Name-passing process calculi : operational models and structural operational semantics

rule induction. Together, an abstract rule of this form and a U-structured B-coalgebra, (X ,h : UX → BUX ), provide parameterised recursion data (C ,Σ,T,UX ,BU T̃X , fρ,X , gX ,h) where the morphisms fρ,X , gX ,h are defined as follows. fρ,X = Σ(TUX × BU T̃X ) Σ(uX×... ) −−−−−→ Σ(UT̃X × BU T̃X ) ρ T̃ X −→ BU T̃ T̃ X BUμ̃X −→ BU T̃X gX ,h = UX h −→ BUX BUη̃X −−−→ BU T̃X . (6.2.12) Hence an abstract rule ρ gives rise to an operator Tρ which assigns to each U-structured B-coalgebra (X ,h) a U-structured B-coalgebra with carrier T̃ X , and with structure given by Tρh= UT̃X uX −→ TUX (C ,Σ,T,UX ,BU T̃X , fρ,X ,gX ,h) ♯ −−−−−−−−−−−−−−−→ BU T̃X . (6.2.13) Theorem 6.2.14. The operator Tρ defined in equation 6.2.13 defines a strict lifting of the monad T̃ along the forgetful functor from the category of U-structured B-coalgebras. Proof. The forgetful functor (U ,B)-Coalg → D is faithful, so we make use of the characterisation highlighted in Remark 6.2.4. We begin by proving requirement (1) of that remark: for any coalgebra homomorphism α : (X ,h) → (Y, k), the morphism T̃α in D is a map of coalgebras, i.e. the following diagram commutes in C .