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import Mathlib
import SymbolicModalLogic.FixedPoints
class Modalities (Λ : Type) where
signature : Λ → Type
signature_fintype : ∀ l, Fintype (signature l) := by intro; infer_instance
dual : Λ → Λ
dualSignature {α : Type} : (l : Λ) → (signature l → α) → (signature (dual l) → α)
dual_involution : ∀ l, dual (dual l) = l
inductive Formula (Λ : Type) [Modalities Λ] where
| top : Formula Λ
| bot : Formula Λ
| or : Formula Λ → Formula Λ → Formula Λ
| and : Formula Λ → Formula Λ → Formula Λ
| modality : (l : Λ) → (args : Modalities.signature l → Formula Λ) → Formula Λ
| var : ℕ → Formula Λ
| lfp : Formula Λ → Formula Λ
| gfp : Formula Λ → Formula Λ
namespace Formula
scoped notation "⊤" => Formula.top
scoped notation "⊥" => Formula.bot
scoped infixl:30 " ∨ " => Formula.or
scoped infixl:35 " ∧ " => Formula.and
scoped notation "^" n:max => Formula.var n
scoped notation "μ, " φ => Formula.lfp φ
scoped notation "ν, " φ => Formula.gfp φ
variable {Λ : Type} [Modalities Λ]
def not : Formula Λ → Formula Λ
| .top => ⊥
| .bot => ⊤
| .or φ ψ => (not φ) ∧ (not ψ)
| .and φ ψ => (not φ) ∨ (not ψ)
| .modality l args => .modality (Modalities.dual l) (Modalities.dualSignature l fun a ↦ not (args a))
| .var x => ^x
| .lfp φ => ν, not φ
| .gfp φ => μ, not φ
scoped notation "¬" φ:60 => Formula.not φ
scoped infixl:25 " ⇒ " => fun φ ψ ↦ ¬φ ∨ ψ
def up : Formula Λ → (start : ℕ := 1) → Formula Λ
| .top, _ => ⊤
| .bot, _ => ⊥
| .or φ ψ, start => up φ start ∨ up ψ start
| .and φ ψ, start => up φ start ∧ up ψ start
| .modality l args, start => .modality l fun a ↦ up (args a) start
| .var x, start => if x < start then ^x else ^(x + 1)
| .lfp φ, start => μ, up φ (start + 1)
| .gfp φ, start => ν, up φ (start + 1)
scoped notation "⇑ " φ => Formula.up φ
def subst : Formula Λ → (ℕ × Formula Λ) → Formula Λ
| .top, _ => ⊤
| .bot, _ => ⊥
| .or φ ψ, σ => subst φ σ ∨ subst ψ σ
| .and φ ψ, σ => subst φ σ ∧ subst ψ σ
| .modality l args, σ => .modality l fun a ↦ subst (args a) σ
| .var x, (y, ψ) => if x < y then ^x else
if x > y then ^(x - 1) else
ψ
| .lfp φ, (y, ψ) => μ, subst φ (y + 1, ⇑ψ)
| .gfp φ, (y, ψ) => ν, subst φ (y + 1, ⇑ψ)
macro_rules
| `($φ[$ψ/^$n]) => `(Formula.subst $φ ($n, $ψ))
end Formula
section Semantics
class Model (Λ : Type) [Modalities Λ] (F : Type → Type) [Functor F] [LawfulFunctor F] where
lifting {X : Type} (l : Λ) : (args : Modalities.signature l → Set X) → Set (F X)
lifting_natural {X Y : Type} (f : X → Y) (l : Λ) (args : Modalities.signature l → Set Y) :
(lifting l fun a ↦ f ⁻¹' (args a)) = ((Functor.map f) ⁻¹' (lifting l args))
lifting_monotone {X : Type} (l : Λ) (args args' : Modalities.signature l → Set X) :
(∀ a, args a ⊆ args' a) → lifting l args ⊆ lifting l args'
lifting_dual {X : Type} (l : Λ) (args : Modalities.signature l → Set X) :
lifting (Modalities.dual l) (Modalities.dualSignature l args) = Set.compl (lifting l fun a ↦ Set.compl (args a))
variable {Λ : Type} [Modalities Λ]
variable {F : Type → Type} [Functor F] [LawfulFunctor F] [Model Λ F]
def Environment (X : Type) : Type := ℕ → Set X
namespace Environment
variable {X : Type}
def empty : Environment X := λ _ ↦ ∅
def up (xs : Environment X) : Environment X
| 0 => ∅
| n + 1 => xs n
def subst (xs : Environment X) (n : ℕ) (x : Set X) : Environment X :=
λ m ↦ if m = n then x else xs m
scoped notation "⇑ " φ => Environment.up φ
def add (x : Set X) (xs : Environment X) : Environment X := subst (⇑ xs) 0 x
infixr:67 " :: " => Environment.add
variable {xs : Environment X} {x y : Set X} {n m : ℕ}
@[simp]
lemma subst_eq : subst xs n x n = x := by simp [subst]
@[simp]
lemma subst_neq : m ≠ n → subst xs n x m = xs m := by intro h; simp [subst, h]
@[simp]
lemma subst_subst_eq : subst (subst xs n x) n y = subst xs n y := by
