A Gentle Introduction to Multiparty Asynchronous Session Types

Abstract

This article provides a gentle introduction to multiparty session types, a class of behavioural types specifically targeted at describing protocols in distributed systems based on asynchronous communication. The type system ensures well-typed processes to enjoy non-trivial properties, including communication safety, protocol fidelity, as well as progress. The adoption of multiparty session types can positively affect the whole software lifecycle, from design to deployment, improving software reliability and reducing its development costs.

A Properties of the Communication Type System

This appendix completes the description of the communication type system given in Sect. 4. Auxiliary lemmas, in particular inversion lemmas, are the content of Sect. A.1. Lastly Sect. A.2 proves subject reduction.

1.1 A.1 Auxiliary Lemmas

We start with inversion lemmas which can be easily shown by induction on derivations.

Lemma 2

(Inversion Lemma for Pure Processes).

• 1. If $\varGamma \vdash u:{S}$, then $u:{S}\in \varGamma$.

• 2. If $\varGamma \vdash {\mathsf{true}}:{S}$, then ${S}={\mathsf{bool}}$.

• 3. If $\varGamma \vdash {\mathsf{false}}:{S}$, then ${S}={\mathsf{bool}}$.

• 4. If $\varGamma \vdash {e_1\ {\mathsf{and}}\ e_2}:{S}$, then $\varGamma \vdash e_1:{\mathsf{bool}}$ and $\varGamma \vdash e_2:{\mathsf{bool}}$ and ${S}={\mathsf{bool}}$.

• 5. If ${\varGamma \vdash {{\overline{\,{ a}\,}[{\mathtt {p}}]({y})}.{ P}}\triangleright {\varDelta }}$, then ${\varGamma \vdash { a}:{ \mathsf {{G}}}}$ and ${\varGamma \vdash { P}\triangleright {\varDelta },{y}{:}\mathsf {{G}}\upharpoonright {\mathtt {p}}}$ and ${\mathtt {p}}={\text {mp}(\mathsf {{G}})}$.

• 6. If ${\varGamma \vdash {{{ a}[{\mathtt {p}}]({y})}.{ P}}\triangleright {\varDelta }}$, then ${\varGamma \vdash { a}:{ \mathsf {{G}}}}$ and ${\varGamma \vdash { P}\triangleright {\varDelta },{y}{:}\mathsf {{G}}\upharpoonright {\mathtt {p}}}$ and ${\mathtt {p}}<{\text {mp}(\mathsf {{G}})}$.

• 7. If ${\varGamma \vdash {{c}!\langle {\mathtt {p}},{e}\rangle .{ P}}\triangleright {\varDelta }}$, then ${\varDelta }={\varDelta }',{c}:{!}\langle {\mathtt {p}},{{{S}}}\rangle .{T}$ and ${\varGamma \vdash {{e}}:{{{S}}}}$ and ${\varGamma \vdash { P}\triangleright {\varDelta }',{c}:{T}}$.

• 8. If ${\varGamma \vdash {{c}?( {\mathtt {q}},{x}).{ P}}\triangleright {\varDelta }}$, then ${\varDelta }={\varDelta }',{c}:{?( {\mathtt {q}},{{{S}}} )}.{T}$ and ${\varGamma ,{{{x}}}:{{{S}}}\vdash { P}\triangleright {\varDelta }',{c}:{T}}$.

• 9. If ${\varGamma \vdash {{c}!\langle \! \langle {\mathtt {p}},{c}'\rangle \!\rangle .{ P}}\triangleright {\varDelta }}$, then ${\varDelta }={\varDelta }',{c}:{!}\langle {\mathtt {p}},\mathsf {{T}}\rangle .{T},{c}':\mathsf {{T}}$ and ${\varGamma \vdash { P}\triangleright {\varDelta }',{c}:{T}}$.

• 10. If ${\varGamma \vdash {{c}?(\!({\mathtt {q}},{y})\!).{ P}}\triangleright {\varDelta }}$, then ${\varDelta }={\varDelta }',{c}:{?( {\mathtt {q}},\mathsf {{T}} )}.{T}$ and ${\varGamma \vdash { P}\triangleright {\varDelta }',{c}:{T},{y}:\mathsf {{T}}}$.

