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import OAI.LinearAlgebra.MatrixMultiplication.Entropy.FiniteAverage
import OAI.LinearAlgebra.MatrixMultiplication.Recovery.OrbitCounting
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.Canonicalization
import OAI.LinearAlgebra.MatrixMultiplication.Recovery.EquivOrbitTransport
import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedMaskOrbitFilter
import Mathlib.Data.Fintype.Perm
namespace OAI
/-! Finite orbit symmetries, masks and exact recovery operations. -/
open scoped BigOperators
noncomputable section
namespace MatrixMultiplication.GroupOrbitProjection
open PermutationMatching
attribute [local instance] Classical.propDecidable Classical.decEq
section UniformRestriction
variable {H S : Type*} [Fintype H] [Fintype S] [DecidableEq H] [DecidableEq S]
def curryFamily (X : H × S → Type*) : (∀ c, X c) ≃ (∀ h s, X (h, s)) where
toFun x h s := x (h, s)
invFun x c := x c.1 c.2
left_inv _ := rfl
right_inv _ := rfl
theorem average_subtype_restrict {X : H → Type*} [∀ h, Fintype (X h)]
[∀ h, Nonempty (X h)] (p : H → Prop) [DecidablePred p]
(f : (∀ h : {h // p h}, X h.val) → ℝ) :
average (fun x : ∀ h, X h => f (fun h => x h.val)) = average f := by
change average (fun x =>
(fun z : (∀ h : {h // p h}, X h.val) × (∀ h : {h // ¬p h}, X h.val) => f z.1)
((Equiv.piEquivPiSubtypeProd p X) x)) = average f
rw [average_equiv (Equiv.piEquivPiSubtypeProd p X)
(fun z : (∀ h : {h // p h}, X h.val) × (∀ h : {h // ¬p h}, X h.val) => f z.1),
average_prod_fst]
theorem average_history_restrict {X : H × S → Type*} [∀ c, Fintype (X c)]
[∀ c, Nonempty (X c)] (p : H → Prop) [DecidablePred p]
(f : (∀ c : {h // p h} × S, X (c.1.val, c.2)) → ℝ) :
average (fun x : ∀ c, X c => f (fun c => x (c.1.val, c.2))) = average f := by
calc
_ = average (fun x : ∀ h s, X (h, s) => f (fun c => x c.1.val c.2)) :=
average_equiv (curryFamily X) _
_ = average (fun x : ∀ h : {h // p h}, ∀ s, X (h.val, s) =>
f (fun c => x c.1 c.2)) :=
average_subtype_restrict (X := fun h => ∀ s, X (h, s)) p
(fun x => f (fun c => x c.1 c.2))
_ = average f :=
average_equiv (curryFamily (fun c : {h // p h} × S => X (c.1.val, c.2))).symm f
theorem average_half_history_restrict {X : H × S → Type*} [∀ c, Fintype (X c)]
[∀ c, Nonempty (X c)] (p : H → Prop) [DecidablePred p]
(f : ((∀ c : {h // p h} × S, X (c.1.val, c.2)) ×
(∀ c : {h // p h} × S, X (c.1.val, c.2))) → ℝ) :
average (fun g : (∀ c, X c) × (∀ c, X c) =>
f ((fun c => g.1 (c.1.val, c.2)), (fun c => g.2 (c.1.val, c.2)))) =
average f := by
rw [average_prod_average]
calc
_ = average (fun x : ∀ c, X c =>
average (fun y : ∀ c : {h // p h} × S, X (c.1.val, c.2) =>
f ((fun c => x (c.1.val, c.2)), y))) := by
apply average_congr
intro x
exact average_history_restrict (X := X) p
(fun y => f ((fun c => x (c.1.val, c.2)), y))
_ = average (fun x : ∀ c : {h // p h} × S, X (c.1.val, c.2) =>
average (fun y => f (x, y))) :=
average_history_restrict (X := X) p (fun x => average (fun y => f (x, y)))
_ = average f := (average_prod_average f).symm
end UniformRestriction
