feat(LTS): labeled transition systems as coalgebras

Cslib.lean

@@ -78,6 +78,7 @@ public import Cslib.Foundations.Semantics.LTS.Divergence
 public import Cslib.Foundations.Semantics.LTS.Execution
 public import Cslib.Foundations.Semantics.LTS.HasTau
 public import Cslib.Foundations.Semantics.LTS.LTSCat.Basic
+public import Cslib.Foundations.Semantics.LTS.LTSCat.Coalgebra
 public import Cslib.Foundations.Semantics.LTS.Notation
 public import Cslib.Foundations.Semantics.LTS.OmegaExecution
 public import Cslib.Foundations.Semantics.LTS.Relation

Cslib/Foundations/Semantics/LTS/LTSCat/Basic.lean

@@ -53,7 +53,8 @@ A morphism between two labelled transition systems consists of (1) a function on
 states, (2) a partial function on labels, and a proof that (1) preserves each
 transition along (2).
 -/
-structure LTS.Morphism (lts₁ lts₂ : LTSCat) : Type where
+@[ext]
+structure LTS.Morphism (lts₁ lts₂ : LTSCat.{u, v}) : Type (max u v) where
   /-- Mapping of states of `lts₁` to states of `lts₂` -/
   stateMap : lts₁.State → lts₂.State
   /-- Mapping of labels of `lts₁` to labels of `lts₂` -/
@@ -63,7 +64,7 @@ structure LTS.Morphism (lts₁ lts₂ : LTSCat) : Type where
     lts₁.lts.Tr s l s' → (withIdle lts₂.lts).Tr (stateMap s) (labelMap l) (stateMap s')
 
 /-- The identity LTS morphism. -/
-def LTS.Morphism.id (lts : LTSCat) : LTS.Morphism lts lts where
+def LTS.Morphism.id (lts : LTSCat.{u, v}) : LTS.Morphism lts lts where
   stateMap := _root_.id
   labelMap := pure
   labelMap_tr _ _ _ := _root_.id
@@ -72,7 +73,8 @@ def LTS.Morphism.id (lts : LTSCat) : LTS.Morphism lts lts where
 
 We use Kleisli composition to define this.
 -/
-def LTS.Morphism.comp {lts₁ lts₂ lts₃} (f : LTS.Morphism lts₁ lts₂) (g : LTS.Morphism lts₂ lts₃) :
+def LTS.Morphism.comp {lts₁ lts₂ lts₃ : LTSCat.{u, v}}
+    (f : LTS.Morphism lts₁ lts₂) (g : LTS.Morphism lts₂ lts₃) :
     LTS.Morphism lts₁ lts₃ where
   stateMap := g.stateMap ∘ f.stateMap
   labelMap := f.labelMap >=> g.labelMap
@@ -84,7 +86,7 @@ def LTS.Morphism.comp {lts₁ lts₂ lts₃} (f : LTS.Morphism lts₁ lts₂) (g
     cases hμ : μ l with grind
 
 /-- Finally, we prove that these form a category. -/
-instance : CategoryTheory.Category LTSCat where
+instance : CategoryTheory.Category LTSCat.{u, v} where
   Hom := LTS.Morphism
   id := LTS.Morphism.id
   comp := LTS.Morphism.comp