• 11. If ${\varGamma \vdash {{c} \oplus {\langle {\mathtt {p}}, l_j\rangle }.{ P}}\triangleright {\varDelta }}$, then ${\varDelta }={\varDelta }',{c}:\oplus \langle {\mathtt {p}},\{l_i:{T}_i\}_{i\in I} \rangle$ and ${\varGamma \vdash { P}\triangleright {\varDelta }',{c}:}$ ${{T}_j}$ and $j\in I$.

• 12. If ${\varGamma \vdash {{c} \& ({{\mathtt {p}}},{\{L_i : { P}_i\}_{i \in I}})}\triangleright {\varDelta }}$, then ${\varDelta }={\varDelta }', {c}: \& ({\mathtt {p}},\{l_i:{T}_i\}_{i\in I})$ and ${\varGamma \vdash { P}_i\triangleright {\varDelta }',{c}:{T}_i}$ $\forall i\in I$.

• 13. If ${\varGamma \vdash { P}{\ |\ }{Q}\triangleright {\varDelta }}$, then ${\varDelta }={\varDelta }',{\varDelta }''$ and ${\varGamma \vdash { P}\triangleright {\varDelta }}'$ and ${\varGamma \vdash {Q}\triangleright {\varDelta }''}$.

• 14. If ${\varGamma \vdash {\mathsf{if}} ~{{e}}~{\mathsf{then}} ~{{ P}}~{\mathsf{else}} ~{{Q}}\triangleright {\varDelta }}$, then ${\varGamma \vdash {e}:{\mathsf{bool}}}$ and ${\varGamma \vdash { P}\triangleright {\varDelta }}$ and ${\varGamma \vdash {Q}\triangleright {\varDelta }}$.

• 15. If ${\varGamma \vdash {\mathbf {0}}\triangleright {\varDelta }}$, then $\varDelta \ {\mathsf{end}}\ \text {only}$.

• 16. If $\varGamma \vdash (\nu a) P \triangleright \varDelta$, then $\varGamma ,a:\mathsf {{G}}\vdash P\triangleright \varDelta$.

• 17. If ${\varGamma \vdash {{X}\langle {{{e}}},{{{c}}}\rangle }\triangleright {\varDelta }}$, then $\varGamma =\varGamma ', {X : S\;T}$ and ${\varDelta }={\varDelta }', {{{c}}}:{{{T}}}$ and ${\varGamma \vdash {{{e}}}:{{{S}}}}$ and $\varDelta '\ {\mathsf{end}}\ \text {only}$.

• 18. If ${\varGamma \vdash {{\mathsf{def}} \ {{{X}({x}, {y})}={ P}}\ {\mathsf{in}}\ }{{Q}}\triangleright {\varDelta }}$, then ${\varGamma , {{X}: {{{S}}}\;\mathbf t }, {x}:{{{S}}}\vdash { P}\triangleright \{{y}:{{{{T}}}}\}}$ and ${\varGamma , {{X}: {{{S}}}\;\mu \mathbf t .{{{{T}}}}}\vdash {Q}\triangleright {\varDelta }}$.

Lemma 3

(Inversion Lemma for Processes).

• 1. If ${\varGamma \vdash _{\varSigma } { P} \triangleright {\varDelta }}$ and ${ P}$ is a pure process, then $\varSigma =\emptyset$ and ${\varGamma \vdash { P}\triangleright {\varDelta }}$.

• 2. If ${\varGamma \vdash _{\varSigma } {{ s} : {h}} \triangleright {\varDelta }}$, then $\varSigma = {\{{ s}\}}$.

• 3. If ${\varGamma \vdash _{{\{{ s}\}}} {{ s} : \phi } \triangleright {\varDelta }}$, then ${\varDelta }~\mathtt{end}\,only$.