section OrbitFraction
variable {G H X Y : Type*} [Group G] [Group H] [Fintype G] [Fintype H]
[MulAction G X] [MulAction H Y]
theorem orbit_fraction_of_uniform_projection (project : X → Y) (shift : G → H)
(hsmul : ∀ g x, project (g • x) = shift g • project x)
(huniform : ∀ f : H → ℝ, average (fun g => f (shift g)) = average f)
(x : X) (bad : Y → Prop) :
(((OrbitCounting.orbitSet (G := G) x).filter (fun v => bad (project v))).card : ℝ) /
(OrbitCounting.orbitSet (G := G) x).card =
(((OrbitCounting.orbitSet (G := H) (project x)).filter bad).card : ℝ) /
(OrbitCounting.orbitSet (G := H) (project x)).card := by
rw [← OrbitCounting.bad_shift_fraction (G := G) x (fun v => bad (project v)),
← OrbitCounting.bad_shift_fraction (G := H) (project x) bad]
rw [← average_indicator_eq, ← average_indicator_eq]
simp_rw [hsmul]
exact huniform (fun g => if bad (g • project x) then (1 : ℝ) else 0)
theorem rejected_fraction_of_uniform_projection (project : X → Y) (shift : G → H)
(hsmul : ∀ g x, project (g • x) = shift g • project x)
(huniform : ∀ f : H → ℝ, average (fun g => f (shift g)) = average f)
(x : X) (bad : Y → Prop) :
((rejectedFilter (OrbitCounting.orbitSet (G := G) x)
(fun v => bad (project v))).card : ℝ) /
(OrbitCounting.orbitSet (G := G) x).card =
((rejectedFilter (OrbitCounting.orbitSet (G := H) (project x)) bad).card : ℝ) /
(OrbitCounting.orbitSet (G := H) (project x)).card := by
simpa only [rejectedFilter, Finset.filter_congr_decidable] using
orbit_fraction_of_uniform_projection project shift hsmul huniform x bad
end OrbitFraction
section ConjugatedProjection
variable {G H RX RY CX CY : Type*} [Group G] [Group H]
[MulAction G CX] [MulAction H CY]
theorem conjugated_projection_smul (ex : RX ≃ CX) (ey : RY ≃ CY)
(raw : RX → RY) (canonical : CX → CY) (shift : G → H)
(hc : ∀ x, ey (raw x) = canonical (ex x))
(hs : ∀ g x, canonical (g • x) = shift g • canonical x) :
letI : MulAction G RX := EquivOrbitTransport.action ex
letI : MulAction H RY := EquivOrbitTransport.action ey
∀ g x, raw (g • x) = shift g • raw x := by
let : MulAction G RX := EquivOrbitTransport.action ex
let : MulAction H RY := EquivOrbitTransport.action ey
intro g x
apply ey.injective
rw [hc, EquivOrbitTransport.equiv_smul ex, hs,
EquivOrbitTransport.equiv_smul ey, hc]
end ConjugatedProjection
def transportedOrbit {G RX CX : Type*} [Group G] [Fintype G] [MulAction G CX]
(ex : RX ≃ CX) (x : RX) : Finset RX :=
@OrbitCounting.orbitSet G RX _ (EquivOrbitTransport.action ex) _ _ x
theorem transported_projection_fraction {G H RX RY CX CY : Type*}
[Group G] [Group H] [Fintype G] [Fintype H]
[MulAction G CX] [MulAction H CY]
(ex : RX ≃ CX) (ey : RY ≃ CY)
(raw : RX → RY) (canonical : CX → CY) (shift : G → H)
(hc : ∀ x, ey (raw x) = canonical (ex x))
(hs : ∀ g x, canonical (g • x) = shift g • canonical x)
(huniform : ∀ f : H → ℝ, average (fun g => f (shift g)) = average f)
(x : RX) (bad : RY → Prop) :
((rejectedFilter (transportedOrbit (G := G) ex x)
(fun v => bad (raw v))).card : ℝ) / (transportedOrbit (G := G) ex x).card =
((rejectedFilter (transportedOrbit (G := H) ey (raw x)) bad).card : ℝ) /
