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|
import SymbolicModalLogic.MuCalculus
import Mathlib
inductive FixedPointOp where
| μ
| ν
deriving DecidableEq
variable {Λ : Type} [Modalities Λ]
open Formula
inductive IsAlternationDepthBound : FixedPointOp → ℕ → Formula Λ → Prop where
| top {η : FixedPointOp} : IsAlternationDepthBound η 0 .top
| bot {η : FixedPointOp} : IsAlternationDepthBound η 0 .bot
| var {η : FixedPointOp} (x : ℕ) : IsAlternationDepthBound η 0 (.var x)
| or {η : FixedPointOp} {n : ℕ} {φ ψ : Formula Λ} :
IsAlternationDepthBound η n φ →
IsAlternationDepthBound η n ψ →
IsAlternationDepthBound η n (.or φ ψ)
| and {η : FixedPointOp} {n : ℕ} {φ ψ : Formula Λ} :
IsAlternationDepthBound η n φ →
IsAlternationDepthBound η n ψ →
IsAlternationDepthBound η n (.and φ ψ)
| modality {η : FixedPointOp} {n : ℕ} {l : Λ} {args : Modalities.signature l → Formula Λ} :
(∀ l, IsAlternationDepthBound η n (args l)) →
IsAlternationDepthBound η n (.modality l args)
| lfp {n : ℕ} {φ : Formula Λ} :
IsAlternationDepthBound .ν n φ →
IsAlternationDepthBound .ν n (μ, φ)
| gfp {n : ℕ} {φ : Formula Λ} :
IsAlternationDepthBound .μ n φ →
IsAlternationDepthBound .μ n (ν, φ)
| subst {η : FixedPointOp} {n : ℕ} {φ ψ : Formula Λ} {x : ℕ} :
IsAlternationDepthBound η n φ →
IsAlternationDepthBound η n ψ →
IsAlternationDepthBound η n (φ.subst (x, ψ))
| up {η₁ η₂ : FixedPointOp} {n : ℕ} {φ : Formula Λ} :
IsAlternationDepthBound η₁ n φ →
IsAlternationDepthBound η₂ (n + 1) φ
def IsAlternationDepth (n : ℕ) (φ : Formula Λ) : Prop :=
IsAlternationDepthBound .μ n φ ∧ IsAlternationDepthBound .ν n φ
example : IsAlternationDepth 2 ((μ, ν, ^0 ∧ μ, ^2 ∨ ^0) : Formula Λ) := by
suffices IsAlternationDepthBound .ν 1 ((ν, ^0 ∧ μ, ^2 ∨ ^0) : Formula Λ) by
constructor
· apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.lfp
exact this
· apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.lfp
exact this
rw [show ((ν, ^0 ∧ μ, ^2 ∨ ^0) : Formula Λ) = (ν, ^0 ∧ ^1).subst (0, (μ, ^1 ∨ ^0)) from rfl]
apply IsAlternationDepthBound.subst
· apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.gfp
apply IsAlternationDepthBound.and
· apply IsAlternationDepthBound.var
· apply IsAlternationDepthBound.var
· apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.lfp
apply IsAlternationDepthBound.or
· apply IsAlternationDepthBound.var
· apply IsAlternationDepthBound.var
example : IsAlternationDepth 1 ((ν, ^0) : Formula Λ) := by
constructor
· apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.gfp
apply IsAlternationDepthBound.var
· apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.gfp
apply IsAlternationDepthBound.var
example : IsAlternationDepth 1 ((μ, ^0) : Formula Λ) := by
constructor
· apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.lfp
apply IsAlternationDepthBound.var
· apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.lfp
apply IsAlternationDepthBound.var
example : IsAlternationDepth 1 ((ν, ν, ^0 ∧ ^1) : Formula Λ) := by
constructor
· apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.gfp
apply IsAlternationDepthBound.gfp
apply IsAlternationDepthBound.and
· apply IsAlternationDepthBound.var
· apply IsAlternationDepthBound.var
· apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.gfp
apply IsAlternationDepthBound.gfp
apply IsAlternationDepthBound.and
