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2f887db | 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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | import Mathlib
import OAI.Computability.MinUncut.Search.FiniteEnumeration
import OAI.Computability.MinUncut.Estimates.WeightedTestLaw
import OAI.Computability.MinUncut.Encoding.QuestionEncoding
namespace OAI
section
namespace MinUncut.Outer
open MinUncut.Inner MinUncut.FiniteGaussian MinUncut.FiniteProof OuterSmoothness
open MinUncutGames.Foundations.Hastad.SourceOccurrences
namespace SampleEnumeration
def fin (n : ℕ) : Enumeration (Fin n) := Enumeration.ofEncoding (Encoding.fin n)
def field : Enumeration F₂ := Enumeration.ofEncoding fieldEncoding
def point (m n : ℕ) : Enumeration (Point m n) := (fin m).function (fin n)
def row (m n : ℕ) : Enumeration (Row m n) :=
(fin m).sigma (fun i => ((fin m).subtype (fun j => j≠i)).function (fin n))
local instance codeEq (m n : ℕ) : DecidableEq (Code m n) :=
Subtype.instDecidableEq
def code (m n : ℕ) : Enumeration (Code m n) :=
((row m n).function field).image (faceSum m n).rangeRestrict (by
intro z
obtain ⟨f,hf⟩ := z.property
exact ⟨f,Subtype.ext hf⟩)
local instance codeType (m n : ℕ) : Fintype (Code m n) :=
Fintype.ofList (code m n).values (code m n).complete
abbrev Slots {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) :=
Σ j, Fin (secondSlotCount (Q j))
def slots {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) : Enumeration (Slots Q) :=
(fin t).sigma (fun j => fin (secondSlotCount (Q j)))
def formsPiEquiv (A : Type) [Fintype A] [DecidableEq A] :
Forms (A → F₂) ≃ F₂ × (A → F₂) :=
(AffineMap.toConstProdLinearMap F₂).toEquiv.trans
(Equiv.prodCongr (Equiv.refl F₂) (LinearEquiv.piRing F₂ F₂ A F₂).toEquiv)
def formsCurryEquiv (A : Type) (B : A → Type) :
Forms (∀a, B a → F₂) ≃ Forms ((Σa, B a) → F₂) :=
AffineEquiv.arrowCongrEquiv
(LinearEquiv.piCurry F₂ (fun (_ : A) (_ : B _) => F₂)).symm.toAffineEquiv
(AffineEquiv.refl F₂ F₂)
def formsEquiv {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) :
Forms (SecondAlphabet Q) ≃ F₂ × (Slots Q → F₂) :=
(formsCurryEquiv (Fin t) (fun j => Fin (secondSlotCount (Q j)))).trans (formsPiEquiv (Slots Q))
local instance formEq {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) :
DecidableEq (Forms (SecondAlphabet Q)) :=
fun f g => decidable_of_iff (formsEquiv Q f=formsEquiv Q g) (formsEquiv Q).injective.eq_iff
def forms {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) : Enumeration (Forms (SecondAlphabet Q)) :=
(field.pair ((slots Q).function field)).image (formsEquiv Q).symm (formsEquiv Q).symm.surjective
def arrays {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) (m n : ℕ) :
Enumeration (FaceArray (SecondAlphabet Q) m n) := (row m n).function (forms Q)
def coordinates (m n : ℕ) : Enumeration (TestCoordinates m n) :=
Enumeration.ofEncoding ((point m n).encoding.sum ((point m n).encoding.sum (code m n).encoding))
def localSample {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) (m n : ℕ) (g : GridData) :
Enumeration (LocalSample (SecondAlphabet Q) m n g) := by
letI : DecidableEq (Forms (SecondAlphabet Q)) := formEq Q
letI : DecidableEq (FaceArray (SecondAlphabet Q) m n) := inferInstanceAs (DecidableEq (Row m n → Forms (SecondAlphabet Q)))
exact (arrays Q m n).pair ((arrays Q m n).pair ((code m n).pair ((coordinates m n).function (fin g.L))))
def finsets (t : ℕ) : Enumeration (Finset (Fin t)) :=
((fin t).function (Enumeration.ofEncoding Encoding.bool)).image
(fun f => Finset.univ.filter (fun j => f j=true)) (by
intro S
refine ⟨fun j => decide (j∈S),?_⟩
ext j
simp)
def fixedSets (t k : ℕ) : Enumeration (FixedSets (Fin t) k) :=
(finsets t).subtype (fun H => H.card=k)
def outerSample (t s k : ℕ) : Enumeration (OuterSample (Fin t) (Fin s) k) :=
(fixedSets t k).pair (((fin t).function (fin s)).pair ((fin t).function (fin 3)))
local instance localEq {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) (m n : ℕ) (g : GridData) :
DecidableEq (LocalSample (SecondAlphabet Q) m n g) := by
letI : DecidableEq (Forms (SecondAlphabet Q)) := formEq Q
letI : DecidableEq (FaceArray (SecondAlphabet Q) m n) :=
inferInstanceAs (DecidableEq (Row m n → Forms (SecondAlphabet Q)))
exact fun f h => decidable_of_iff
(f.1=h.1 ∧ f.2.1=h.2.1 ∧ f.2.2.1=h.2.2.1 ∧ f.2.2.2=h.2.2.2)
(by constructor <;> intro he <;> aesop)
def second {N s t : ℕ} (eqs : Fin s → Equation (Fin N)) {k : ℕ}
(o : OuterSample (Fin t) (Fin s) k) : Fin t → SecondQuestion (Fin N) :=
fun j => if j∈o.1.val then Sum.inl ((eqs (o.2.1 j)).names (o.2.2 j)) else Sum.inr (eqs (o.2.1 j))
lemma second_eq {N s t : ℕ} (eqs : Fin s → Equation (Fin N)) {k : ℕ}
(o : OuterSample (Fin t) (Fin s) k) : second eqs o=sampleSecond eqs o := by
funext j
simp [second,sampleSecond,secondQuestion,sampleFirst,sampleHidden,hiddenSet,slotQuestion]
def semanticLocal {N s t : ℕ} (eqs : Fin s → Equation (Fin N)) {k : ℕ}
(o : OuterSample (Fin t) (Fin s) k) (m n : ℕ) (g : GridData) :
Enumeration (LocalSample (SecondAlphabet (sampleSecond eqs o)) m n g) :=
(second_eq eqs o) ▸ localSample (second eqs o) m n g
local instance semanticLocalEq {N s t : ℕ} (eqs : Fin s → Equation (Fin N)) {k : ℕ}
(o : OuterSample (Fin t) (Fin s) k) (m n : ℕ) (g : GridData) :
DecidableEq (LocalSample (SecondAlphabet (sampleSecond eqs o)) m n g) :=
(second_eq eqs o) ▸ localEq (second eqs o) m n g
def elementary {N s t : ℕ} (eqs : Fin s → Equation (Fin N)) (k m n : ℕ) (g : GridData) :
Enumeration (Elementary (I := Fin t) eqs k m n g) :=
(outerSample t s k).sigma (fun o => semanticLocal eqs o m n g)
def tests : Enumeration Test :=
Enumeration.ofList [.first,.second,.third,.fourth] (by intro j; cases j <;> simp)
def all {N s t : ℕ} (eqs : Fin s → Equation (Fin N)) (k m n : ℕ) (g : GridData) :
Enumeration (Test × Elementary (I := Fin t) eqs k m n g) :=
tests.pair (elementary eqs k m n g)
end SampleEnumeration
end MinUncut.Outer
end
end OAI
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