(transportedOrbit (G := H) ey (raw x)).card :=
@rejected_fraction_of_uniform_projection G H RX RY _ _ _ _
(EquivOrbitTransport.action ex) (EquivOrbitTransport.action ey)
raw shift (conjugated_projection_smul ex ey raw canonical shift hc hs) huniform x bad
section CompleteHistories
variable {H S : Type*} [Fintype H] [Fintype S] [DecidableEq H] [DecidableEq S]
(P X Y : H × S → Type*) [∀ c, Fintype (P c)] [∀ c, DecidableEq (P c)]
def projectPair (p : H → Prop) (w : CompleteWordPair P X Y) :
CompleteWordPair (fun c : {h // p h} × S => P (c.1.val, c.2))
(fun c => X (c.1.val, c.2)) (fun c => Y (c.1.val, c.2)) where
left c i := w.left (c.1.val, c.2) i
right c i := w.right (c.1.val, c.2) i
def projectShifts (p : H → Prop) (g : HalfClassPermutations P) :
HalfClassPermutations (fun c : {h // p h} × S => P (c.1.val, c.2)) :=
((fun c => g.1 (c.1.val, c.2)), (fun c => g.2 (c.1.val, c.2)))
omit [Fintype H] [Fintype S] [DecidableEq H] [DecidableEq S]
[∀ c, Fintype (P c)] [∀ c, DecidableEq (P c)] in
theorem projectPair_smul (p : H → Prop) (g : HalfClassPermutations P)
(w : CompleteWordPair P X Y) :
projectPair P X Y p (g • w) = projectShifts P p g • projectPair P X Y p w := rfl
theorem complete_orbit_projection_fraction (p : H → Prop) [DecidablePred p]
(w : CompleteWordPair P X Y)
(bad : CompleteWordPair (fun c : {h // p h} × S => P (c.1.val, c.2))
(fun c => X (c.1.val, c.2)) (fun c => Y (c.1.val, c.2)) → Prop) :
(((OrbitCounting.orbitSet (G := HalfClassPermutations P) w).filter
(fun v => bad (projectPair P X Y p v))).card : ℝ) /
(OrbitCounting.orbitSet (G := HalfClassPermutations P) w).card =
(((OrbitCounting.orbitSet
(G := HalfClassPermutations (fun c : {h // p h} × S => P (c.1.val, c.2)))
(projectPair P X Y p w)).filter bad).card : ℝ) /
(OrbitCounting.orbitSet
(G := HalfClassPermutations (fun c : {h // p h} × S => P (c.1.val, c.2)))
(projectPair P X Y p w)).card := by
apply orbit_fraction_of_uniform_projection (projectPair P X Y p) (projectShifts P p)
· exact projectPair_smul P X Y p
· intro f
exact average_half_history_restrict (X := fun c => Equiv.Perm (P c)) p f
end CompleteHistories
section UnionBound
variable {G X I : Type*} [Group G] [Fintype G] [MulAction G X] [Fintype I]
theorem orbit_union_fraction_le (x : X) (bad : I → X → Prop) (bound : I → ℝ)
(hbound : ∀ i,
(((OrbitCounting.orbitSet (G := G) x).filter (bad i)).card : ℝ) /
(OrbitCounting.orbitSet (G := G) x).card ≤ bound i) :
(((OrbitCounting.orbitSet (G := G) x).filter (fun v => ∃ i, bad i v)).card : ℝ) /
(OrbitCounting.orbitSet (G := G) x).card ≤ ∑ i, bound i := by
rw [← OrbitCounting.bad_shift_fraction (G := G) x (fun v => ∃ i, bad i v),
← average_indicator_eq]
calc
_ ≤ average (fun g : G => ∑ i, if bad i (g • x) then (1 : ℝ) else 0) := by
apply average_mono
intro g
split_ifs with hex
· obtain ⟨i, hi⟩ := hex
have hterm : (if bad i (g • x) then (1 : ℝ) else 0) = 1 := ite_eq_left hi
exact hterm.symm.le.trans (Finset.single_le_sum
(f := fun j : I => if bad j (g • x) then (1 : ℝ) else 0)
(fun j _ => by split_ifs <;> norm_num) (Finset.mem_univ i))
