attempts for relate.instDecidable

Pdl/Distance.lean

@@ -26,6 +26,11 @@ structure DecidableKripkeModel (W :Type) where
   decrel : ∀ n, DecidableRel (M.Rel n)
   decval : ∀ w n, Decidable (M.val w n)
 
+-- Possibly useful reference:
+-- Martin Lange (2005): *Model checking propositional dynamic logic with all extras*
+-- Journal of Applied Logic
+-- https://doi.org/10.1016/j.jal.2005.08.002
+
 mutual
 instance evaluate.instDecidable (Mod : DecidableKripkeModel W) w φ
     : Decidable (evaluate Mod.M w φ) := by
@@ -51,8 +56,32 @@ instance evaluate.instDecidable (Mod : DecidableKripkeModel W) w φ
     all_goals
       apply isFalse; simp; tauto
   case box α φ =>
-    simp
-    sorry
+    simp only [evaluate]
+    let allW := Mod.enum.toList
+    let reachable := allW.filter
+      (fun v => @decide (relate Mod.M α w v) (relate.instDecidable Mod α w v))
+    let hyp := reachable.all
+      (fun v => @decide (evaluate Mod.M v φ) (evaluate.instDecidable Mod v φ))
+    by_cases hyp
+    case pos yes =>
+      apply isTrue
+      intro v w_v
+      have : v ∈ allW := Mod.enum.mem_toList v
+      have : v ∈ reachable := by
+        unfold_let reachable
+        simp only [List.mem_filter, decide_eq_true_eq]
+        tauto
+      unfold_let hyp at yes
+      simp_all
+    case neg no =>
+      apply isFalse
+      push_neg
+      unfold_let hyp at no
+      simp at no
+      rcases no with ⟨v, v_in_r, not_v_φ⟩
+      aesop
+  termination_by
+    sizeOf φ
 
 instance relate.instDecidable (Mod : DecidableKripkeModel W) α v w
     : Decidable (relate Mod.M α v w) := by
@@ -61,10 +90,39 @@ instance relate.instDecidable (Mod : DecidableKripkeModel W) α v w
     simp only [relate]
     apply Mod.decrel
   case sequence α β =>
-    have IH1 := relate.instDecidable Mod α v
-    have IH1 := relate.instDecidable Mod β
-    -- do we need more here? iterate over all `W` or so?
+    have IHα := relate.instDecidable Mod α
+    have IHβ := relate.instDecidable Mod β
+    let allW := Mod.enum.toList
+    let α_img := allW.filter (fun x => @decide (relate Mod.M α v x) (IHα v x))
+    let α_β_fun := fun x => allW.filter (fun y => @decide (relate Mod.M β x y) (IHβ x y))
+    let α_β_img := (α_img.map α_β_fun).join
+    simp only [relate]
+    -- the following part works, but somehow relies on `Classical.propDecidable`???
     sorry
+    /-
+    by_cases w ∈ α_β_img
+    case pos yes =>
+      apply isTrue
+      unfold_let α_β_img at yes
+      simp only [List.mem_join, List.mem_map, exists_exists_and_eq_and] at yes
+      rcases yes with ⟨y, y_in, w_in⟩
+      use y
+      unfold_let α_img at y_in
+      unfold_let α_β_fun at w_in
+      simp only [List.mem_filter, decide_eq_true_eq] at *
+      tauto
+    case neg no =>
+      apply isFalse
+      push_neg
+      intro y v_y
+      unfold_let α_β_img at no
+      unfold_let α_β_fun at no
+      unfold_let α_img at no
+      unfold_let allW at no
+      simp only [List.mem_join, List.mem_map, List.mem_filter, FinEnum.mem_toList,
+        decide_eq_true_eq, true_and, exists_exists_and_eq_and, not_exists, not_and] at no
+      exact no y v_y
+    -/
   case union α β =>
     have IH1 := relate.instDecidable Mod α v w
     have IH1 := relate.instDecidable Mod β v w
@@ -82,6 +140,7 @@ instance relate.instDecidable (Mod : DecidableKripkeModel W) α v w
     -- QUESTION: would it be enough to have decidable
     --           transitive closures of atomic programs
     --           to then compute all other PDL stars?
+    -- FOR NOW: we have `FinEnum`, that should be enough.
     sorry
   case test τ =>
     simp only [relate]
@@ -91,6 +150,9 @@ instance relate.instDecidable (Mod : DecidableKripkeModel W) α v w
     · apply isTrue; tauto
     all_goals
       apply isFalse; tauto
+  termination_by
+    sizeOf α
+
 end
 
 def distance {W} (Mod : DecidableKripkeModel W) (v w : W)
@@ -101,13 +163,25 @@ def distance {W} (Mod : DecidableKripkeModel W) (v w : W)
   have := Mod.deceq v w
   exact (if v = w ∧ evaluate Mod.M v τ then some 0 else none)
 | α⋓β => min (distance Mod v w α) (distance Mod v w β)
-| α;'β => sorry -- min { ... | u ∈ W } -- need map or enum over W here :-/
-| ∗α => sorry -- min { ... | u ∈ W } -- need map or enum over W here :-/
+| α;'β =>
+    let allW := Mod.enum.toList
+    let α_β_distOf := fun x => Nat.add <$> distance Mod v x α <*> distance Mod x w β
+    (allW.map α_β_distOf).reduceOption.minimum?
+| ∗α =>
+    let allW := Mod.enum.toList
+    -- This will be ugly, but essentially we need something like the matrix method?
+    let maxN := allW.length
+    sorry -- min { ... | u ∈ W } -- need map or enum over W here :-/
+termination_by
+  α => sizeOf α
 
 def distance_list {W} (Mod : DecidableKripkeModel W) (v w : W) : (δ : List Program) → Option Nat
 | [] => have := Mod.deceq v w
         if v = w then some 0 else none
-| (α::δ) => sorry -- min { ... + ... | u ∈ W } need map or enum over W here :-/
+| (α::δ) => -- similar to α;'β case in `distance`
+    let allW := Mod.enum.toList
+    let α_δ_distOf := fun x => Nat.add <$> distance Mod v x α <*> distance_list Mod x w δ
+    (allW.map α_δ_distOf).reduceOption.minimum?
 
 theorem distance_iff_relate (Mod : DecidableKripkeModel W) :
     (distance Mod v w α).isSome ↔ relate Mod.M α v w := by