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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 | import Mathlib.Data.List.NodupEquivFin
import OAI.Combinatorics.BalancedRyser.Local.Local
import OAI.Combinatorics.BalancedRyser.Local.SmallReusePermutation
namespace OAI
namespace Balanced.Local.EqualityPattern
open Finset
def classSet {b κ : ℕ} (P : EqualityPattern b κ) (i : Fin b) :
Finset (Fin (4 * b - κ)) := Finset.univ.image (P.classOf i)
@[simp]
theorem card_classSet {b κ : ℕ} (P : EqualityPattern b κ) (i : Fin b) :
(P.classSet i).card = 4 := by
rw [classSet, Finset.card_image_of_injective _ (P.withinTuple i)]
simp
theorem biUnion_classSet {b κ : ℕ} (P : EqualityPattern b κ) :
Finset.univ.biUnion P.classSet = Finset.univ := by
apply Finset.eq_univ_of_forall
intro a
obtain ⟨⟨i, j⟩, hij⟩ := P.everyClass a
apply Finset.mem_biUnion.mpr
refine ⟨i, Finset.mem_univ _, ?_⟩
exact Finset.mem_image.mpr ⟨j, Finset.mem_univ _, hij⟩
@[simp]
theorem card_biUnion_classSet {b κ : ℕ} (P : EqualityPattern b κ) :
(Finset.univ.biUnion P.classSet).card = 4 * b - κ := by
rw [P.biUnion_classSet]
simp
@[simp]
theorem sum_card_classSet {b κ : ℕ} (P : EqualityPattern b κ) :
(∑ i : Fin b, (P.classSet i).card) = 4 * b := by
simp [Nat.mul_comm]
theorem classSet_reuse_le {b κ : ℕ} (P : EqualityPattern b κ) :
(∑ i : Fin b, (P.classSet i).card) -
(Finset.univ.biUnion P.classSet).card ≤ κ := by
rw [P.sum_card_classSet, P.card_biUnion_classSet]
omega
theorem classSet_reuse_eq {b κ : ℕ} (P : EqualityPattern b κ) (hκ : κ ≤ 4 * b) :
(∑ i : Fin b, (P.classSet i).card) -
(Finset.univ.biUnion P.classSet).card = κ := by
rw [P.sum_card_classSet, P.card_biUnion_classSet]
omega
theorem exists_smallReuseOrder {b κ : ℕ} (P : EqualityPattern b κ) (hκ : κ ≤ 2) :
∃ l : List (Fin b), l.Nodup ∧ l.toFinset = Finset.univ ∧
Balanced.SmallReuseOrder P.classSet l := by
apply Balanced.exists_smallReuseOrder P.classSet Finset.univ
· intro i _ j _ hij
exact P.pairOverlap i j hij
· exact P.classSet_reuse_le.trans hκ
theorem exists_processing_perm {b κ : ℕ} (P : EqualityPattern b κ) (hκ : κ ≤ 2) :
∃ π : Equiv.Perm (Fin b), ∀ i : Fin b,
(P.classSet (π i) ∩ (Finset.univ.filter (fun j : Fin b => j < i)).biUnion
(fun j => P.classSet (π j))).card ≤ 1 := by
obtain ⟨l, hnd, hfull, horder⟩ := P.exists_smallReuseOrder hκ
exact Balanced.exists_perm_of_list_tail_overlap P.classSet l hnd hfull
horder.tail_overlap
theorem generatorSet_prescribedTuple {F : Type*} [DecidableEq F]
{b κ : ℕ} (P : EqualityPattern b κ) (a : DirectionAssignment F b κ)
(bits : Bits b) (i : Fin b) :
generatorSet (prescribedTuple P a bits i) = (P.classSet i).image a.2 := by
simp [generatorSet, prescribedTuple, classSet, Finset.image_image, Function.comp_def]
theorem prescribed_generator_overlap_card {F : Type*} [DecidableEq F]
{b κ : ℕ} (P : EqualityPattern b κ) (a : DirectionAssignment F b κ)
(hinj : Function.Injective a.2) (bits : Bits b) (i : Fin b) (s : Finset (Fin b)) :
((generatorSet (prescribedTuple P a bits i)) ∩
s.biUnion (fun j => generatorSet (prescribedTuple P a bits j))).card =
(P.classSet i ∩ s.biUnion P.classSet).card := by
simp_rw [P.generatorSet_prescribedTuple]
rw [← Finset.biUnion_image, ← Finset.image_inter _ _ hinj,
Finset.card_image_of_injective _ hinj]
theorem exists_uniform_direction_order {b κ : ℕ}
(P : EqualityPattern b κ) (hκ : κ ≤ 2) :
∃ l : List (Fin b), l.Nodup ∧ l.toFinset = Finset.univ ∧
∀ (F : Type*) [DecidableEq F] (a : DirectionAssignment F b κ),
Eligible P a → ∀ bits : Bits b,
Balanced.SmallReuseOrder (fun i => generatorSet (prescribedTuple P a bits i)) l := by
obtain ⟨l, hnd, hfull, horder⟩ := P.exists_smallReuseOrder hκ
refine ⟨l, hnd, hfull, fun F _ a ha bits => ?_⟩
simpa only [P.generatorSet_prescribedTuple] using horder.image a.2 ha.2.1
theorem exists_direction_processing_perm {b κ : ℕ}
(P : EqualityPattern b κ) (hκ : κ ≤ 2) :
∃ π : Equiv.Perm (Fin b), ∀ (F : Type*) [DecidableEq F]
(a : DirectionAssignment F b κ), Eligible P a → ∀ (bits : Bits b) (i : Fin b),
(generatorSet (prescribedTuple P a bits (π i)) ∩
(Finset.univ.filter (fun j : Fin b => j < i)).biUnion
(fun j => generatorSet (prescribedTuple P a bits (π j)))).card ≤ 1 := by
obtain ⟨π, hπ⟩ := P.exists_processing_perm hκ
refine ⟨π, fun F _ a ha bits i => ?_⟩
have hclass := hπ i
have h := P.prescribed_generator_overlap_card a ha.2.1 bits (π i)
((Finset.univ.filter (fun j : Fin b => j < i)).image π)
simp only [Finset.image_biUnion] at h
rw [h]
exact hclass
end Balanced.Local.EqualityPattern
end OAI
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