• 4. If ${\varGamma }\vdash _{\{{ s}\}} {{{ s}} : {{h} \cdot {({{\mathtt {q}}},{{\mathtt {p}}},{v})}}} \triangleright {{\varDelta }}$, then ${\varDelta }\approx {\varDelta }';\{{{ s}[{\mathtt {q}}]} : \;!\langle {{\mathtt {p}}},{S}\rangle \}$ and ${\varGamma } \vdash _{\{{ s}\}} {{{ s}} : {h}} \triangleright {{\varDelta }'}$ and ${\varGamma }\vdash {v}:{S}$.

• 5. If ${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{{{h}} \cdot {({\mathtt {q}},{{\mathtt {p}}},{{ s}'[{\mathtt {p}}']})}}}} \triangleright {\varDelta }}$, then ${{\varDelta }\approx ({\varDelta }'{;}{\{{{ s}[{\mathtt {q}}]} :\;{!}\langle {\mathtt {p}},\mathsf {{T}}\rangle \}}),{{ s}'[{\mathtt {p}}']}:\mathsf {{T}}}$ and ${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{h}}} \triangleright {\varDelta }'}$.

• 6. If ${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{{{h}} \cdot {({\mathtt {q}},{\mathtt {p}},l)}}}} \triangleright {\varDelta }}$, then ${{\varDelta }\approx {\varDelta }'{;}{\{{{ s}[{\mathtt {q}}]} :\oplus \langle {\mathtt {p}},l\rangle \}}}$ and ${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{h}}} \triangleright {\varDelta }'}$.

• 7. If ${\varGamma \vdash _{\varSigma } { P}{\ |\ }{Q} \triangleright {\varDelta }}$, then $\varSigma = \varSigma _1\cup \varSigma _2$ and $\varSigma _1\cap \varSigma _2=\emptyset$ and ${\varDelta }= {\varDelta }_1\,{*}\,{\varDelta }_2$ and ${\varGamma \vdash _{\varSigma _1} { P} \triangleright {\varDelta }_1}$ and ${\varGamma \vdash _{\varSigma _2} {Q} \triangleright {\varDelta }_2}$.

• 8. If ${\varGamma \vdash _{\varSigma } (\nu { s}){ P} \triangleright {\varDelta }}$, then $\varSigma =\varSigma '\setminus { s}$ and ${\varDelta }={\varDelta }'\setminus { s}$ and ${\mathsf {co}({{\varDelta }'},{{ s}})}$ and ${\varGamma \vdash _{\varSigma '} { P} \triangleright {\varDelta }'}$.

• 9. If ${\varGamma \vdash _{\varSigma } (\nu { a}){ P} \triangleright {\varDelta }}$, then ${\varGamma ,{ a}:\mathsf {{G}} \vdash _{\varSigma } { P} \triangleright {\varDelta }}$.

• 10. If ${\varGamma \vdash _{\varSigma } {{\mathsf{def}} \ {{{X}({x}, {y})}={ P}}\ {\mathsf{in}}\ }{{Q}} \triangleright {\varDelta }}$, then ${\varGamma , {{X}: {{{S}}}\;\mathbf t }, {x}:{{{S}}}\vdash { P}\triangleright {y}:{{{{T}}}}}$ and ${\varGamma , {{X}: {{{S}}}\;\mu \mathbf t .{{{{T}}}}} \vdash _{\varSigma } {Q} \triangleright {\varDelta }}$.

The following lemma allows to characterise the types due to the messages which occur in queues. The proof is standard by induction on the lengths of queues.

Lemma 4

• 1. If ${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{{{h}}_1 \cdot {{({\mathtt {q}},{\mathtt {p}},{ v})} \cdot {{h}}_2}}}} \triangleright {\varDelta }}$, then ${\varDelta }= {\varDelta }_1\,{*}\,{\{{{ s}[{\mathtt {q}}]}:\;{!}\langle {\mathtt {p}},{{{S}}}\rangle \}}\,{*}\,{\varDelta }_2$ and ${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{h}}}_i \triangleright {\varDelta }_i}$ ($i=1,2$) and ${\varGamma \vdash v:{{{S}}}}$.

