| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineInitial |
| import OAI.Combinatorics.MatchingCount.Complexity.OnlineScheduleCost |
|
|
| namespace OAI |
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|
| namespace MatchingFPRAS.OnlineInitial |
| open TreeTyped TreeFraction OnlineSchedule Polynomial NumericBounds |
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| lemma work_root : WorkBound rootRealizer id := by |
| unfold rootRealizer |
| apply work_then_polynomial _ polynomial_findRoot |
| apply work_pair _ _ (4*X+1) (fun n=>by simpa using weight_nat_numeric n) |
| · exact work_of_polynomial (polynomial_composition (polynomial_pair polynomial_identity (polynomial_constant _)) |
| polynomial_subtract) (4*X+1) (fun n=>by simpa using weight_nat_numeric n) |
| · exact work_composition (work_of_polynomial polynomial_identity (4*X+1) |
| (fun n=>by simpa using weight_nat_numeric n)) fuel_numeric_work X (by simp) |
| lemma work_initialScale : WorkBound initialScaleRealizer id := by |
| unfold initialScaleRealizer |
| apply work_then_polynomial _ polynomial_div |
| apply work_pair _ _ (4*X+1) (fun n=>by simpa using weight_nat_numeric n) |
| · exact work_of_polynomial (polynomial_constant _) (4*X+1) (fun n=>by simpa using weight_nat_numeric n) |
| · exact work_pair work_root (work_of_polynomial (polynomial_constant _) (4*X+1) |
| (fun n=>by simpa using weight_nat_numeric n)) (4*X+1) (fun n=>by simpa using weight_nat_numeric n) |
| lemma polynomial_scaleVector : PolynomialTime scaleVectorRealizer := by |
| unfold scaleVectorRealizer |
| apply polynomial_congr |
| apply work_then_polynomial _ (polynomial_mapWith (polynomial_pair polynomial_second polynomial_first)) |
| apply work_pair _ polynomial_identity X (by simp) |
| exact work_composition polynomial_listLength work_initialScale X (fun vs=>by simpa using list_length_le_weight vs) |
| lemma polynomial_count : PolynomialTime countRealizer := by |
| unfold countRealizer nRealizer Realizer.fst Realizer.snd |
| apply polynomial_composition _ polynomial_sampleCount |
| tree_cost |
| lemma polynomial_seedD : PolynomialTime seedDRealizer := by |
| unfold seedDRealizer Realizer.fst |
| apply polynomial_pair polynomial_first |
| apply polynomial_pair polynomial_second |
| apply polynomial_composition _ polynomial_distortion |
| tree_cost |
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| def configNumeric (x : Config) : ℕ := x.1.1.1.1.length+x.1.2+x.2.1+x.2.2+ |
| ladderSchedule (x.2,x.1.2)+Transport.ladderQ x.2.1 x.2.2 |
| def configSize (x : Config) : ℕ := configNumeric x+weight x |
| def seedSize (x : Seed) : ℕ := x.1.1.length+natCeil (inverse x.2.1)+configNumeric (config x)+weight x |
| lemma weight_configSize (x : Config) : weight x ≤ configSize x := by unfold configSize;omega |
| lemma weight_seedSize (x : Seed) : weight x ≤ seedSize x := by unfold seedSize;omega |
| lemma config_fields (x : Config) : x.1.1.1.1.length ≤ configSize x ∧ x.1.2 ≤ configSize x ∧ |
| x.2.1 ≤ configSize x ∧ x.2.2 ≤ configSize x ∧ |
| ladderSchedule (x.2,x.1.2) ≤ configSize x ∧Transport.ladderQ x.2.1 x.2.2 ≤ configSize x := by |
| unfold configSize configNumeric;omega |
| lemma seed_fields (x : Seed) : x.1.1.length ≤ seedSize x ∧ natCeil (inverse x.2.1) ≤ seedSize x ∧ |
| coolingSchedule (x.1.1.length,x.2.1) ≤ seedSize x ∧configNumeric (config x) ≤ seedSize x := by |
| unfold seedSize configNumeric config seedD seedN seedK |
| dsimp only |
| omega |
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| lemma work_seedK : WorkBound seedKRealizer seedSize := by |
| have lift {A : Type} [Codec A] {f : Seed→A} {F : Realizer f} (hF : PolynomialTime F) : |
| WorkBound F seedSize := work_of_polynomial hF X (fun x=>by simpa using weight_seedSize x) |
| unfold seedKRealizer |
| apply work_congr |
| apply work_pair (lift polynomial_identity) _ X (fun x=>by simpa using weight_seedSize x) |
| apply work_comp_bound (lift (polynomial_pair (polynomial_composition (polynomial_fst polynomial_first) |
| polynomial_listLength) (polynomial_fst polynomial_second))) work_coolingSchedule |
| have hw:=NumericBounds.weight (lift (polynomial_pair (polynomial_composition (polynomial_fst polynomial_first) |
| polynomial_listLength) (polynomial_fst polynomial_second))) |
| exact add (add (of_le (fun x=>(seed_fields x).1)) (of_le (fun x=>(seed_fields x).2.1))) hw |
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| def seedNSize (x : Seed×ℕ) : ℕ := x.1.1.1.length+x.2+weight x |
| lemma work_seedN : WorkBound seedNRealizer seedNSize := by |
| have lift {A : Type} [Codec A] {f : (Seed×ℕ)→A} {F : Realizer f} (hF : PolynomialTime F) : |
| WorkBound F seedNSize := work_of_polynomial hF X (fun x=>by simp [seedNSize]) |
| unfold seedNRealizer |
| apply work_congr |
| apply work_pair (lift polynomial_identity) _ X (fun x=>by simp [seedNSize]) |
| apply work_comp_bound (lift (polynomial_pair (polynomial_composition (polynomial_fst (polynomial_fst polynomial_first)) |
| polynomial_listLength) polynomial_second)) work_vertexNumber |
| have hw:=NumericBounds.weight (lift (polynomial_pair (polynomial_composition (polynomial_fst (polynomial_fst polynomial_first)) |
| polynomial_listLength) polynomial_second)) |
| exact add (add (of_le (fun x=>by dsimp only [Function.comp_apply]; unfold seedNSize;omega)) (of_le (fun x=>by dsimp only [Function.comp_apply]; unfold seedNSize;omega))) hw |
| lemma work_config : WorkBound configRealizer seedSize := by |
| unfold configRealizer |
| apply work_then_polynomial _ polynomial_seedD |
| apply work_comp_bound work_seedK work_seedN |
| have hw:=NumericBounds.weight work_seedK |
| exact add (add (of_le (fun x=>(seed_fields x).1)) (of_le (fun x=>(seed_fields x).2.2.1))) hw |
| lemma bound_configSize : Bound (configSize ∘ config) seedSize := |
| add (of_le (fun x=>(seed_fields x).2.2.2)) (NumericBounds.weight work_config) |
| end MatchingFPRAS.OnlineInitial |
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| end OAI |
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