· exact Finset.sum_nonneg (fun i _ => by split_ifs <;> norm_num)
_ = ∑ i, average (fun g : G => if bad i (g • x) then (1 : ℝ) else 0) :=
average_sum _ _
_ ≤ ∑ i, bound i := by
apply Finset.sum_le_sum
intro i _
rw [average_indicator_eq, OrbitCounting.bad_shift_fraction (G := G) x (bad i)]
exact hbound i
end UnionBound
section FinsetUnion
variable {X I : Type*} [Fintype I]
theorem finset_union_fraction_le (samples : Finset X) (bad : I → X → Prop)
(bound : I → ℝ)
(hbound : ∀ i, (((samples.filter (bad i)).card : ℝ) / samples.card) ≤ bound i) :
(((samples.filter (fun v => ∃ i, bad i v)).card : ℝ) / samples.card) ≤
∑ i, bound i := by
have hset : samples.filter (fun v => ∃ i, bad i v) =
Finset.univ.biUnion (fun i => samples.filter (bad i)) := by
ext v
simp only [Finset.mem_filter, Finset.mem_biUnion, Finset.mem_univ, true_and]
exact ⟨fun ⟨hv, i, hi⟩ => ⟨i, hv, hi⟩, fun ⟨i, hv, hi⟩ => ⟨hv, i, hi⟩⟩
have hcard : ((samples.filter (fun v => ∃ i, bad i v)).card : ℝ) ≤
∑ i, ((samples.filter (bad i)).card : ℝ) := by
rw [hset, ← Nat.cast_sum]
exact Nat.cast_le.mpr Finset.card_biUnion_le
calc
_ ≤ (∑ i, ((samples.filter (bad i)).card : ℝ)) / samples.card :=
div_le_div_of_nonneg_right hcard (Nat.cast_nonneg _)
_ = ∑ i, ((samples.filter (bad i)).card : ℝ) / samples.card := Finset.sum_div _ _ _
_ ≤ ∑ i, bound i := Finset.sum_le_sum (fun i _ => hbound i)
theorem rejected_union_fraction_le (samples : Finset X) (bad : I → X → Prop)
(bound : I → ℝ)
(hbound : ∀ i, ((rejectedFilter samples (bad i)).card : ℝ) / samples.card ≤ bound i) :
((rejectedFilter samples (fun v => ∃ i, bad i v)).card : ℝ) / samples.card ≤
∑ i, bound i := by
have hh (i : I) : ((samples.filter (bad i)).card : ℝ) / samples.card ≤ bound i := by
simpa only [rejectedFilter, Finset.filter_congr_decidable] using hbound i
have heq : samples.filter (fun v => ∃ i, bad i v) =
rejectedFilter samples (fun v => ∃ i, bad i v) := by
ext v
simp only [rejectedFilter, Finset.mem_filter]
rw [← heq]
exact finset_union_fraction_le samples bad bound hh
end FinsetUnion
section GroupedHistories
variable {H S I : Type*} [Fintype H] [Fintype S] [Fintype I]
[DecidableEq H] [DecidableEq S] [DecidableEq I]
(P X Y : H × S → Type*) [∀ c, Fintype (P c)] [∀ c, DecidableEq (P c)]
theorem complete_group_union_fraction_le (group : H → I)
(w : CompleteWordPair P X Y)
(bad : ∀ i, CompleteWordPair
(fun c : {h // group h = i} × S => P (c.1.val, c.2))
(fun c => X (c.1.val, c.2)) (fun c => Y (c.1.val, c.2)) → Prop)
(bound : I → ℝ)
(hbound : ∀ i,
(((OrbitCounting.orbitSet
(G := HalfClassPermutations (fun c : {h // group h = i} × S => P (c.1.val, c.2)))
(projectPair P X Y (fun h => group h = i) w)).filter (bad i)).card : ℝ) /
(OrbitCounting.orbitSet
(G := HalfClassPermutations (fun c : {h // group h = i} × S => P (c.1.val, c.2)))
(projectPair P X Y (fun h => group h = i) w)).card ≤ bound i) :
(((OrbitCounting.orbitSet (G := HalfClassPermutations P) w).filter
(fun v => ∃ i, bad i (projectPair P X Y (fun h => group h = i) v))).card : ℝ) /
(OrbitCounting.orbitSet (G := HalfClassPermutations P) w).card ≤ ∑ i, bound i := by