  Vice versa ${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{h}}}_i \triangleright {\varDelta }_i}$ ($i=1,2$) and ${\varGamma \vdash v:{{{S}}}}$ imply

  $${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{{{h}}_1 \cdot {{({\mathtt {q}},{\mathtt {p}},{ v})} \cdot {{h}}_2}}}} \triangleright {\varDelta }_1{*}{\{{{ s}[{\mathtt {q}}]} :\;{!}\langle {\mathtt {p}},{{{S}}}\rangle \}}{*}{\varDelta }_2}.$$

• 2. If ${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{{{h}}_1 \cdot {{({\mathtt {q}},{{\mathtt {p}}},{{ s}'[{\mathtt {p}}']})} \cdot {{h}}_2}}}} \triangleright {\varDelta }}$, then ${\varDelta }= ({\varDelta }_1{*}{\{{{ s}[{\mathtt {q}}]} :\;{!}\langle {\mathtt {p}},\mathsf {{T}}\rangle \}}{*}{\varDelta }_2),{{ s}'[{\mathtt {p}}']}:\mathsf {{T}}$ and ${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{h}}}_i \triangleright {\varDelta }_i}$ ($i=1,2$).

  Vice versa ${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{h}}}_i \triangleright {\varDelta }_i}$ ($i=1,2$) imply

  $${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{{{h}}_1 \cdot {{({\mathtt {q}},{{\mathtt {p}}},{{ s}'[{\mathtt {p}}']})} \cdot {{h}}_2}}}} \triangleright ({\varDelta }_1{*}{\{{{ s}[{\mathtt {q}}]} :\;{!}\langle {\mathtt {p}},\mathsf {{T}}\rangle \}}{*}{\varDelta }_2),{{ s}'[{\mathtt {p}}']}:\mathsf {{T}}}.$$

• 3. If ${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{{{h}}_1 \cdot {{({\mathtt {q}},{\mathtt {p}},l)} \cdot {{h}}_2}}}} \triangleright {\varDelta }}$, then ${\varDelta }= {\varDelta }_1{*}{\{{{ s}[{\mathtt {q}}]} :\oplus \langle {\mathtt {p}},l\rangle \}}{*}{\varDelta }_2$ and ${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{h}}}_i \triangleright {\varDelta }_i}$ ($i=1,2$). para Vice versa ${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{h}}}_i \triangleright {\varDelta }_i}$ ($i=1,2$) imply

  $${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{{{h}}_1 \cdot {{({\mathtt {q}},{\mathtt {p}},l)} \cdot {{h}}_2}}}} \triangleright {\varDelta }_1{*}{\{{{ s}[{\mathtt {q}}]} :\oplus \langle {\mathtt {p}},l\rangle \}}{*}{\varDelta }_2}.$$

We end this subsection with two classical results: type preservation under substitution and under equivalence of processes.

Lemma 5

(Substitution Lemma).

• 1. If ${\varGamma ,{{{x}}}:{{{S}}}\vdash { P}\triangleright {\varDelta }}$ and ${\varGamma \vdash {{v}}:{{{S}}}}$, then ${\varGamma \vdash { P}{\{{{{ v}}} / {{{x}}}\}}\triangleright {\varDelta }}$.

• 2. If ${\varGamma \vdash { P}\triangleright {\varDelta },{y}:{T}}$, then ${\varGamma \vdash { P}{\{{{ s}[{\mathtt {p}}]}/{y}\}}\triangleright {\varDelta },{{ s}[{\mathtt {p}}]}:{T}}$.

Proof

Standard induction on type derivations, with a case analysis on the last applied rule.   $\square$

Theorem 2

(Type Preservation Under Equivalence). If ${\varGamma \vdash _{\varSigma } { P} \triangleright {\varDelta }}$ and ${ P}\equiv { P}'$, then ${\varGamma \vdash _{\varSigma } { P}' \triangleright {\varDelta }}$.

Proof

By induction on $\equiv$. We only consider some interesting cases (the other cases are straightforward).

• ${ P}{\ |\ }{\mathbf {0}}\equiv { P}$. First we assume ${\varGamma \vdash _{\varSigma } { P} \triangleright {\varDelta }}$. From ${\varGamma \vdash _{\emptyset } {\mathbf {0}} \triangleright \emptyset }$ by applying (GPar) to these two sequents we obtain ${\varGamma \vdash _{\varSigma } { P}|{\mathbf {0}} \triangleright {\varDelta }}$.

