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namespace OAI
namespace Balanced.AlignedListProbability
open Local PartialAlignment
noncomputable section
open scoped Classical
variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
def installCurrentTupleOffsets (T : FourTuple F)
(o : Fin 4 → Bool → F) (a : MergedPairs F) : MergedPairs F :=
Function.extend T.generator o a
omit [Field F] [Fintype F] [DecidableEq F] in
@[simp] theorem installCurrentTupleOffsets_generator
(T : FourTuple F) (hT : Function.Injective T.generator)
(o : Fin 4 → Bool → F) (a : MergedPairs F) (j : Fin 4) :
installCurrentTupleOffsets T o a (T.generator j) = o j :=
hT.extend_apply o a j
omit [Field F] [Fintype F] in
theorem installCurrentTupleOffsets_outside (T : FourTuple F)
(o : Fin 4 → Bool → F) (a : MergedPairs F) (d : Direction F)
(hd : d ∉ generatorSet T) :
installCurrentTupleOffsets T o a d = a d := by
apply Function.extend_apply'
intro h
obtain ⟨j, rfl⟩ := h
exact hd (Finset.mem_image.mpr ⟨j, Finset.mem_univ _, rfl⟩)
omit [Field F] [Fintype F] in
theorem installCurrentTupleOffsets_agree_old
(T : FourTuple F) (hT : Function.Injective T.generator)
(D : Finset (Direction F)) (o : Fin 4 → Bool → F) (a : MergedPairs F)
(ho : ∀ j ∈ exposedTupleIndices T D, o j = a (T.generator j)) :
∀ d ∈ D, installCurrentTupleOffsets T o a d = a d := by
intro d hd
by_cases hg : d ∈ generatorSet T
· obtain ⟨j, _, rfl⟩ := Finset.mem_image.mp hg
rw [installCurrentTupleOffsets_generator T hT]
exact ho j (by simpa only [exposedTupleIndices, Finset.mem_filter,
Finset.mem_univ, true_and] using hd)
· exact installCurrentTupleOffsets_outside T o a d hg
def generateCurrentState
(T : FourTuple F) (hT : Function.Injective T.generator)
(g : Direction F) (hg : ∀ j, T.generator j ≠ g) (c : F)
(D : Finset (Direction F)) (hD : (exposedTupleIndices T D).card ≤ 1)
(a : MergedPairs F) (x : CurrentFreeCoordinates T D) : MergedPairs F :=
installCurrentTupleOffsets T (generateCurrentTupleOffsets T hT g hg c D hD a x) a
theorem generateCurrentState_agree_old
(T : FourTuple F) (hT : Function.Injective T.generator)
(g : Direction F) (hg : ∀ j, T.generator j ≠ g) (c : F)
(D : Finset (Direction F)) (hD : (exposedTupleIndices T D).card ≤ 1)
(a : MergedPairs F) (x : CurrentFreeCoordinates T D) :
∀ d ∈ D, generateCurrentState T hT g hg c D hD a x d = a d := by
apply installCurrentTupleOffsets_agree_old T hT D
exact ((alignedFreeCoordinatesEquiv T.generator
(TupleAlignment.adjacent_ne_of_injective T hT) T.side g hg c
(exposedTupleIndices T D) hD (fun j => a (T.generator j))).symm x).1.2
theorem generateCurrentState_aligned
(T : FourTuple F) (hT : Function.Injective T.generator)
(g : Direction F) (hg : ∀ j, T.generator j ≠ g) (c : F)
(D : Finset (Direction F)) (hD : (exposedTupleIndices T D).card ≤ 1)
(a : MergedPairs F) (x : CurrentFreeCoordinates T D) :
∀ j, OnLine (tuplePoint (generateCurrentState T hT g hg c D hD a x) T j) (g,c) := by
intro j
have h := ((alignedFreeCoordinatesEquiv T.generator
(TupleAlignment.adjacent_ne_of_injective T hT) T.side g hg c
(exposedTupleIndices T D) hD (fun k => a (T.generator k))).symm x).2 j
simpa only [generateCurrentState, tuplePoint, constituent,
installCurrentTupleOffsets_generator T hT,
TupleAlignment.tuplePositionEquiv_apply, generateCurrentTupleOffsets] using h
omit [Field F] [Fintype F] in
theorem installCurrentTupleOffsets_reconstruct
(T : FourTuple F) (hT : Function.Injective T.generator)
(D : Finset (Direction F)) (a M : MergedPairs F)
(hfix : ∀ d ∈ D, M d = a d) :
∀ d ∈ D ∪ generatorSet T,
M d = installCurrentTupleOffsets T (fun j => M (T.generator j)) a d := by