apply orbit_union_fraction_le
intro i
rw [complete_orbit_projection_fraction]
exact hbound i
end GroupedHistories
open JointPopulation JointCanonicalization
variable {H : Type*} [Fintype H] [DecidableEq H]
(counts : H → Shape → ℕ) (L R : H → Type*)
def groupCounts (p : H → Prop) : {h // p h} → Shape → ℕ := fun h => counts h.val
def projectTarget (p : H → Prop) (e : Target counts) : Target (groupCounts counts p) :=
fun h => e h.val
def projectRaw (p : H → Prop) (w : RawPairs counts L R) :
RawPairs (groupCounts counts p) (fun h => L h.val) (fun h => R h.val) :=
fun h => w h.val
omit [Fintype H] [DecidableEq H] in
theorem pairEquiv_projectRaw (p : H → Prop) [DecidablePred p] (e : Target counts)
(w : RawPairs counts L R) :
pairEquiv (groupCounts counts p) (fun h => L h.val) (fun h => R h.val)
(projectTarget counts p e) (projectRaw counts L R p w) =
projectPair (ClassPositions counts) (fun c => L c.1) (fun c => R c.1) p
(pairEquiv counts L R e w) := rfl
omit [Fintype H] [DecidableEq H] in
theorem projectRaw_pairEquiv_symm (p : H → Prop) [DecidablePred p] (e : Target counts)
(v : CanonicalPairs counts L R) :
projectRaw counts L R p ((pairEquiv counts L R e).symm v) =
(pairEquiv (groupCounts counts p) (fun h => L h.val) (fun h => R h.val)
(projectTarget counts p e)).symm
(projectPair (ClassPositions counts) (fun c => L c.1) (fun c => R c.1) p v) := by
exact (Equiv.eq_symm_apply
(pairEquiv (groupCounts counts p) (fun h => L h.val) (fun h => R h.val)
(projectTarget counts p e))).mpr
((pairEquiv_projectRaw counts L R p e ((pairEquiv counts L R e).symm v)).trans
(congrArg
(projectPair (ClassPositions counts) (fun c => L c.1) (fun c => R c.1) p)
((pairEquiv counts L R e).apply_symm_apply v)))
def rawOrbit (e : Target counts) (w : RawPairs counts L R) :
Finset (RawPairs counts L R) :=
transportedOrbit (G := HalfClassPermutations (ClassPositions counts))
(pairEquiv counts L R e) w
theorem raw_orbit_projection_fraction (p : H → Prop) [DecidablePred p] (e : Target counts)
(w : RawPairs counts L R)
(bad : RawPairs (groupCounts counts p) (fun h => L h.val) (fun h => R h.val) → Prop) :
((rejectedFilter (rawOrbit counts L R e w)
(fun v => bad (projectRaw counts L R p v))).card : ℝ) /
(rawOrbit counts L R e w).card =
((rejectedFilter (rawOrbit (groupCounts counts p) (fun h => L h.val) (fun h => R h.val)
(projectTarget counts p e) (projectRaw counts L R p w)) bad).card : ℝ) /
(rawOrbit (groupCounts counts p) (fun h => L h.val) (fun h => R h.val)
(projectTarget counts p e) (projectRaw counts L R p w)).card := by
exact transported_projection_fraction
(G := HalfClassPermutations (ClassPositions counts))
(H := HalfClassPermutations (ClassPositions (groupCounts counts p)))
(RX := RawPairs counts L R)
(RY := RawPairs (groupCounts counts p) (fun h => L h.val) (fun h => R h.val))
(CX := CanonicalPairs counts L R)
(CY := CanonicalPairs (groupCounts counts p) (fun h => L h.val) (fun h => R h.val))
(pairEquiv counts L R e)
(pairEquiv (groupCounts counts p) (fun h => L h.val) (fun h => R h.val)
