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import OAI.Computability.VertexCover.PCP.AlphabetTableEnumeration
import OAI.Computability.VertexCover.PCP.AlphabetTableQueries
namespace OAI
namespace UniqueGames.Foundations.PCP.AlphabetTable.Relations
open Hastad
open Queries
variable {q : Nat}
def sameAddress (selfLoop : Bool) : RawQuery q → RawQuery q → Bool
| .inl (side, f), .inl (side', g) => ((side == side') || selfLoop) && decide (f = g)
| .inr f, .inr g => decide (f = g)
| _, _ => false
@[simp] theorem sameAddress_refl (selfLoop : Bool) (x : RawQuery q) :
sameAddress selfLoop x x = true := by
rcases x with ⟨side, f⟩ | f <;> simp [sameAddress]
theorem globalize_eq_iff {V E : Type*} [DecidableEq V]
(tail head : V) (edge : E) (selfLoop : Bool)
(hloop : selfLoop = decide (tail = head)) (x y : RawQuery q) :
globalize tail head edge x = globalize tail head edge y ↔
sameAddress selfLoop x y = true := by
rcases x with ⟨side, f⟩ | f <;> rcases y with ⟨side', g⟩ | g
· cases side <;> cases side'
· simp [globalize, sameAddress]
· simp [globalize, sameAddress, hloop]
· simp [globalize, sameAddress, hloop,
show head = tail ↔ tail = head from eq_comm]
· simp [globalize, sameAddress]
· simp [globalize, sameAddress]
· simp [globalize, sameAddress]
· simp [globalize, sameAddress]
def eqProfile (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
(i j : Fin 6) : Bool := sameAddress selfLoop (queries i) (queries j)
@[simp] theorem eqProfile_refl (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
(i : Fin 6) : eqProfile selfLoop queries i i = true := sameAddress_refl _ _
def canonicalSlot (selfLoop : Bool) (queries : Fin 6 → RawQuery q) (i : Fin 6) : Fin 6 :=
Fin.find (fun j => eqProfile selfLoop queries j i = true) ⟨i, eqProfile_refl _ _ _⟩
theorem canonicalSlot_profile (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
(i : Fin 6) : eqProfile selfLoop queries (canonicalSlot selfLoop queries i) i = true := by
unfold canonicalSlot
exact Fin.find_spec (p := fun j => eqProfile selfLoop queries j i = true)
⟨i, eqProfile_refl _ _ _⟩
theorem canonicalSlot_le (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
(i j : Fin 6) (h : eqProfile selfLoop queries j i = true) :
canonicalSlot selfLoop queries i ≤ j :=
Fin.find_le_of_pos ⟨i, eqProfile_refl _ _ _⟩ h
def RepeatedConsistent (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
(label : QueryIncidence.Label 6) : Prop :=
∀ i j, eqProfile selfLoop queries i j = true → label i = label j
instance repeatedConsistentDecidable (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
(label : QueryIncidence.Label 6) : Decidable (RepeatedConsistent selfLoop queries label) := by
unfold RepeatedConsistent
infer_instance
def incidenceAccept (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
(accept : QueryIncidence.Label 6 → Bool) (slot : Fin 6)
(left right : QueryIncidence.Label 6) : Bool :=
decide (accept left = true ∧ RepeatedConsistent selfLoop queries left) &&
decide (left (canonicalSlot selfLoop queries slot) = right 0)
def predicate (selfLoop : Bool) (queries : Fin 6 → RawQuery q)
(accept : QueryIncidence.Label 6 → Bool) (slot : Fin 6) (orientation : Bool)
(a b : QueryIncidence.Label 6) : Bool :=
if orientation then incidenceAccept selfLoop queries accept slot b a
else incidenceAccept selfLoop queries accept slot a b
def relation (P : Fin q → Fin q → Bool) (selfLoop : Bool)
(event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) (orientation : Bool) :
GraphTables.RelationTable :=
GraphTables.relationOf fun a b =>
predicate selfLoop (query P event) (AssignmentTester.eventAccepts event.1)
slot orientation (Enumeration.labelEquiv.symm a) (Enumeration.labelEquiv.symm b)
@[simp] theorem relationAt_relation (P : Fin q → Fin q → Bool) (selfLoop : Bool)
(event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) (orientation : Bool)
(a b : GraphTables.Label) :
GraphTables.relationAt (relation P selfLoop event slot orientation) a b =
predicate selfLoop (query P event) (AssignmentTester.eventAccepts event.1)
slot orientation (Enumeration.labelEquiv.symm a) (Enumeration.labelEquiv.symm b) := by
exact GraphTables.relationAt_relationOf _ _ _