intro d hd
by_cases hg : d ∈ generatorSet T
· obtain ⟨j, _, rfl⟩ := Finset.mem_image.mp hg
rw [installCurrentTupleOffsets_generator T hT]
· rw [installCurrentTupleOffsets_outside T _ a d hg]
exact hfix d ((Finset.mem_union.mp hd).resolve_right hg)
omit [Field F] in
theorem conditionedOffsetProbability_congr_on (A E G : MergedPairs F → Prop)
(h : ∀ M, A M → (E M ↔ G M)) :
conditionedOffsetProbability A E = conditionedOffsetProbability A G := by
have he : (fun M => A M ∧ E M) = (fun M => A M ∧ G M) := by
funext M
apply propext
by_cases ha : A M
· simp only [ha, true_and, h M ha]
· simp only [ha, false_and]
unfold conditionedOffsetProbability
rw [he]
theorem current_state_generated_probability
(T : FourTuple F) (hT : Function.Injective T.generator)
(g : Direction F) (hg : ∀ j, T.generator j ≠ g) (c : F)
(D : Finset (Direction F)) (hD : (exposedTupleIndices T D).card ≤ 1)
(a : MergedPairs F) (E : MergedPairs F → Prop)
(hE : ∀ M N, (∀ d ∈ D ∪ generatorSet T, M d = N d) → (E M ↔ E N)) :
conditionedOffsetProbability
(fun M => (∀ d ∈ D, M d = a d) ∧ ∀ j, OnLine (tuplePoint M T j) (g,c)) E =
((PMF.uniformOfFintype (CurrentFreeCoordinates T D)).toOuterMeasure
{x | E (generateCurrentState T hT g hg c D hD a x)}).toReal := by
let G : (Fin 4 → Bool → F) → Prop := fun o => E (installCurrentTupleOffsets T o a)
have hc := conditionedOffsetProbability_congr_on
(fun M => (∀ d ∈ D, M d = a d) ∧ ∀ j, OnLine (tuplePoint M T j) (g,c))
E (fun M => G (fun j => M (T.generator j)))
(fun M hm => hE M _ (installCurrentTupleOffsets_reconstruct T hT D a M hm.1))
exact hc.trans (current_tuple_generated_probability T hT g hg c D hD a G)
theorem generateCurrentState_freePoint
(T : FourTuple F) (hT : Function.Injective T.generator)
(g : Direction F) (hg : ∀ j, T.generator j ≠ g) (c : F)
(D : Finset (Direction F)) (hD : (exposedTupleIndices T D).card ≤ 1)
(a : MergedPairs F) (x : CurrentFreeCoordinates T D)
(j : {j : Fin 4 // j ∈ freePositions (exposedTupleIndices T D)}) :
tuplePoint (generateCurrentState T hT g hg c D hD a x) T j.1 =
((TupleAlignment.linePointEquiv g c).symm (x j)).1 := by
have h := alignedPartial_generated_free T.generator
(TupleAlignment.adjacent_ne_of_injective T hT) T.side g hg c
(exposedTupleIndices T D) hD (fun k => a (T.generator k)) x j
simpa only [generateCurrentState, tuplePoint, constituent,
installCurrentTupleOffsets_generator T hT,
TupleAlignment.tuplePositionEquiv_apply, generateCurrentTupleOffsets] using h
theorem generateCurrentState_old_left
(T : FourTuple F) (hT : Function.Injective T.generator)
(g : Direction F) (hg : ∀ j, T.generator j ≠ g) (c : F)
(D : Finset (Direction F)) (hD : (exposedTupleIndices T D).card ≤ 1)
(a : MergedPairs F) (x : CurrentFreeCoordinates T D)
(j : Fin 4) (hj : T.generator j ∈ D) :
tuplePoint (generateCurrentState T hT g hg c D hD a x) T j =
(TupleAlignment.lineEvalEquiv (T.generator j) g (hg j)).symm
(a (T.generator j) (T.side j), c) := by
have h := alignedPartial_position_old_left T.generator
(TupleAlignment.adjacent_ne_of_injective T hT) T.side g hg c
(exposedTupleIndices T D) (fun k => a (T.generator k))
((alignedFreeCoordinatesEquiv T.generator
(TupleAlignment.adjacent_ne_of_injective T hT) T.side g hg c
(exposedTupleIndices T D) hD (fun k => a (T.generator k))).symm x)
j (by simpa only [exposedTupleIndices, Finset.mem_filter,
Finset.mem_univ, true_and] using hj)
simpa only [generateCurrentState, tuplePoint, constituent,
installCurrentTupleOffsets_generator T hT,
TupleAlignment.tuplePositionEquiv_apply, generateCurrentTupleOffsets] using h