  For the converse direction assume ${\varGamma \vdash _{\varSigma } { P}|{\mathbf {0}} \triangleright {\varDelta }}$. Using 3(7) we obtain: ${\varGamma \vdash _{\varSigma _1} { P} \triangleright {\varDelta }_1}$, ${\varGamma \vdash _{\varSigma _2} {\mathbf {0}} \triangleright {\varDelta }_2}$, where ${\varDelta }={\varDelta }_1{*}{\varDelta }_2,\ \varSigma =\varSigma _1\cup \varSigma _2$ and $\varSigma _1\cap \varSigma _2=\emptyset$. Using 3(1) we get $\varSigma _2=\emptyset$, which implies $\varSigma =\varSigma _1$, and ${\varGamma \vdash {\mathbf {0}}\triangleright {\varDelta }_2}$. Using 2(15) we get ${\varDelta }_2\ {\mathsf{end}}\ \text {only}$ which implies ${{\varDelta }_1\approx {\varDelta }_1 {*}{\varDelta }_2}$, so we conclude ${\varGamma \vdash _{\varSigma } { P} \triangleright {\varDelta }_1 {*}{\varDelta }_2}$ by applying (Equiv).

• ${ P}{\ |\ }{Q}\equiv {Q}{\ |\ }{ P}$. By the symmetry of the rule we have to show only one direction. Suppose ${\varGamma \vdash _{\varSigma } { P}{\ |\ }{Q} \triangleright {\varDelta }}$. Using 3(7) we obtain ${\varGamma \vdash _{\varSigma _1} { P} \triangleright {\varDelta }_1}$, ${\varGamma \vdash _{\varSigma _2} {Q} \triangleright {\varDelta }_2}$, where ${\varDelta }={\varDelta }_1{*}{\varDelta }_2$, $\varSigma =\varSigma _1\cup \varSigma _2$ and $\varSigma _1\cap \varSigma _2=\emptyset$. Using (GPar) we get ${\varGamma \vdash _{\varSigma } {Q}{\ |\ }{ P} \triangleright {\varDelta }_2{*}{\varDelta }_1}$. Thanks to the commutativity of ${*}$, we get ${\varDelta }_2{*}{\varDelta }_1={\varDelta }$ and so we are done.

• ${ P}{\ |\ }({Q}{\ |\ }R) \equiv ({ P}{\ |\ }{Q}){\ |\ }R$. Suppose ${\varGamma \vdash _{\varSigma } { P}{\ |\ }({Q}{\ |\ }R) \triangleright {\varDelta }}$. Using 3(7) we obtain ${\varGamma \vdash _{\varSigma _1} { P} \triangleright {\varDelta }_1}$, ${\varGamma \vdash _{\varSigma _2} {Q}{\ |\ }R \triangleright {\varDelta }_2}$, where ${\varDelta }={\varDelta }_1{*}{\varDelta }_2$, $\varSigma =\varSigma _1\cup \varSigma _2$ and $\varSigma _1\cap \varSigma _2=\emptyset$. Using 3(7) we obtain ${\varGamma \vdash _{\varSigma _{21}} {Q} \triangleright {\varDelta }_{21}}$, ${\varGamma \vdash _{\varSigma _{22}} R \triangleright {\varDelta }_{22}}$ where ${\varDelta }_2={\varDelta }_{21}{*}{\varDelta }_{22},\ \varSigma _2=\varSigma _{21}\cup \varSigma _{22}$ and $\varSigma _{21}\cap \varSigma _{22}=\emptyset$. Using (GPar) we get ${\varGamma \vdash _{\varSigma _1\cup \varSigma _{21}} { P}{\ |\ }{Q} \triangleright {\varDelta }_1{*}{\varDelta }_{21}}$. Using (GPar) again we get ${\varGamma \vdash _{\varSigma } ({ P}{\ |\ }{Q}){\ |\ }R \triangleright {\varDelta }_1{*}{\varDelta }_{21}{*}{\varDelta }_{22}}$ and so we are done by the associativity of ${*}$. The proof for the other direction is similar.

