PDL-Tableaux (Section 4)

Projections

def formProjection : ℕ → Formula → Option Formula

Equations

• formProjection x✝ (⌈·B⌉φ) = if (x✝ == B) = true then some φ else none
• formProjection x✝¹ x✝ = none

def projection : ℕ → List Formula → List Formula

Equations

• projection x✝¹ x✝ = (List.map (fun (x : Formula) => formProjection x✝¹ x) x✝).reduceOption

@[simp]

theorem proj {A : ℕ} {X : List Formula} {g : Formula} : g ∈ projection A X ↔ (⌈·A⌉g) ∈ X

Loading and Loaded Histories

def boxesOf : Formula → List Program × Formula

Equations

• boxesOf (⌈prog⌉nextf) = match boxesOf nextf with | (rest, endf) => (prog :: rest, endf)
• boxesOf x✝ = ([], x✝)

@[reducible, inline]

abbrev History : Type

A history is a list of Sequents. This only tracks "big" steps, hoping we do not need steps within a local tableau. The head of the first list is the newest Sequent.

Equations

• History = List Sequent

def rep (Hist : History) (X : Sequent) : Prop

We have a repeat iff the history contains a node that is setEqTo the current node.

Equations

• rep Hist X = ∃ Y ∈ Hist, Y.multisetEqTo X

instance instDecidableRep {H : History} {X : Sequent} : Decidable (rep H X)

Equations

• instDecidableRep = id (List.rec (isFalse ⋯) (fun (head : Sequent) (tail : List Sequent) (tail_ih : Decidable (∃ Y ∈ tail, Y.multisetEqTo X)) => ⋯.mpr instDecidableOr) H)

def LoadedPathRepeat (Hist : History) (X : Sequent) : Type

A lpr means we can go k steps back in the history to reach an equal node, and all nodes on the way are loaded. Note: k=0 means the first element of Hist is the companion.

Equations

• LoadedPathRepeat Hist X = { k : Fin (List.length Hist) // (List.get Hist k).multisetEqTo X ∧ ∀ m ≤ k, (List.get Hist m).isLoaded = true }

theorem LoadedPathRepeat_comp_isLoaded {Hist : History} {X : Sequent} (lpr : LoadedPathRepeat Hist X) : (List.get Hist ↑lpr).isLoaded = true

theorem LoadedPathRepeat_rep_isLoaded {Hist : History} {X : Sequent} (lpr : LoadedPathRepeat Hist X) : X.isLoaded = true

instance instDecidableNonemptyLoadedPathRepeat {H : History} {X : Sequent} : Decidable (Nonempty (LoadedPathRepeat H X))

Equations

• instDecidableNonemptyLoadedPathRepeat = if h : ∃ (k : Fin (List.length H)), (List.get H k).multisetEqTo X ∧ ∀ m ≤ k, (List.get H m).isLoaded = true then isTrue ⋯ else isFalse ⋯

The PDL rules

inductive PdlRule (Γ Δ : Sequent) : Type

A rule to go from Γ to Δ. Note the four variants of the modal rule.

• loadL {L : List Formula} {δ : List Program} {α : Program} {φ : Formula} {R : List Formula} : (~⌈⌈δ⌉⌉⌈α⌉φ) ∈ L → PdlRule (L, R, none) (L.erase (~⌈⌈δ⌉⌉⌈α⌉φ), R, some (Sum.inl (~'⌊⌊δ⌋⌋⌊α⌋AnyFormula.normal φ)))
• loadR {R : List Formula} {δ : List Program} {α : Program} {φ : Formula} {L : List Formula} : (~⌈⌈δ⌉⌉⌈α⌉φ) ∈ R → PdlRule (L, R, none) (L, R.erase (~⌈⌈δ⌉⌉⌈α⌉φ), some (Sum.inr (~'⌊⌊δ⌋⌋⌊α⌋AnyFormula.normal φ)))
• freeL {L R : List Formula} {δ : List Program} {α : Program} {φ : Formula} : PdlRule (L, R, some (Sum.inl (~'⌊⌊δ⌋⌋⌊α⌋AnyFormula.normal φ))) (List.insert (~⌈⌈δ⌉⌉⌈α⌉φ) L, R, none)
• freeR {L R : List Formula} {δ : List Program} {α : Program} {φ : Formula} : PdlRule (L, R, some (Sum.inr (~'⌊⌊δ⌋⌋⌊α⌋AnyFormula.normal φ))) (L, List.insert (~⌈⌈δ⌉⌉⌈α⌉φ) R, none)
• modL {L R : List Formula} {A : ℕ} {X : Sequent} {ξ : AnyFormula} : X = (L, R, some (Sum.inl (~'⌊·A⌋ξ))) → PdlRule X (match ξ with | AnyFormula.normal φ => ((~φ) :: projection A L, projection A R, none) | AnyFormula.loaded χ => (projection A L, projection A R, some (Sum.inl (~'χ))))
• modR {L R : List Formula} {A : ℕ} {X : Sequent} {ξ : AnyFormula} : X = (L, R, some (Sum.inr (~'⌊·A⌋ξ))) → PdlRule X (match ξ with | AnyFormula.normal φ => (projection A L, (~φ) :: projection A R, none) | AnyFormula.loaded χ => (projection A L, projection A R, some (Sum.inr (~'χ))))

inductive Tableau : History → Sequent → Type

The Tableau [parent, grandparent, ...] child type.

A closed tableau for X is either of:

• a local tableau for X followed by closed tableaux for all end nodes,
• a PDL rule application
• a successful loaded repeat (MB condition six)

• loc {Hist : History} {X : Sequent} (nrep : ¬rep Hist X) (nbas : ¬X.basic) (lt : LocalTableau X) (next : (Y : Sequent) → Y ∈ endNodesOf lt → Tableau (X :: Hist) Y) : Tableau Hist X
• pdl {Hist : History} {X Y : Sequent} (nrep : ¬rep Hist X) (bas : X.basic) (r : PdlRule X Y) (next : Tableau (X :: Hist) Y) : Tableau Hist X
• lrep {X : Sequent} {Hist : History} (lpr : LoadedPathRepeat Hist X) : Tableau Hist X

def Tableau.isLrep {Hist : History} {X : Sequent} : Tableau Hist X → Prop

Equations

• (Tableau.loc nrep nbas lt next).isLrep = False
• (Tableau.pdl nrep bas r next).isLrep = False
• (Tableau.lrep lpr).isLrep = True

inductive provable : Formula → Prop

• byTableauL {φ : Formula} : ∀ (a : Tableau [] ([~φ], [], none)), provable φ
• byTableauR {φ : Formula} : ∀ (a : Tableau [] ([], [~φ], none)), provable φ

def inconsistent : Sequent → Prop

A Sequent is inconsistent if there exists a closed tableau for it.

Equations

• inconsistent x✝ = Nonempty (Tableau [] x✝)

def consistent : Sequent → Prop

A Sequent is consistent iff it is not inconsistent.

Equations

• consistent x✝ = ¬inconsistent x✝