Cslib/Foundations/Semantics/LTS/LTSCat/Coalgebra.lean

@@ -0,0 +1,193 @@
+/-
+Copyright (c) 2026 Tanner Duve (Logical Intelligence). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Tanner Duve
+-/
+
+module
+
+public import Cslib.Foundations.Semantics.LTS.Bisimulation
+public import Cslib.Foundations.Semantics.LTS.LTSCat.Basic
+public import Mathlib.CategoryTheory.Endofunctor.Algebra
+public import Mathlib.CategoryTheory.Types.Basic
+public import Mathlib.CategoryTheory.Functor.FullyFaithful
+
+/-!
+# Labelled transition systems as coalgebras
+
+For a label set `L`, a labelled transition system on `X` is the same datum as a
+coalgebra `α : X → Set (L × X)` for the `Set`-endofunctor `F_L X = Set (L × X)`.
+This file defines `F_L` as an endofunctor of `Type u`, exhibits LTSs as
+coalgebras via Mathlib's `CategoryTheory.Endofunctor.Coalgebra`, and proves
+that Mathlib's coalgebra morphisms coincide with functional bisimulations on
+the underlying LTSs.
+
+## Main definitions
+
+- `LTS.Endo`: the endofunctor `F_L X = Set (L × X)` on `Type u`.
+- `LTS.Coalgebra.toLTSCat`: the forgetful functor from coalgebras of `F_L` to
+  `LTSCat` that maps a functional bisimulation to its underlying simulation.
+
+## Main results
+
+- `LTS.Coalgebra.instFaithful`: `toLTSCat` is faithful.
+- `LTS.Coalgebra.graphBisimEquiv`: coalgebra morphisms `V ⟶ W` are in bijection
+  with the state maps `V.V → W.V` whose graph is a functional bisimulation
+  between the underlying LTSs.
+
+## References
+
+* [J. Rutten, *Universal coalgebra: a theory of systems*][Rutten2000]
+* [G. Winskel and M. Nielsen, *Models for concurrency*][WinskelNielsen1995]
+-/
+
+@[expose] public section
+
+namespace Cslib
+
+universe u
+
+open CategoryTheory TypeCat Endofunctor
+
+namespace LTS
+
+/-- Action on morphisms of the LTS endofunctor: image of a set under
+`(l, x) ↦ (l, f x)`. -/
+def Endo.mapFn (Label : Type u) {X Y : Type u} (f : X → Y) (S : Set (Label × X)) :
+    Set (Label × Y) :=
+  {p | ∃ x : X, (p.1, x) ∈ S ∧ f x = p.2}
+
+namespace Endo
+
+variable {Label : Type u} {X Y Z : Type u}
+
+@[simp] lemma mem_mapFn (f : X → Y) (S : Set (Label × X)) (p : Label × Y) :
+    p ∈ mapFn Label f S ↔ ∃ x : X, (p.1, x) ∈ S ∧ f x = p.2 := Iff.rfl
+
+lemma mapFn_id : mapFn Label (_root_.id : X → X) = _root_.id := by
+  funext S
+  ext ⟨l, x⟩
+  constructor
+  · rintro ⟨_, hw, rfl⟩
+    exact hw
+  · exact fun hx => ⟨x, hx, rfl⟩
+
+lemma mapFn_comp (f : X → Y) (g : Y → Z) :
+    mapFn Label (g ∘ f) = mapFn Label g ∘ mapFn Label f := by
+  funext S
+  ext ⟨l, z⟩
+  constructor
+  · rintro ⟨x, hx, rfl⟩
+    exact ⟨f x, ⟨x, hx, rfl⟩, rfl⟩
+  · rintro ⟨y, ⟨x, hx, rfl⟩, rfl⟩
+    exact ⟨x, hx, rfl⟩
+
+end Endo
+
+/-- The `Set`-endofunctor `F_L X = Set (L × X)` on `Type u`. -/
+@[reducible] def Endo (Label : Type u) : Type u ⥤ Type u where
+  obj X := Set (Label × X)
+  map {_ _} f := ofHom (Endo.mapFn Label f)
+  map_id _ := by aesop
+  map_comp f g := by aesop
+
+/-- The category of LTS-coalgebras for a fixed label set, as Mathlib's
+`Endofunctor.Coalgebra` applied to `Endo Label`. Morphisms are the strict
+commuting-square coalgebra morphisms, which are precisely the functional
+bisimulations on the underlying LTSs (see `Coalgebra.graph_isBisimulation` and
+`Coalgebra.homOfGraphBisim`). -/
+abbrev Coalgebra (Label : Type u) : Type (u + 1) := Endofunctor.Coalgebra (Endo Label)
+
+namespace Coalgebra
+
+variable {Label : Type u}
+
+/-- The LTS underlying a coalgebra of `Endo Label`. -/
+abbrev toLTS (V : Coalgebra Label) : LTS V.V Label where
+  Tr s l s' := (l, s') ∈ (V.str : V.V → _) s