• ${{ s} : {{{{h}}_1 \cdot {{{({\mathtt {q}},{\mathtt {p}}, { v})} \cdot {({\mathtt {q}}',{\mathtt {p}}', { v}')}} \cdot {{h}}_2}}} } \equiv {{ s} : {{{{h}}_1 \cdot { {{({\mathtt {q}}',{\mathtt {p}}', { v}')} \cdot {({\mathtt {q}},{\mathtt {p}}, { v})}} \cdot {{h}}_2}}} }$ where ${\mathtt {p}}\ne {\mathtt {p}}'$ or ${\mathtt {q}}\ne {\mathtt {q}}'$. We assume ${\mathtt {p}}\ne {\mathtt {p}}'$ and ${\mathtt {q}}= {\mathtt {q}}'$, the proof in the case ${\mathtt {q}}\ne {\mathtt {q}}'$ being similar and simpler. If ${\varGamma \vdash _{\varSigma } {{ s} : {{{{h}}_1 \cdot {{{({\mathtt {q}},{\mathtt {p}}, { v})} \cdot {({\mathtt {q}},{\mathtt {p}}', { v}')}} \cdot {{h}}_2}}}} \triangleright {\varDelta }}$, then $\varSigma = {\{{ s}\}}$ by Lemma 3(2). This implies ${\varDelta }= {\varDelta }_1{*}{\{{{ s}[{\mathtt {q}}]} :\;{!}\langle {\mathtt {p}},{{{S}}}\rangle ;\;{!}\langle {\mathtt {p}}',{{{S}}}'\rangle \}}{*}{\varDelta }_2$ and ${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{h}}}_i \triangleright {\varDelta }_i}$ ($i=1,2$) and ${\varGamma \vdash v:{{{S}}}}$ and ${\varGamma \vdash v':{{{S}}}'}$ by Lemma 4(1). By the same lemma we can derive

  $${\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{{{h}}_1 \cdot {{{({\mathtt {q}},{\mathtt {p}}', { v}')} \cdot {({\mathtt {q}},{\mathtt {p}}, { v})}} \cdot {{h}}_2}}}} \triangleright {\varDelta }_1{*}{\{{{ s}[{\mathtt {q}}]} :\;{!}\langle {\mathtt {p}}',{{{S}}}'\rangle ;\;{!}\langle {\mathtt {p}},{{{S}}}\rangle \}}{*}{\varDelta }_2},$$

  and we conclude using rule (Equiv), since by definition

  $${{\varDelta }_1{*}{\{{{ s}[{\mathtt {q}}]} :\;{!}\langle {\mathtt {p}}',{{{S}}}'\rangle ;\;{!}\langle {\mathtt {p}},{{{S}}}\rangle \}}{*}{\varDelta }_2\approx {\varDelta }}.$$

   $\square$

1.2 A.2 Proof of Subject Reduction

We show the Main Lemma first and then the Subject Reduction Theorem.

Lemma 1 (Main Lemma). Let ${\varGamma \vdash _{\varSigma } { P} \triangleright {\varDelta }}$, and ${{ P}{\longrightarrow }{ P}'}$ be obtained by any reduction rule different from [Ctxt], [Str], and ${\varDelta }\,{*}\,{\varDelta }_0$ be consistent, for some ${\varDelta }_0$. Then there is ${\varDelta }'$ such that ${\varGamma \vdash _{\varSigma } { P}' \triangleright {\varDelta }'}$ and ${{{\varDelta }}~\Rightarrow ^*~{{\varDelta }'}}$ and ${\varDelta }'\,{*}\,{\varDelta }_0$ is consistent.

Proof

The proof is by cases on process reduction rules. We only consider some paradigmatic cases.

•  [Init] ${{{ a}[1]( {y})}.{ P}_1} {\ |\ }...{\ |\ }{{\overline{\,{ a}\,}[n]( {y})}.{ P}_n}$ ${\longrightarrow }(\nu { s})( { P}_{1}{\{{{ s}[1]}/{y}_{1}\}} {\ |\ }...{\ |\ }{ P}_n{\{{{ s}[n]}/ {y}\}}{\ |\ }s :~\phi )$.

