openAI-math-dataset / lean /OAI /Combinatorics /MatchingCount /Algorithm /TreeNumericStatistics.lean
| import Mathlib | |
| import OAI.Combinatorics.MatchingCount.Sampling.TreeNumericMarking | |
| import OAI.Combinatorics.MatchingCount.Algorithm.TreeStageStatisticsCorrect | |
| namespace OAI | |
| namespace MatchingFPRAS.TreeTyped.Numbered | |
| variable {E A : Type} [Fintype E] [LinearOrder E] | |
| def extend (d : A) (f : E→A) (i : ℕ) : A := | |
| if h:i<Fintype.card E then f (UnitRefresh.orderedEnum E ⟨i,h⟩) else d | |
| lemma extend_index (d : A) (f : E→A) (e : E) : extend d f (index e)=f e := by | |
| unfold extend index | |
| rw [dite_eq_left ((UnitRefresh.orderedEnum E).symm e).isLt] | |
| exact congrArg f ((UnitRefresh.orderedEnum E).apply_symm_apply e) | |
| end MatchingFPRAS.TreeTyped.Numbered | |
| namespace MatchingFPRAS.NumericMarking | |
| open TreeTyped TreeFraction TreeTyped.Numbered ExecutableMarking ExecutableStatistics | |
| variable {E : Type} [Fintype E] [LinearOrder E] | |
| lemma numericMarks_eq (s : Finset E) (m : E→Fin 3) : | |
| numericMarks s m=marksRep (Numbered.set s) (extend 0 m) := by | |
| simp only [numericMarks,marksRep,sort_set,List.map_map,Function.comp_def,extend_index] | |
| lemma zero_numbered (s : Finset E) (m : E→Fin 3) : | |
| (∀ a∈Numbered.set s,extend 0 m a=0) ↔ ∀ e∈s,m e=0 := by | |
| constructor | |
| · intro h e he | |
| have hh:=h (index e) (Finset.mem_map.mpr ⟨e,he,rfl⟩) | |
| simpa only [extend_index] using hh | |
| · intro h a ha | |
| obtain ⟨e,he,rfl⟩:=Finset.mem_map.mp ha | |
| exact (extend_index 0 m e).trans (h e he) | |
| lemma scalarA_numbered {S : ℕ} (s : Fin S→Finset E) (m : Fin S→E→Fin 3) (i : Fin S) : | |
| scalarA (fun j=>Numbered.set (s j)) (fun j=>extend 0 (m j)) i=scalarA s m i := by | |
| by_cases h : ∀ e∈s i,m i e=0 | |
| · have hh := (zero_numbered (s i) (m i)).mpr h | |
| simp only [scalarA,ite_eq_left h,ite_eq_left hh] | |
| · have hh : ¬∀e∈Numbered.set (s i),extend 0 (m i) e=0 := fun hh=>h ((zero_numbered _ _).mp hh) | |
| simp only [scalarA,ite_eq_right h,ite_eq_right hh] | |
| lemma scalarD_numbered {S : ℕ} (s : Fin S→Finset E) (m : Fin S→E→Fin 3) (r : E→ℚ) (i : Fin S) : | |
| scalarD (fun j=>Numbered.set (s j)) (fun j=>extend 0 (m j)) (extend 1 r) i=scalarD s m r i := by | |
| by_cases h : ∀ e∈s i,m i e=0 | |
| · have hh := (zero_numbered (s i) (m i)).mpr h | |
| simp only [scalarD,ite_eq_left h,ite_eq_left hh] | |
| simp only [Numbered.set,Finset.prod_map,Numbered.embedding,Function.Embedding.coeFn_mk,extend_index] | |
| · have hh : ¬∀e∈Numbered.set (s i),extend 0 (m i) e=0 := fun hh=>h ((zero_numbered _ _).mp hh) | |
| simp only [scalarD,ite_eq_right h,ite_eq_right hh] | |
| abbrev Reweights := List (ℕ×Fraction) | |
| def reweight (x : Reweights×ℕ) : Fraction := ExecutableLadder.lookup (1,1) x | |
| def reweightRealizer : Realizer reweight := ExecutableLadder.lookupRealizer (1,1) natEqual | |
| lemma polynomial_reweight : PolynomialTime reweightRealizer := ExecutableLadder.polynomial_lookup (1,1) polynomial_natEqual | |
| abbrev Inputs := (Reweights×(ℕ×List ℕ))×List (Marks ℕ) | |
| def statistics (x : Inputs) : Fraction×List (ℕ×Fraction) := estimates reweight x | |
| def statisticsRealizer : Realizer statistics := estimatesRealizer reweightRealizer natEqual | |
| lemma polynomial_statistics : PolynomialTime statisticsRealizer := polynomial_estimates polynomial_reweight polynomial_natEqual | |
| theorem statistics_post : ∃ p : Polynomial ℕ,∀ x : Inputs,∀ t : ℕ, | |
| p.eval (TreePost.packed x).length≤t → ∀ coins : Fin t→Bool, | |
| (run (TreePost.machine statisticsRealizer.program) (TreePost.packed x) coins).state=none ∧ | |
| (run (TreePost.machine statisticsRealizer.program) (TreePost.packed x) coins).tape.right₀= | |
| Turing.ListBlank.mk (TreePost.packed (statistics x)) := | |
| TreePost.polynomial_compiles statisticsRealizer polynomial_statistics | |
| end MatchingFPRAS.NumericMarking | |
| end OAI | |