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import OAI.Computability.VertexCover.PCP.AlphabetTableQueries
import OAI.Computability.VertexCover.Reduction.CompatibleSingleton
namespace OAI
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namespace VertexCover.LabelCover
abbrev PositionPair (d : ℕ) := {jk : Fin d × Fin d // jk.1 < jk.2}
abbrev Seeds (Φ : LabelCover) (d : ℕ) := PositionPair d → Fin Φ.M
def query (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) (j : Fin d) : Φ.Query d :=
⟨j, (fun k => Φ.left (seed ⟨(j, k.1), k.2⟩)),
(fun k => Φ.right (seed ⟨(k.1, j), k.2⟩))⟩
theorem query_left_mem (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
(j k : Fin d) (hjk : j < k) :
Φ.left (seed ⟨(j, k), hjk⟩) ∈ (Φ.query seed j).scopeU := by
exact Finset.mem_image.mpr ⟨⟨k, hjk⟩, Finset.mem_univ _, rfl⟩
theorem query_right_mem (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
(j k : Fin d) (hjk : j < k) :
Φ.right (seed ⟨(j, k), hjk⟩) ∈ (Φ.query seed k).scopeV := by
exact Finset.mem_image.mpr ⟨⟨j, hjk⟩, Finset.mem_univ _, rfl⟩
theorem query_injective (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) :
Function.Injective (Φ.query seed) := by
intro j k h
exact congrArg Sigma.fst h
namespace Query
theorem scope_card_le {Φ : LabelCover} {d : ℕ} (i : Φ.Query d) :
i.scopeU.card + i.scopeV.card ≤ d - 1 := by
have hU : i.scopeU.card ≤ d - 1 - i.1.val := by
calc
i.scopeU.card ≤ Fintype.card {k : Fin d // i.1 < k} := Finset.card_image_le
_ = (Finset.Ioi i.1).card := by simp [Fintype.card_subtype, Finset.filter_lt_eq_Ioi]
_ = d - 1 - i.1.val := Fin.card_Ioi i.1
have hV : i.scopeV.card ≤ i.1.val := by
calc
i.scopeV.card ≤ Fintype.card {k : Fin d // k < i.1} := Finset.card_image_le
_ = (Finset.Iio i.1).card := by simp [Fintype.card_subtype, Finset.filter_gt_eq_Iio]
_ = i.1.val := Fin.card_Iio i.1
have := i.1.isLt
omega
theorem localLabel_card_le {Φ : LabelCover} {d : ℕ} (i : Φ.Query d) :
Fintype.card i.LocalLabel ≤ (max Φ.qU Φ.qV)^(d - 1) := by
let q := max Φ.qU Φ.qV
have hq : 0 < q := lt_of_lt_of_le Φ.qU_pos (le_max_left _ _)
calc
Fintype.card i.LocalLabel ≤ Fintype.card i.Assignment := Fintype.card_subtype_le _
_ = Φ.qU^i.scopeU.card * Φ.qV^i.scopeV.card := by
simp [Assignment, Fintype.card_prod, ]
_ ≤ q^i.scopeU.card * q^i.scopeV.card :=
Nat.mul_le_mul (Nat.pow_le_pow_left (le_max_left _ _) _)
(Nat.pow_le_pow_left (le_max_right _ _) _)
_ = q^(i.scopeU.card + i.scopeV.card) := (pow_add _ _ _).symm
_ ≤ q^(d - 1) := Nat.pow_le_pow_right hq i.scope_card_le
end Query
def localOfSatisfying (Φ : LabelCover) (A : Φ.Labeling)
(hA : ∀ c, Φ.Satisfies A c) {d : ℕ} (i : Φ.Query d) : i.LocalLabel :=
⟨(fun x => A.1 x.1, fun y => A.2 y.1), fun c _ _ => hA c⟩
def honestSelection (Φ : LabelCover) (A : Φ.Labeling)
(hA : ∀ c, Φ.Satisfies A c) (d : ℕ) : Finset (Φ.Coordinate d) :=
Finset.univ.image (fun i => ⟨i, Φ.localOfSatisfying A hA i⟩)
theorem honestSelection_compatible (Φ : LabelCover) (A : Φ.Labeling)
(hA : ∀ c, Φ.Satisfies A c) (d : ℕ) :
Φ.Compatible (Φ.honestSelection A hA d) := by
classical
have shape : ∀ p ∈ Φ.honestSelection A hA d,
p = ⟨p.1, Φ.localOfSatisfying A hA p.1⟩ := by
intro p hp
obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hp
rfl
refine ⟨?_, ?_, ?_, ?_⟩
· intro p hp q hq heq
calc
p = ⟨p.1, Φ.localOfSatisfying A hA p.1⟩ := shape p hp
_ = ⟨q.1, Φ.localOfSatisfying A hA q.1⟩ :=
congrArg (fun i => (⟨i, Φ.localOfSatisfying A hA i⟩ : Φ.Coordinate d)) heq
_ = q := (shape q hq).symm
· intro p hp q hq x hx hy
have ep := shape p hp
have eq := shape q hq
cases p with | mk i a =>
cases q with | mk j b =>
simp only [Sigma.mk.inj_iff, heq_eq_eq, true_and] at ep eq
subst a; subst b; rfl
· intro p hp q hq y hx hy
have ep := shape p hp
have eq := shape q hq
cases p with | mk i a =>
cases q with | mk j b =>
simp only [Sigma.mk.inj_iff, heq_eq_eq, true_and] at ep eq
subst a; subst b; rfl
· intro p hp q hq c hx hy
have ep := shape p hp
have eq := shape q hq
cases p with | mk i a =>
cases q with | mk j b =>
simp only [Sigma.mk.inj_iff, heq_eq_eq, true_and] at ep eq
subst a; subst b
exact hA c
end VertexCover.LabelCover
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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