  By hypothesis ${\varGamma \vdash _{\varSigma } {{{ a}[1]( {y})}.{ P}_1}{\ |\ }{{{ a}[2]({y}_2)}.{ P}_2}{\ |\ }\ldots {\ |\ }{{\overline{\,{ a}\,}[n]( {y})}.{ P}_n} \triangleright {\varDelta }}$; then, since the redex is a pure process, $\varSigma =\emptyset$ and ${\varGamma \vdash {{{ a}[1]( {y})}.{ P}_1}{\ |\ }{{{ a}[2]({y}_2)}.{ P}_2}{\ |\ }\ldots \triangleright }$ ${\vdash {\ |\ }{{\overline{\,{ a}\,}[n]( {y})}.{ P}_n}\triangleright {\varDelta }}$ by Lemma 3(1). Using Lemma 2(13) on all the processes in parallel we have

  $$\begin{aligned}&{\varGamma \vdash {{{ a}[i]( {y})}.{ P}_i}\triangleright {\varDelta }_i}\quad (1\le i\le n-1) \end{aligned}$$

  (1)
  $$\begin{aligned}&{\varGamma \vdash {{\overline{\,{ a}\,}[n]( {y})}.{ P}_n}\triangleright {\varDelta }_n} \end{aligned}$$

  (2)

  where ${\varDelta }= \bigcup _{i=1}^n {\varDelta }_i$. Using Lemma 2(6) on (1) we have

  $$\begin{aligned}&\quad \qquad \qquad \quad \ {\varGamma \vdash { a}:{ \mathsf {{G}}}}\nonumber \\&{\varGamma \vdash { P}_i\triangleright {\varDelta }_i, {y}{:}\mathsf {{G}}\upharpoonright i} \quad (1\le i\le n-1). \end{aligned}$$

  (3)

  Using Lemma 2(5) on (2) we have

  $$\begin{aligned}&\qquad \,{\varGamma \vdash { a}:{ \mathsf {{G}}}}\nonumber \\&{\varGamma \vdash { P}_n\triangleright {\varDelta }_n, {y}:\mathsf {{G}}\upharpoonright n} \end{aligned}$$

  (4)

  and ${\text {mp}(\mathsf {{G}})}= n$. Using Lemma 5(2) on (4) and (3) we have

  $$\begin{aligned} {\varGamma \vdash { P}_i{\{{{ s}[i]}/ {y}\}}\triangleright {\varDelta }_i,{{ s}[i]}:\mathsf {{G}}\upharpoonright i} \quad (1\le i\le n). \end{aligned}$$

  (5)

  Using (Par) on all the processes of (5) we have

  $$\begin{aligned}&{\varGamma \vdash { P}_1{\{{{ s}[1]}/ {y}\}} | ... | { P}_n{\{{{ s}[n]}/ {y}\}}\triangleright \bigcup _{i=1}^n ({\varDelta }_i,{{ s}[i]}:\mathsf {{G}}\upharpoonright i)}. \end{aligned}$$

  (6)

  Note that $\bigcup _{i=1}^n ({\varDelta }_i,{{ s}[i]}{:}\mathsf {{G}}\upharpoonright i) = {\varDelta },{{ s}[1]}{:}{G}\upharpoonright 1,\ldots ,{{ s}[n]}{:}\mathsf {{G}}\upharpoonright n$. Using (GInit), (QInit) and (GPar) on (6) we derive

  $$\begin{aligned}&{\varGamma \vdash _{\{{ s}\}} { P}_1{\{{{ s}[1]}/ {y}\}} | ... | { P}_n{\{{{ s}[n]}/ {y}\}}{\ |\ }s : \phi \triangleright {\varDelta },{{ s}[1]}{:}\mathsf {{G}}\upharpoonright 1,\ldots ,{{ s}[n]}{:}\mathsf {{G}}\upharpoonright n}. \end{aligned}$$

  (7)

