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f0f05de | 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 | import OAI.Combinatorics.BalancedRyser.Local.AlignedGlobalProduct
import OAI.Combinatorics.BalancedRyser.Local.LocalUnusedOffsetProbability
namespace OAI
namespace Balanced.Local.AlignedGlobal
open AlignedUnused TupleAlignment RootChildPartition
noncomputable section
attribute [local instance] Classical.propDecidable
variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
variable {b κ : ℕ}
def refreshUnused (P : EqualityPattern b κ) (a : DirectionAssignment F b κ)
(M : MergedPairs F) (u : UnusedArray P a) : MergedPairs F :=
(split P a).symm ((split P a M).1, u)
omit [Field F] [Fintype F] in
theorem refreshUnused_used (P : EqualityPattern b κ)
(a : DirectionAssignment F b κ) (M : MergedPairs F) (u : UnusedArray P a)
(d : Direction F) (hd : d ∈ usedDirections P a) :
refreshUnused P a M u d = M d := by
exact congrFun (congrArg Prod.fst
((split P a).apply_symm_apply ((split P a M).1, u))) ⟨d, hd⟩
omit [Field F] [Fintype F] in
theorem refreshUnused_unused (P : EqualityPattern b κ)
(a : DirectionAssignment F b κ) (M : MergedPairs F) (u : UnusedArray P a)
(d : {d : Direction F // d ∉ usedDirections P a}) :
refreshUnused P a M u d.1 = u d := by
exact congrFun (congrArg Prod.snd
((split P a).apply_symm_apply ((split P a M).1, u))) d
def refreshAligned (P : EqualityPattern b κ) (a : DirectionAssignment F b κ)
(bits : Bits b) (p : Point F) (M : AlignedOffsets P a bits p)
(u : UnusedArray P a) : AlignedOffsets P a bits p :=
⟨refreshUnused P a M.1 u,
(aligned_agree P a bits p M.1 (refreshUnused P a M.1 u)
(fun d hd => (refreshUnused_used P a M.1 u d hd).symm)).mp M.2⟩
omit [Fintype F] in
theorem coordinates_refreshAligned (P : EqualityPattern b κ)
(R : Finset (Fin b)) (a : DirectionAssignment F b κ)
(bits : Bits b) (p : Point F) (M : AlignedOffsets P a bits p)
(u : UnusedArray P a) :
coordinates P R a bits p (refreshAligned P a bits p M u) =
((coordinates P R a bits p M).1, u) := by
apply Prod.ext
· funext ij
apply congrArg (linePointEquiv (a.1 ij.1.1) (lineEval (a.1 ij.1.1) p))
apply Subtype.ext
exact UnusedOffsetProbability.prescribedPoint_eq_of_used_agreement
_ _ (refreshUnused_used P a M.1 u) ij.1
· funext d
exact refreshUnused_unused P a M.1 u d
theorem generated_refresh (P : EqualityPattern b κ) (hκ : κ ≤ 2)
(R : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ)
(a : DirectionAssignment F b κ) (ha : Eligible P a)
(bits : Bits b) (p : Point F) (x : FreeIndex P R → F)
(u v : UnusedArray P a) :
refreshAligned P a bits p
((coordinateEquiv P hκ R hD a ha bits p).symm (x,u)) v =
(coordinateEquiv P hκ R hD a ha bits p).symm (x,v) := by
apply (coordinateEquiv P hκ R hD a ha bits p).injective
rw [coordinateEquiv_apply, coordinates_refreshAligned]
change ((coordinateEquiv P hκ R hD a ha bits p
((coordinateEquiv P hκ R hD a ha bits p).symm (x,u))).1, v) = _
simp only [Equiv.apply_symm_apply]
theorem generated_used_agree (P : EqualityPattern b κ) (hκ : κ ≤ 2)
(R : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ)
(a : DirectionAssignment F b κ) (ha : Eligible P a)
(bits : Bits b) (p : Point F) (x : FreeIndex P R → F)
(u v : UnusedArray P a) (d : Direction F) (hd : d ∈ usedDirections P a) :
((coordinateEquiv P hκ R hD a ha bits p).symm (x,u)).1 d =
((coordinateEquiv P hκ R hD a ha bits p).symm (x,v)).1 d := by
rw [← generated_refresh P hκ R hD a ha bits p x u v]
exact (refreshUnused_used P a _ v d hd).symm
def generatedZero (P : EqualityPattern b κ) (hκ : κ ≤ 2)
(R : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ)
(a : DirectionAssignment F b κ) (ha : Eligible P a)
(bits : Bits b) (p : Point F) (x : FreeIndex P R → F) : MergedPairs F :=
((coordinateEquiv P hκ R hD a ha bits p).symm (x, fun _ _ => 0)).1
theorem alignedLaw_map_used {O : Type*}
(P : EqualityPattern b κ) (hκ : κ ≤ 2)
(R : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ)
(a : DirectionAssignment F b κ) (ha : Eligible P a)
(bits : Bits b) (p : Point F) (obs : MergedPairs F → O)
(hobs : ∀ M N, (∀ d ∈ usedDirections P a, M d = N d) → obs M = obs N) :
(alignedLaw P hκ a ha bits p).map obs =
(PMF.uniformOfFintype (FreeIndex P R → F)).map
(fun x => obs (generatedZero P hκ R hD a ha bits p x)) := by
rw [alignedLaw_eq_uniform_coordinates P hκ R hD a ha bits p, PMF.map_comp]
have hf : (obs ∘ (fun z =>
((coordinateEquiv P hκ R hD a ha bits p).symm z).1)) =
(fun x => obs (generatedZero P hκ R hD a ha bits p x)) ∘ Prod.fst := by
funext z
apply hobs
intro d hd
exact generated_used_agree P hκ R hD a ha bits p z.1 z.2 (fun _ _ => 0) d hd
rw [hf, ← PMF.map_comp, uniform_map_fst]
theorem conditioned_probability_eq_uniform_free
(P : EqualityPattern b κ) (hκ : κ ≤ 2)
(R : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ)
(a : DirectionAssignment F b κ) (ha : Eligible P a)
(bits : Bits b) (p : Point F) (E : MergedPairs F → Prop)
(hE : ∀ M N, (∀ d ∈ usedDirections P a, M d = N d) → (E M ↔ E N)) :
conditionedOffsetProbability (Aligned P a bits p) E =
((PMF.uniformOfFintype (FreeIndex P R → F)).toOuterMeasure
{x | E (generatedZero P hκ R hD a ha bits p x)}).toReal := by
rw [conditioned_probability_eq_uniform_coordinates P hκ R hD a ha bits p E]
have heq : {z | E ((coordinateEquiv P hκ R hD a ha bits p).symm z).1} =
Prod.fst ⁻¹' {x | E (generatedZero P hκ R hD a ha bits p x)} := by
ext z
exact hE _ _ (generated_used_agree P hκ R hD a ha bits p z.1 z.2 (fun _ _ => 0))
rw [heq]
have hm := congrArg (fun μ : PMF (FreeIndex P R → F) =>
(μ.toOuterMeasure {x | E (generatedZero P hκ R hD a ha bits p x)}).toReal)
(uniform_map_fst (α := FreeIndex P R → F) (β := UnusedArray P a))
simpa only [PMF.toOuterMeasure_map_apply] using hm
end
end Balanced.Local.AlignedGlobal
end OAI
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