theorem generateCurrentState_old_right
(T : FourTuple F) (hT : Function.Injective T.generator)
(g : Direction F) (hg : ∀ j, T.generator j ≠ g) (c : F)
(D : Finset (Direction F)) (hD : (exposedTupleIndices T D).card ≤ 1)
(a : MergedPairs F) (x : CurrentFreeCoordinates T D)
(j : Fin 4) (hj : T.generator (j+1) ∈ D) :
tuplePoint (generateCurrentState T hT g hg c D hD a x) T j =
(TupleAlignment.lineEvalEquiv (T.generator (j+1)) g (hg (j+1))).symm
(a (T.generator (j+1)) (!(T.side (j+1))), c) := by
have h := alignedPartial_position_old_right T.generator
(TupleAlignment.adjacent_ne_of_injective T hT) T.side g hg c
(exposedTupleIndices T D) (fun k => a (T.generator k))
((alignedFreeCoordinatesEquiv T.generator
(TupleAlignment.adjacent_ne_of_injective T hT) T.side g hg c
(exposedTupleIndices T D) hD (fun k => a (T.generator k))).symm x)
j (by simpa only [exposedTupleIndices, Finset.mem_filter,
Finset.mem_univ, true_and] using hj)
simpa only [generateCurrentState, tuplePoint, constituent,
installCurrentTupleOffsets_generator T hT,
TupleAlignment.tuplePositionEquiv_apply, generateCurrentTupleOffsets] using h
theorem generateCurrentState_listAligned {ι : Type*} [DecidableEq ι]
(T : ι → FourTuple F) (L : ι → Line F) (pre : List ι) (i : ι)
(hi : Function.Injective (T i).generator)
(hg : ∀ j, (T i).generator j ≠ (L i).1)
(hD : (exposedTupleIndices (T i)
(pre.toFinset.biUnion (fun k => generatorSet (T k)))).card ≤ 1)
(a : MergedPairs F) (ha : ListAligned T L pre a)
(x : CurrentFreeCoordinates (T i)
(pre.toFinset.biUnion (fun k => generatorSet (T k)))) :
ListAligned T L (i :: pre)
(generateCurrentState (T i) hi (L i).1 hg (L i).2
(pre.toFinset.biUnion (fun k => generatorSet (T k))) hD a x) := by
intro k hk
rcases List.mem_cons.mp hk with hki | hk
· subst k
exact generateCurrentState_aligned (T i) hi (L i).1 hg (L i).2 _ hD a x
· have hp := (listAligned_agree T L pre _ a
(generateCurrentState_agree_old (T i) hi (L i).1 hg (L i).2 _ hD a x)).mpr ha
exact hp k hk
theorem full_aligned_state_generated_probability {ι : Type*} [DecidableEq ι]
(T : ι → FourTuple F) (L : ι → Line F) (post pre : List ι) (i : ι)
(hT : ∀ k ∈ post ++ i :: pre, Function.Injective (T k).generator)
(hg : ∀ k ∈ post ++ i :: pre, ∀ j, (T k).generator j ≠ (L k).1)
(horder : SmallReuseOrder (fun k => generatorSet (T k)) (post ++ i :: pre))
(a : MergedPairs F) (ha : ListAligned T L pre a)
(E : MergedPairs F → Prop)
(hE : ∀ M N, (∀ d ∈ (i :: pre).toFinset.biUnion (fun k => generatorSet (T k)),
M d = N d) → (E M ↔ E N)) :
let D := pre.toFinset.biUnion (fun k => generatorSet (T k))
conditionedOffsetProbability
(fun M => ListAligned T L (post ++ i :: pre) M ∧ ∀ d ∈ D, M d = a d) E =
((PMF.uniformOfFintype (CurrentFreeCoordinates (T i) D)).toOuterMeasure
{x | E (generateCurrentState (T i) (hT i (by simp)) (L i).1
(hg i (by simp)) (L i).2 D
(step_exposed_count T post pre i (hT i (by simp)) horder) a x)}).toReal := by
let D := pre.toFinset.biUnion (fun k => generatorSet (T k))
have hdep (M N : MergedPairs F)
(h : ∀ d ∈ D ∪ generatorSet (T i), M d = N d) : E M ↔ E N := by
apply hE M N
intro d hd
apply h d
simpa only [D, List.toFinset_cons, Finset.biUnion_insert, Finset.union_comm] using hd
have hs := conditioned_alignment_step T L post pre i hT hg horder a ha E hE
have hc := current_state_generated_probability (T i) (hT i (by simp)) (L i).1
(hg i (by simp)) (L i).2 D
(step_exposed_count T post pre i (hT i (by simp)) horder) a E hdep
exact hs.trans (by simpa only [and_comm, Prod.eta] using hc)
end
end Balanced.AlignedListProbability
end OAI
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