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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 | import OAI.Combinatorics.BalancedRyser.Local.GlobalPartialAlignment
import OAI.Combinatorics.BalancedRyser.Local.Local
import OAI.Combinatorics.BalancedRyser.Local.LowReuseOrder
import OAI.Combinatorics.BalancedRyser.Probability.FiniteUniformFibers
namespace OAI
namespace Balanced.AlignedListProbability
open TupleAlignment PartialAlignment ConditionalRestriction
open scoped BigOperators
noncomputable section
attribute [local instance] Classical.propDecidable
variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
theorem partial_alignment_probability_eq (d : Fin 4 → Direction F)
(hd : ∀ i, d i ≠ d (i + 1)) (σ : Fin 4 → Bool)
(g : Direction F) (hg : ∀ i, d i ≠ g) (c : F)
(J : Finset (Fin 4)) (hJ : J.card ≤ 1) (a : Fin 4 → Bool → F) :
((PMF.uniformOfFintype (PartialOffsets J a)).toOuterMeasure
{o | ∀ i, OnLine (tuplePositionEquiv d hd σ o.1 i) (g,c)}).toReal =
(Fintype.card F : ℝ)⁻¹ ^ 4 := by
rcases J.eq_empty_or_nonempty with rfl | ⟨j, hj⟩
· let e : PartialOffsets ∅ a ≃ (Fin 4 → Bool → F) :=
Equiv.subtypeUnivEquiv (by simp)
have h := congrArg (fun law => (law.toOuterMeasure
{o | ∀ i, OnLine (tuplePositionEquiv d hd σ o i) (g,c)}).toReal)
(uniform_map_equiv e)
rw [PMF.toOuterMeasure_map_apply] at h
exact h.trans (alignedOffsets_outer_probability d hd σ g c)
· have heq : J = {j} := by
apply Finset.eq_singleton_iff_unique_mem.mpr
exact ⟨hj, fun k hk => Finset.card_le_one.mp hJ k hk j hj⟩
subst J
let e : PartialOffsets {j} a ≃ ChildOffsetsAt j (a j) :=
Equiv.subtypeEquivRight (by intro o; simp)
have h := congrArg (fun law => (law.toOuterMeasure
{o : ChildOffsetsAt j (a j) |
∀ i, OnLine (tuplePositionEquiv d hd σ o.1 i) (g,c)}).toReal)
(uniform_map_equiv e)
rw [PMF.toOuterMeasure_map_apply] at h
exact h.trans (alignedChildOffsetsAt_pmf_probability d hd σ g j (hg j) c (a j))
theorem global_conditional_alignment_probability_eq (T : FourTuple F)
(hT : Function.Injective T.generator) (g : Direction F)
(hg : ∀ i, T.generator i ≠ g) (c : F)
(D : Finset (Direction F)) (hD : (exposedTupleIndices T D).card ≤ 1)
(a : MergedPairs F) :
(((PMF.uniformOfFintype (MergedPairs F)).filter {M | ∀ d, d ∈ D → M d = a d}
(fixed_supported (fun d => d ∈ D) a)).toOuterMeasure
{M | ∀ i, OnLine (tuplePoint M T i) (g,c)}).toReal =
(Fintype.card F : ℝ)⁻¹ ^ 4 := by
let hd := adjacent_ne_of_injective T hT
let E : Set (Fin 4 → Bool → F) :=
{o | ∀ i, OnLine (tuplePositionEquiv T.generator hd T.side o i) (g,c)}
have h := congrArg (fun law => (law.toOuterMeasure E).toReal)
(global_exposed_tuple_law T hT D a)
rw [PMF.toOuterMeasure_map_apply] at h
calc
_ = (((PMF.uniformOfFintype (Fin 4 → Bool → F)).filter
{o | ∀ i ∈ exposedTupleIndices T D, o i = a (T.generator i)}
(partialOffsets_supported (exposedTupleIndices T D)
(fun i => a (T.generator i)))).toOuterMeasure E).toReal := h
_ = _ := by
rw [partialOffsets_conditional_law, PMF.toOuterMeasure_map_apply]
exact partial_alignment_probability_eq T.generator hd T.side g hg c
(exposedTupleIndices T D) hD (fun i => a (T.generator i))
theorem canonical_conditional_alignment_probability_eq (T : FourTuple F)
(hT : Function.Injective T.generator) (g : Direction F)
(hg : ∀ i, T.generator i ≠ g) (c : F)
(D : Finset (Direction F)) (hD : (exposedTupleIndices T D).card ≤ 1)
(a : MergedPairs F) :
(((offsetLaw F).filter {M | ∀ d, d ∈ D → M d = a d}
(canonical_exposure_supported D a)).toOuterMeasure
{M | ∀ i, OnLine (tuplePoint M T i) (g,c)}).toReal =
(Fintype.card F : ℝ)⁻¹ ^ 4 := by
simpa only [offsetLaw_eq_uniform] using
global_conditional_alignment_probability_eq T hT g hg c D hD a
omit [Field F] [Fintype F] in
theorem card_exposedTupleIndices (T : FourTuple F)
(hT : Function.Injective T.generator) (D : Finset (Direction F)) :
(exposedTupleIndices T D).card = (generatorSet T ∩ D).card := by
have heq : (exposedTupleIndices T D).image T.generator = generatorSet T ∩ D := by
ext d
simp only [exposedTupleIndices, generatorSet, Finset.mem_image,
Finset.mem_filter, Finset.mem_univ, true_and, Finset.mem_inter]
