openAI-math-dataset / lean /OAI /Combinatorics /MatchingCount /Complexity /OnlineInitialDataCost.lean
| import Mathlib | |
| import OAI.Combinatorics.MatchingCount.Complexity.OnlineInitialCost | |
| namespace OAI | |
| namespace MatchingFPRAS.OnlineInitial | |
| open TreeTyped TreeFraction OnlineSchedule Polynomial NumericBounds ExecutableSource ExecutableScaling | |
| lemma work_data : WorkBound dataRealizer configSize := by | |
| have lift {A : Type} [Codec A] {f : Config→A} {F : Realizer f} (hF : PolynomialTime F) : | |
| WorkBound F configSize := work_of_polynomial hF X (fun x=>by simpa using weight_configSize x) | |
| have wb : ∀ x : Config,weight x≤X.eval (configSize x) := fun x=>by simpa using weight_configSize x | |
| have _workK : WorkBound (first.snd : Realizer (fun x : Config=>x.1.2)) configSize := lift (by unfold Realizer.snd; tree_cost) | |
| have hk : Bound (fun x : Config=>x.1.2) configSize := of_le (fun x=>(config_fields x).2.1) | |
| have hN : Bound (fun x : Config=>x.2.1) configSize := of_le (fun x=>(config_fields x).2.2.1) | |
| have hD : Bound (fun x : Config=>x.2.2) configSize := of_le (fun x=>(config_fields x).2.2.2.1) | |
| have hT : Bound (fun x : Config=>ladderSchedule (x.2,x.1.2)) configSize := of_le (fun x=>(config_fields x).2.2.2.2.1) | |
| have hq : Bound (fun x : Config=>Transport.ladderQ x.2.1 x.2.2) configSize := of_le (fun x=>(config_fields x).2.2.2.2.2) | |
| unfold dataRealizer Realizer.fst Realizer.snd | |
| apply work_congr | |
| apply work_pair _ _ X wb | |
| · apply work_pair _ _ X wb | |
| · exact lift (by tree_cost) | |
| · apply work_pair _ _ X wb | |
| · apply work_pair _ (lift (by tree_cost)) X wb | |
| apply work_pair _ _ X wb | |
| · apply work_comp_bound (lift (by apply polynomial_addNat (polynomial_mulNat (polynomial_constant _) (by tree_cost)) (polynomial_constant _))) fuel_numeric_work | |
| exact add (mul (const 2 _) hk) (const 2 _) | |
| · apply work_comp_bound (lift (by apply polynomial_mulNat (polynomial_constant _) (by tree_cost))) fuel_numeric_work | |
| exact mul (const 2 _) hk | |
| · apply work_pair _ _ X wb | |
| · apply work_comp_bound _ fuel_numeric_work hT | |
| apply work_comp_bound (lift (by tree_cost)) work_ladderSchedule | |
| exact add (add (add hN hD) hk) (NumericBounds.weight (lift (by tree_cost))) | |
| · apply work_comp_bound (lift (polynomial_composition polynomial_second polynomial_q)) fuel_numeric_work hq | |
| · exact lift (polynomial_composition (by tree_cost) polynomial_scaleVector) | |
| lemma work_trialInput : WorkBound trialInputRealizer configSize := by | |
| have lift {A : Type} [Codec A] {f : Config→A} {F : Realizer f} (hF : PolynomialTime F) : | |
| WorkBound F configSize := work_of_polynomial hF X (fun x=>by simpa using weight_configSize x) | |
| have wb : ∀ x : Config,weight x≤X.eval (configSize x) := fun x=>by simpa using weight_configSize x | |
| unfold trialInputRealizer Realizer.snd | |
| apply work_pair _ _ X wb | |
| · exact work_comp_bound (lift (by tree_cost)) fuel_numeric_work (of_le (fun x=>(config_fields x).2.1)) | |
| · apply work_pair _ _ X wb | |
| · exact work_pair work_data (lift polynomial_count) X wb | |
| · apply work_pair _ (lift (polynomial_constant _)) X wb | |
| apply work_comp_bound (lift (by unfold nRealizer Realizer.fst;tree_cost)) pairingNumber_numeric_work | |
| exact of_le (fun x=>(config_fields x).1) | |
| lemma work_runInput : WorkBound runInputRealizer configSize := by | |
| unfold runInputRealizer Realizer.fst Realizer.snd | |
| apply work_congr | |
| apply work_pair _ work_trialInput X (fun x=>by simpa using weight_configSize x) | |
| apply work_of_polynomial (polynomial_composition (by tree_cost) polynomial_repetitions) X | |
| exact fun x=>by simpa using weight_configSize x | |
| lemma work_initializeInput : WorkBound initializeInputRealizer seedSize := | |
| work_comp_bound work_config work_runInput bound_configSize | |
| end MatchingFPRAS.OnlineInitial | |
| end OAI | |