openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Queries /QueryDependencyCosts.lean
| import OAI.Combinatorics.EditApproximation.Queries.QueryDependencyExpansion | |
| import OAI.Combinatorics.EditApproximation.Geometry.RankedMemoSimulation | |
| namespace OAI | |
| /-! Weighted read counts are the actual same-index and earlier-index | |
| lengths used by the dependency expansion, after identifying its values | |
| with the local mathematical recurrence. -/ | |
| namespace EditApproximation | |
| theorem list_filter_length_eq_sum {κ : Type*} (keys : List κ) (predicate : κ → Bool) : | |
| (keys.filter predicate).length = (keys.map fun key => if predicate key then 1 else 0).sum := by | |
| induction keys with | |
| | nil => rfl | |
| | cons key keys ih => | |
| cases hp : predicate key <;> simp [hp, ih, Nat.add_comm] | |
| section Queries | |
| variable {κ : Type*} (depth index : κ → ℕ) | |
| (program : (key : κ) → FiniteQuery {child : κ // depth child + index child < depth key + index key}) | |
| (expected : κ → ℚ) | |
| (hstep : ∀ key, (program key).value (fun child => expected child.val) = expected key) | |
| include hstep | |
| theorem queryRequestList_expected (key : κ) : | |
| queryRequestList depth index program key = | |
| ((program key).keys (fun child => expected child.val)).map Subtype.val := by | |
| unfold queryRequestList | |
| simp only [rankedProgramValue_eq (fun key => depth key + index key) program expected hstep] | |
| theorem querySameRequests_length_eq_cost (key : κ) : | |
| (querySameRequests depth index program key).length = | |
| (program key).cost (fun child => expected child.val) | |
| (fun child => if ¬ index child.val < index key then 1 else 0) := by | |
| unfold querySameRequests | |
| rw [queryRequestList_expected depth index program expected hstep, | |
| list_filter_length_eq_sum, FiniteQuery.cost_eq_sum, List.map_map] | |
| simp only [Function.comp_def, Bool.not_eq_true', decide_eq_false_iff_not] | |
| theorem queryEarlierRequests_length_eq_cost (key : κ) : | |
| (queryEarlierRequests depth index program key).length = | |
| (program key).cost (fun child => expected child.val) | |
| (fun child => if index child.val < index key then 1 else 0) := by | |
| unfold queryEarlierRequests | |
| rw [queryRequestList_expected depth index program expected hstep, | |
| list_filter_length_eq_sum, FiniteQuery.cost_eq_sum, List.map_map] | |
| simp only [Function.comp_def, decide_eq_true_eq] | |
| end Queries | |
| end EditApproximation | |
| end OAI | |