(projectTarget counts p e))
(projectRaw counts L R p)
(projectPair (ClassPositions counts) (fun c => L c.1) (fun c => R c.1) p)
(projectShifts (ClassPositions counts) p)
(pairEquiv_projectRaw counts L R p e)
(projectPair_smul (H := H) (S := Shape) (ClassPositions counts)
(fun c => L c.1) (fun c => R c.1) p)
(fun f => average_half_history_restrict (H := H) (S := Shape)
(X := fun c => Equiv.Perm (ClassPositions counts c)) p f) w bad
theorem raw_group_passing_rejection_fraction_le {I : Type*} [Fintype I] [DecidableEq I]
(group : H → I) (e : Target counts) (w : RawPairs counts L R)
(keep : RawPairs counts L R → Prop)
(localKeep assigned : ∀ i : I, RawPairs (groupCounts counts (fun h => group h = i))
(fun h => L h.val) (fun h => R h.val) → Prop)
(hkeep : ∀ v, keep v → ∀ i,
localKeep i (projectRaw counts L R (fun h => group h = i) v))
(bound : I → ℝ)
(hbound : ∀ i,
((rejectedFilter (rawOrbit (groupCounts counts (fun h => group h = i))
(fun h => L h.val) (fun h => R h.val)
(projectTarget counts (fun h => group h = i) e)
(projectRaw counts L R (fun h => group h = i) w))
(fun v => localKeep i v ∧ ¬assigned i v)).card : ℝ) /
(rawOrbit (groupCounts counts (fun h => group h = i))
(fun h => L h.val) (fun h => R h.val)
(projectTarget counts (fun h => group h = i) e)
(projectRaw counts L R (fun h => group h = i) w)).card ≤ bound i) :
((rejectedFilter (rawOrbit counts L R e w)
(fun v => keep v ∧ ¬∀ i, assigned i
(projectRaw counts L R (fun h => group h = i) v))).card : ℝ) /
(rawOrbit counts L R e w).card ≤ ∑ i, bound i := by
let bad : I → RawPairs counts L R → Prop := fun i v =>
localKeep i (projectRaw counts L R (fun h => group h = i) v) ∧
¬assigned i (projectRaw counts L R (fun h => group h = i) v)
have hsub : rejectedFilter (rawOrbit counts L R e w)
(fun v => keep v ∧ ¬∀ i, assigned i
(projectRaw counts L R (fun h => group h = i) v)) ⊆
rejectedFilter (rawOrbit counts L R e w) (fun v => ∃ i, bad i v) := by
intro v hv
simp only [rejectedFilter, Finset.mem_filter] at hv ⊢
obtain ⟨hmem, hk, ha⟩ := hv
obtain ⟨i, hi⟩ := not_forall.mp ha
exact ⟨hmem, i, hkeep v hk i, hi⟩
have hlocal (i : I) :
((rejectedFilter (rawOrbit counts L R e w) (bad i)).card : ℝ) /
(rawOrbit counts L R e w).card ≤ bound i :=
(raw_orbit_projection_fraction counts L R (fun h => group h = i) e w
(fun v => localKeep i v ∧ ¬assigned i v)).le.trans (hbound i)
have hunion : ((rejectedFilter (rawOrbit counts L R e w)
(fun v => ∃ i, bad i v)).card : ℝ) /
(rawOrbit counts L R e w).card ≤ ∑ i, bound i :=
rejected_union_fraction_le (rawOrbit counts L R e w) bad bound hlocal
have hcard : ((rejectedFilter (rawOrbit counts L R e w)
(fun v => keep v ∧ ¬∀ i, assigned i
(projectRaw counts L R (fun h => group h = i) v))).card : ℝ) ≤
((rejectedFilter (rawOrbit counts L R e w) (fun v => ∃ i, bad i v)).card : ℝ) :=
Nat.cast_le.mpr (Finset.card_le_card hsub)
exact (div_le_div_of_nonneg_right hcard
(Nat.cast_nonneg (rawOrbit counts L R e w).card)).trans hunion
end MatrixMultiplication.GroupOrbitProjection
end
end OAI
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