theorem relation_reverse (P : Fin q → Fin q → Bool) (selfLoop : Bool)
(event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) (orientation : Bool)
(a b : GraphTables.Label) :
GraphTables.relationAt (relation P selfLoop event slot (!orientation)) b a =
GraphTables.relationAt (relation P selfLoop event slot orientation) a b := by
simp only [relationAt_relation]
cases orientation <;> rfl
def oldPredicate (old : Vector Bool (q * q)) (a b : Fin q) : Bool :=
old[(finProdFinEquiv (a, b) : Fin (q * q))]
def relationFromTable (old : Vector Bool (q * q)) (selfLoop : Bool)
(event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) (orientation : Bool) :
GraphTables.RelationTable := relation (oldPredicate old) selfLoop event slot orientation
@[simp] theorem oldPredicate_relationAt (old : GraphTables.RelationTable)
(a b : GraphTables.Label) : oldPredicate old a b = GraphTables.relationAt old a b := rfl
theorem relationFromTable_eq (old : GraphTables.RelationTable) (selfLoop : Bool)
(event : AlphabetGraph.LocalEvent (Fin 64)) (slot : Fin 6) (orientation : Bool) :
relationFromTable old selfLoop event slot orientation =
relation (GraphTables.relationAt old) selfLoop event slot orientation := rfl
variable {V E : Type*} [DecidableEq V] [DecidableEq E]
omit [DecidableEq E] in
theorem eqProfile_query_iff (G : ConstraintGraph V E (Fin q)) (e : E)
(event : AlphabetGraph.LocalEvent (Fin q)) (i j : Fin 6) :
eqProfile (decide (G.tail e = G.head e)) (query (G.accepts e) event) i j = true ↔
(AlphabetGraph.verifier G).query (e, event) i =
(AlphabetGraph.verifier G).query (e, event) j := by
rw [← globalize_query G e event i, ← globalize_query G e event j]
exact (globalize_eq_iff (G.tail e) (G.head e) e _ rfl _ _).symm
theorem canonicalSlot_eq (G : ConstraintGraph V E (Fin q)) (e : E)
(event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) :
canonicalSlot (decide (G.tail e = G.head e)) (query (G.accepts e) event) slot =
QueryIncidence.canonicalSlot (AlphabetGraph.verifier G) (e, event) slot := by
unfold canonicalSlot QueryIncidence.canonicalSlot
apply Fin.find_congr'
exact eqProfile_query_iff G e event _ slot
omit [DecidableEq E] in
theorem repeatedConsistent_iff (G : ConstraintGraph V E (Fin q)) (e : E)
(event : AlphabetGraph.LocalEvent (Fin q)) (label : QueryIncidence.Label 6) :
RepeatedConsistent (decide (G.tail e = G.head e)) (query (G.accepts e) event) label ↔
QueryIncidence.RepeatedConsistent (AlphabetGraph.verifier G) (e, event) label := by
constructor
· intro h i j hij
exact h i j ((eqProfile_query_iff G e event i j).mpr hij)
· intro h i j hij
exact h i j ((eqProfile_query_iff G e event i j).mp hij)
omit [DecidableEq V] [DecidableEq E] in
theorem verifier_accepts (G : ConstraintGraph V E (Fin q)) (e : E)
(event : AlphabetGraph.LocalEvent (Fin q)) (label : QueryIncidence.Label 6) :
(AlphabetGraph.verifier G).accepts (e, event) label =
AssignmentTester.eventAccepts event.1 label := by
rcases event with ⟨kind, k, f, g, r₀, r₁, r₂⟩
rfl
theorem incidenceAccept_eq (G : ConstraintGraph V E (Fin q)) (e : E)
(event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6)
(left right : QueryIncidence.Label 6) :
incidenceAccept (decide (G.tail e = G.head e)) (query (G.accepts e) event)
(AssignmentTester.eventAccepts event.1) slot left right =
QueryIncidence.incidenceAccept (AlphabetGraph.verifier G) (by decide) (e, event) slot
left right := by
simp only [incidenceAccept, QueryIncidence.incidenceAccept, QueryIncidence.LeftValid,
verifier_accepts, repeatedConsistent_iff, canonicalSlot_eq,
QueryIncidence.decodeRight, QueryIncidence.zeroSlot]
rfl
theorem relationAt_relation_graph (G : ConstraintGraph V E (Fin q)) (e : E)
(event : AlphabetGraph.LocalEvent (Fin q)) (slot : Fin 6) (orientation : Bool)
(a b : GraphTables.Label) :
GraphTables.relationAt
(relation (G.accepts e) (decide (G.tail e = G.head e)) event slot orientation) a b =
(AlphabetGraph.graph G).accepts (((e, event), slot), orientation)
(Enumeration.labelEquiv.symm a) (Enumeration.labelEquiv.symm b) := by
rw [relationAt_relation]
cases orientation <;>
exact incidenceAccept_eq G e event slot _ _
end UniqueGames.Foundations.PCP.AlphabetTable.Relations
end OAI
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