [bind, hf, reduceCtorEq, false_and, exists_false] | |
| | some p => | |
| rcases p with ⟨a, left⟩ | |
| simp only [bind, hf, Option.some.injEq, Prod.mk.injEq] | |
| constructor | |
| · intro h | |
| exact ⟨a, left, ⟨rfl, rfl⟩, h⟩ | |
| · rintro ⟨a', left', ⟨rfl, rfl⟩, h⟩ | |
| exact h | |
| theorem bind_complete (first : Outcome α) (next : α → ℕ → Outcome β) | |
| (a : α) (left : ℕ) (b : β) (rest : ℕ) | |
| (hf : first.1 = some (a, left)) (hn : (next a left).1 = some (b, rest)) : | |
| (bind first next).1 = some (b, rest) := | |
| (bind_eq_some first next b rest).mpr ⟨a, left, hf, hn⟩ | |
| end LowSpaceSubsetSum.BudgetedSeparatorFlow | |
| end OAI | |