constructor
· rintro ⟨i, hi, rfl⟩
exact ⟨⟨i, rfl⟩, hi⟩
· rintro ⟨⟨i, rfl⟩, hi⟩
exact ⟨i, hi, rfl⟩
rw [← heq, Finset.card_image_of_injective _ hT]
omit [Fintype F] in
theorem tuplePoint_eq_of_agree (T : FourTuple F) (M N : MergedPairs F)
(h : ∀ d ∈ generatorSet T, M d = N d) (j : Fin 4) :
tuplePoint M T j = tuplePoint N T j := by
have hj : T.generator j ∈ generatorSet T := by simp [generatorSet]
have hn : T.generator (j+1) ∈ generatorSet T := by simp [generatorSet]
simp only [tuplePoint, constituent, h _ hj, h _ hn]
def ListAligned {ι : Type*} (T : ι → FourTuple F) (L : ι → Line F)
(l : List ι) (M : MergedPairs F) : Prop :=
∀ i ∈ l, ∀ j, OnLine (tuplePoint M (T i) j) (L i)
omit [Fintype F] in
theorem listAligned_agree {ι : Type*} [DecidableEq ι]
(T : ι → FourTuple F) (L : ι → Line F) (l : List ι)
(M N : MergedPairs F)
(h : ∀ d ∈ l.toFinset.biUnion (fun i => generatorSet (T i)), M d = N d) :
ListAligned T L l M ↔ ListAligned T L l N := by
have hp (i : ι) (hi : i ∈ l) (j : Fin 4) :
tuplePoint M (T i) j = tuplePoint N (T i) j := by
apply tuplePoint_eq_of_agree
intro d hd
exact h d (Finset.mem_biUnion.mpr ⟨i, List.mem_toFinset.mpr hi, hd⟩)
constructor <;> intro ha i hi j
· rw [← hp i hi j]
exact ha i hi j
· rw [hp i hi j]
exact ha i hi j
theorem fixed_alignment_probability_eq (T : FourTuple F)
(hT : Function.Injective T.generator) (g : Direction F)
(hg : ∀ i, T.generator i ≠ g) (c : F)
(D : Finset (Direction F)) (hD : (exposedTupleIndices T D).card ≤ 1)
(a : MergedPairs F) :
((PMF.uniformOfFintype (FixedArray (fun d => d ∈ D) a)).toOuterMeasure
{M | ∀ j, OnLine (tuplePoint M.1 T j) (g,c)}).toReal =
(Fintype.card F : ℝ)⁻¹ ^ 4 := by
have h := global_conditional_alignment_probability_eq T hT g hg c D hD a
rw [uniform_fixed_filter, PMF.toOuterMeasure_map_apply] at h
convert h using 1
exact FiniteUniformFibers.probability_fintype_congr _ _ rfl
theorem alignment_inter_probability_eq (T : FourTuple F)
(hT : Function.Injective T.generator) (g : Direction F)
(hg : ∀ i, T.generator i ≠ g) (c : F)
(D : Finset (Direction F)) (hD : (exposedTupleIndices T D).card ≤ 1)
(E : MergedPairs F → Prop)
(hE : ∀ M N, (∀ d ∈ D, M d = N d) → (E M ↔ E N)) :
((PMF.uniformOfFintype (MergedPairs F)).toOuterMeasure
{M | E M ∧ ∀ j, OnLine (tuplePoint M T j) (g,c)}).toReal =
(Fintype.card F : ℝ)⁻¹ ^ 4 *
((PMF.uniformOfFintype (MergedPairs F)).toOuterMeasure {M | E M}).toReal := by
apply FiniteUniformFibers.fixed_probability_inter D E _ _ hE
intro a
convert fixed_alignment_probability_eq T hT g hg c D hD a using 1
exact FiniteUniformFibers.probability_fintype_congr _ _ rfl
theorem list_alignment_probability {ι : Type*} [DecidableEq ι]
(T : ι → FourTuple F) (L : ι → Line F) (l : List ι)
(hT : ∀ i ∈ l, Function.Injective (T i).generator)
(hg : ∀ i ∈ l, ∀ j, (T i).generator j ≠ (L i).1)
(horder : SmallReuseOrder (fun i => generatorSet (T i)) l) :
((PMF.uniformOfFintype (MergedPairs F)).toOuterMeasure
{M | ListAligned T L l M}).toReal =
((Fintype.card F : ℝ)⁻¹ ^ 4) ^ l.length := by
induction l with
| nil =>
simp [ListAligned]
| cons i l ih =>
have hTi := hT i (by simp)
have hgi := hg i (by simp)
have htailT : ∀ j ∈ l, Function.Injective (T j).generator :=
fun j hj => hT j (by simp [hj])
have htailg : ∀ j ∈ l, ∀ k, (T j).generator k ≠ (L j).1 :=
fun j hj => hg j (by simp [hj])
have hD : (exposedTupleIndices (T i)
(l.toFinset.biUnion (fun j => generatorSet (T j)))).card ≤ 1 := by
rw [card_exposedTupleIndices (T i) hTi]
exact horder.2
have hevent : {M | ListAligned T L (i :: l) M} =
{M | ListAligned T L l M ∧ ∀ j, OnLine (tuplePoint M (T i) j) (L i)} := by
ext M
simp only [ListAligned, List.mem_cons, forall_eq_or_imp, Set.mem_ofPred_eq]
exact and_comm
rw [hevent]
have hstep := alignment_inter_probability_eq (T i) hTi (L i).1 hgi (L i).2
(l.toFinset.biUnion (fun j => generatorSet (T j))) hD (ListAligned T L l)
(listAligned_agree T L l)
simpa only [Prod.eta, ih htailT htailg horder.1, List.length_cons,
pow_succ, mul_comm] using hstep
end
end Balanced.AlignedListProbability
end OAI
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