· apply IsAlternationDepthBound.var
· apply IsAlternationDepthBound.var
example : IsAlternationDepth 2 ((μ, ν, ^0 ∧ ^1) : Formula Λ) := by
constructor
· apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.lfp
apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.gfp
apply IsAlternationDepthBound.and
· apply IsAlternationDepthBound.var
· apply IsAlternationDepthBound.var
· apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.lfp
apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.gfp
apply IsAlternationDepthBound.and
· apply IsAlternationDepthBound.var
· apply IsAlternationDepthBound.var
example : IsAlternationDepth 2 ((ν, μ, ^0 ∧ ^1) : Formula Λ) := by
constructor
· apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.gfp
apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.lfp
apply IsAlternationDepthBound.and
· apply IsAlternationDepthBound.var
· apply IsAlternationDepthBound.var
· apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.gfp
apply IsAlternationDepthBound.up
apply IsAlternationDepthBound.lfp
apply IsAlternationDepthBound.and
· apply IsAlternationDepthBound.var
· apply IsAlternationDepthBound.var
noncomputable def alternationDepth (φ : Formula Λ) : ℕ :=
sInf { n : ℕ | IsAlternationDepth n φ }
private def collectOps : Formula Λ → List FixedPointOp
| .top => []
| .bot => []
| .or φ ψ => collectOps φ ++ collectOps ψ
| .and φ ψ => collectOps φ ++ collectOps ψ
| .modality l args =>
have : FinEnum (Modalities.signature l) := Modalities.signature_finenum l
FinEnum.toList (Modalities.signature l)
|>.flatMap fun a ↦ (collectOps (args a))
| .var _ => []
| .lfp φ => FixedPointOp.μ :: collectOps φ
| .gfp φ => FixedPointOp.ν :: collectOps φ
def ad (φ : Formula Λ) : ℕ :=
let rec count (curr : FixedPointOp) (val : ℕ) : List FixedPointOp → ℕ
| .nil => val
| .cons .μ xs => let val' := if _ : curr = .μ then val else val + 1
count .μ val' xs
| .cons .ν xs => let val' := if _ : curr = .ν then val else val + 1
count .ν val' xs
count .ν 0 (collectOps φ).reverse
private lemma ad_count_add {curr : FixedPointOp} {val n : ℕ} {xs : List FixedPointOp} :
ad.count curr (val + n) xs = (ad.count curr val xs) + n := by
induction xs generalizing val curr with
| nil => simp [ad.count]
| cons x xs ih =>
cases x <;> cases curr
· simp [ad.count, ih]
· simp only [ad.count, reduceCtorEq, ↓reduceDIte]
rw [show val + n + 1 = (val + 1) + n by ring]
apply ih
· simp only [ad.count, reduceCtorEq, ↓reduceDIte]
rw [show val + n + 1 = (val + 1) + n by ring]
apply ih
· simp [ad.count, ih]
lemma ad_isAlternationDepth (φ : Formula Λ) : IsAlternationDepth (ad φ) φ := by
sorry
-- induction φ with
-- | top =>
-- simp only [ad, collectOps, List.reverse_nil, ad.count]
-- constructor <;> apply IsAlternationDepthBound.top
-- | bot =>
-- simp only [ad, collectOps, List.reverse_nil, ad.count]
-- constructor <;> apply IsAlternationDepthBound.bot
-- | or ψ₁ ψ₂ ih₁ ih₂ =>
-- simp [ad, collectOps]
-- | and ψ₁ ψ₂ => sorry
-- | modality l args => sorry
-- | var x => sorry
-- | lfp ψ => sorry
-- | gfp ψ => sorry
lemma alternationDepth_isAlternationDepth (φ : Formula Λ) : IsAlternationDepth (alternationDepth φ) φ := by
show sInf { n : ℕ | IsAlternationDepth n φ } ∈ { n : ℕ | IsAlternationDepth n φ }
apply Nat.sInf_mem
use ad φ
apply ad_isAlternationDepth
|