apply funext
intro m
by_cases h : m = n <;> simp [h]
lemma subst_subst_neq : m ≠ n → subst (subst xs n x) m y = subst (subst xs m y) n x := by
intro hneq
apply funext
intro k
by_cases h₁ : k = m <;> by_cases h₂ : k = n <;> simp [h₁, h₂, hneq, Ne.symm hneq]
lemma subst_up : (⇑(subst xs n x)) = subst (⇑xs) (n + 1) x := by
apply funext
intro m
cases m with
| zero => simp [up]
| succ m => by_cases h : m = n <;> simp [h, up]
lemma subst_add : x :: subst xs n y = subst (x :: xs) (n + 1) y := by
rw [add, add, subst_subst_neq, subst_up]
simp
end Environment
def interp {X : Type} (γ : X → F X) : Formula Λ → Environment X → Set X
| .top, _ => Set.univ
| .bot, _ => ∅
| .or φ ψ, xs => (interp γ φ xs) ∪ (interp γ ψ xs)
| .and φ ψ, xs => (interp γ φ xs) ∩ (interp γ ψ xs)
| .modality l args, xs => γ ⁻¹' Model.lifting l fun a ↦ interp γ (args a) xs
| .var x, xs => xs x
| .lfp φ, xs => FixedPoints.lfp fun S ↦ interp γ φ (S :: xs)
| .gfp φ, xs => FixedPoints.gfp fun S ↦ interp γ φ (S :: xs)
def interp_subst_monotone {X : Type} {γ : X → F X} (φ : Formula Λ) (xs : Environment X) (n : ℕ) :
Monotone fun x ↦ interp γ φ (xs.subst n x) := by
induction φ generalizing xs n with
| top =>
intro S S' hle
rfl
| bot =>
intro S S' hle
rfl
| or φ ψ ihφ ihψ =>
intro S S' hle
simp only [Set.le_eq_subset, Set.union_subset_iff, interp]
constructor
· apply Set.subset_union_of_subset_left
exact ihφ xs n hle
· apply Set.subset_union_of_subset_right
exact ihψ xs n hle
| and φ ψ ihφ ihψ =>
intro S S' hle
simp only [Set.le_eq_subset, Set.subset_inter_iff, interp]
constructor
· intro x ⟨hx, _⟩
exact ihφ xs n hle hx
· intro x ⟨_, hx⟩
exact ihψ xs n hle hx
| modality l args ih =>
intro S S' hle
simp only [Set.le_eq_subset]
apply Set.preimage_mono
apply Model.lifting_monotone
intro a
exact ih a xs n hle
| var x =>
intro S S' hle
by_cases h : x = n
· simp only [interp, h, Environment.subst_eq, Set.le_eq_subset]
exact hle
· simp [h, interp]
| lfp φ ihφ =>
intro S S' hle
simp only [interp, Set.le_eq_subset, FixedPoints.lfp, Set.sInf_eq_sInter, Set.subset_sInter_iff, Set.mem_setOf_eq]
intro U hU x hx
apply hx
calc
interp γ φ (U :: xs.subst n S) = interp γ φ ((U :: xs).subst (n + 1) S) := by rw [Environment.subst_add]
_ ⊆ interp γ φ ((U :: xs).subst (n + 1) S') := ihφ _ _ hle
_ = interp γ φ (U :: xs.subst n S') := by rw [Environment.subst_add]
_ ⊆ U := hU
| gfp φ ihφ =>
intro S S' hle
simp only [interp, Set.le_eq_subset, FixedPoints.gfp, Set.sSup_eq_sUnion, Set.sUnion_subset_iff, Set.mem_setOf_eq]
intro U hU x hx
use U; constructor
· calc
U ⊆ interp γ φ (U :: xs.subst n S) := hU
_ = interp γ φ ((U :: xs).subst (n + 1) S) := by rw [Environment.subst_add]
_ ⊆ interp γ φ ((U :: xs).subst (n + 1) S') := ihφ _ _ hle
_ = interp γ φ (U :: xs.subst n S') := by rw [Environment.subst_add]
· exact hx
lemma interp_lfp_isFixedPt {X : Type} (γ : X → F X) (φ : Formula Λ) (xs : Environment X) :
Function.IsFixedPt (fun S ↦ interp γ φ (S :: xs)) (interp γ (.lfp φ) xs) := by
apply FixedPoints.lfp_isFixedPt
simp [Environment.add, interp_subst_monotone]
lemma interp_lfp_is_least {X : Type} (γ : X → F X) (φ : Formula Λ) (xs : Environment X) :
∀ x, Function.IsFixedPt (fun S ↦ interp γ φ (S :: xs)) x → interp γ (.lfp φ) xs ⊆ x := by
apply FixedPoints.lfp_is_least
lemma interp_gfp_isFixedPt {X : Type} (γ : X → F X) (φ : Formula Λ) (xs : Environment X) :
Function.IsFixedPt (fun S ↦ interp γ φ (S :: xs)) (interp γ (.gfp φ) xs) := by
apply FixedPoints.gfp_isFixedPt
simp [Environment.add, interp_subst_monotone]
lemma interp_gfp_is_greatest {X : Type} (γ : X → F X) (φ : Formula Λ) (xs : Environment X) :
∀ x, Function.IsFixedPt (fun S ↦ interp γ φ (S :: xs)) x → x ⊆ interp γ (.gfp φ) xs := by
apply FixedPoints.gfp_is_greatest
end Semantics
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