+
+@[simp] lemma toLTS_Tr (V : Coalgebra Label) (s : V.V) (l : Label) (s' : V.V) :
+    (toLTS V).Tr s l s' ↔ (l, s') ∈ (V.str : V.V → _) s := Iff.rfl
+
+variable (Label) in
+/-- The forgetful functor `Coalgebra Label ⥤ LTSCat` mapping each functional
+bisimulation to its underlying simulation. -/
+def toLTSCat : Coalgebra Label ⥤ LTSCat.{u, u} where
+  obj V := ({ State := V.V, Label := Label, lts := toLTS V } : LTSCat.{u, u})
+  map {V W} f :=
+    { stateMap := (f.f : V.V → W.V)
+      labelMap := pure
+      labelMap_tr := by
+        intro s s' l h
+        have hcomm := ConcreteCategory.congr_hom f.h s
+        simp_all only [types_comp_apply, Endo, ofHom_apply, withIdle, toLTS]
+        rw [← hcomm]
+        exact ⟨s', h, rfl⟩ }
+  map_id _ := rfl
+  map_comp _ _ := by
+    apply Morphism.ext
+    · rfl
+    · exact (fish_pure pure).symm
+
+/-- `toLTSCat` is faithful: a functional bisimulation is determined by its
+underlying state map. -/
+instance : (toLTSCat Label).Faithful where
+  map_injective {_ _} _ _ h := by
+    apply Endofunctor.Coalgebra.Hom.ext
+    apply Hom.ext
+    apply Fun.ext
+    funext x
+    exact congrFun (congrArg Morphism.stateMap h) x
+
+/-- Mathlib's coalgebra morphisms are precisely the functional bisimulations of
+the underlying LTSs. Forward direction: every coalgebra morphism, viewed as the
+graph of its underlying state map, is an `LTS.IsBisimulation`. -/
+theorem graph_isBisimulation {V W : Coalgebra Label} (f : V ⟶ W) :
+    LTS.IsBisimulation (toLTS V) (toLTS W) (fun s t => (f.f : V.V → W.V) s = t) := by
+  intro s t hst l
+  have hcomm := ConcreteCategory.congr_hom f.h s
+  simp_all only [types_comp_apply, Endo, ofHom_apply]
+  constructor
+  · intro s' h
+    exact ⟨(f.f : V.V → W.V) s', hcomm ▸ ⟨s', h, rfl⟩, rfl⟩
+  · intro t' h
+    obtain ⟨s', hs', rfl⟩ := hcomm ▸ h
+    exact ⟨s', hs', rfl⟩
+
+/-- Backward direction: any function whose graph is an LTS bisimulation is the
+underlying state map of a (unique) coalgebra morphism. -/
+def homOfGraphBisim {V W : Coalgebra Label} (f : V.V → W.V)
+    (h : LTS.IsBisimulation (toLTS V) (toLTS W) (fun s t => f s = t)) : V ⟶ W where
+  f := ofHom f
+  h := by
+    apply Hom.ext
+    apply Fun.ext
+    funext s
+    ext ⟨l, t'⟩
+    simp only [Endo]
+    constructor
+    · rintro ⟨s', hs', rfl⟩
+      obtain ⟨_, ht₂, rfl⟩ := (h rfl l).1 _ hs'
+      exact ht₂
+    · intro ht'
+      obtain ⟨s', hs', rfl⟩ := (h rfl l).2 _ ht'
+      exact ⟨s', hs', rfl⟩
+
+/-- Coalgebra morphisms `V ⟶ W` are in bijection with the state maps `V.V → W.V` whose graph is
+a functional bisimulation between the underlying LTSs. -/
+def graphBisimEquiv {V W : Coalgebra Label} :
+    (V ⟶ W) ≃
+      { f : V.V → W.V //
+          LTS.IsBisimulation (toLTS V) (toLTS W) (fun s t => f s = t) } where
+  toFun φ := ⟨φ.f, graph_isBisimulation φ⟩
+  invFun p := homOfGraphBisim p.1 p.2
+  left_inv φ := by
+    apply Endofunctor.Coalgebra.Hom.ext
+    rfl
+  right_inv _ := rfl
+
+end Coalgebra
+
+end LTS
+
+end Cslib

references.bib

@@ -358,6 +358,17 @@ @book{KearnsVazirani1994
   address      = {Cambridge, MA, USA}
 }
 
+@article{Rutten2000,
+  author       = {Rutten, J. J. M. M.},
+  title        = {Universal coalgebra: a theory of systems},
+  journal      = {Theoretical Computer Science},
+  volume       = {249},
+  number       = {1},
+  pages        = {3--80},
+  year         = {2000},
+  doi          = {10.1016/S0304-3975(00)00056-6}
+}
+
 @incollection{WinskelNielsen1995,
   author       = {Winskel, Glynn and Nielsen, Mogens},
   isbn         = {9780198537809},