File size: 3,732 Bytes
e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | 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
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