  Using (GSRes) on (7) we conclude

  $$\begin{aligned}&{\varGamma \vdash _{\emptyset } (\nu { s})({ P}_1{\{{{ s}[1]}/ {y}\}} | ... | { P}_n{\{{{ s}[n]}/ {y}\}}{\ |\ }s : \phi ) \triangleright {\varDelta }} \end{aligned}$$

  since ${\{{{ s}[1]}{:}\mathsf {{G}}\upharpoonright 1,\ldots ,{{ s}[n]}{:}\mathsf {{G}}\upharpoonright n\}}$ is consistent and

  $$({\varDelta },{{ s}[1]}{:}\mathsf {{G}}\upharpoonright 1,\ldots ,{{ s}[n]}{:}\mathsf {{G}}\upharpoonright n )\setminus {{ s}} = {\varDelta }.$$

•  [Send] ${{{{ s}[{\mathtt {p}}]}!\langle {\mathtt {q}},{e}\rangle .{ P}} {\ |\ }{{ s} : {{h}}}{\longrightarrow }{ P}{\ |\ }{{ s} : {{{{h}} \cdot {({\mathtt {p}},{\mathtt {q}},{ v})}}}}} \ \ ({{e}}\downarrow {{ v}})$.

  By hypothesis, ${\varGamma \vdash _{\varSigma } {{{ s}[{\mathtt {p}}]}!\langle {\mathtt {q}},{e}\rangle .{ P}} {\ |\ }{{ s} : {{h}}} \triangleright {\varDelta }}.$ Using Lemma 3(7), (1), and (2) we have $\varSigma ={\{{ s}\}}$ and

  $$\begin{aligned}&{\varGamma \vdash {{{ s}[{\mathtt {p}}]}!\langle {\mathtt {q}},{e}\rangle .{ P}}\triangleright {\varDelta }_1}\end{aligned}$$

  (8)
  $$\begin{aligned}&{\varGamma \vdash _{\{s\}} {{ s} : {{h}}} \triangleright {\varDelta }_2} \end{aligned}$$

  (9)

  where ${\varDelta }={\varDelta }_2{*}{\varDelta }_1$. Using 2(7) on (8) we have

  $$\begin{aligned}&{\varDelta }_1={\varDelta }_1',{{ s}[{\mathtt {p}}]}:{!}\langle {\mathtt {q}},{{{S}}}\rangle .{T}\nonumber \\&\qquad \ \ \ {\varGamma \vdash {{e}}:{{{S}}}}\end{aligned}$$

  (10)
  $$\begin{aligned}&\ \ {\varGamma \vdash { P}\triangleright {\varDelta }_1',{{ s}[{\mathtt {p}}]}:{T}}. \end{aligned}$$

  (11)

  From (10) by subject reduction on expressions we have

  $$\begin{aligned} {\varGamma \vdash {{v}}:{{{S}}}}. \end{aligned}$$

  (12)

  Using (QSend) on (9) and (12) we derive

  $$\begin{aligned} {\varGamma \vdash _{\{s\}} {{ s} : {{{{h}} \cdot {({\mathtt {p}},{\mathtt {q}},{ v})}}}} \triangleright {\varDelta }_2{;}{\{ {{ s}[{\mathtt {p}}]} : \;{!}\langle {\mathtt {q}},{{{S}}}\rangle \}}}. \end{aligned}$$

  (13)

  Using (GInit) on (11) we derive

  $$\begin{aligned} {\varGamma \vdash _{\emptyset } { P} \triangleright {\varDelta }_1',{{ s}[{\mathtt {p}}]}:{T}} \end{aligned}$$

  (14)

  and then using (GPar) on (14), (13) we conclude

  $$\begin{aligned} {\varGamma \vdash _{\{{ s}\}} { P}{\ |\ }{{ s} : {{{{h}} \cdot {({\mathtt {p}},{\mathtt {q}},{ v})}}}} \triangleright ({\varDelta }_2{;}{\{{{ s}[{\mathtt {p}}]} : \;{!}\langle {\mathtt {q}},{{{S}}}\rangle \}}){*}({\varDelta }_1',{{ s}[{\mathtt {p}}]}:{T})}. \end{aligned}$$

  Note that ${\varDelta }_2{*}({\varDelta }_1',{{ s}[{\mathtt {p}}]}:{!}\langle {\mathtt {q}},{{{S}}}\rangle .{T}) ~\Rightarrow ({\varDelta }_2{;}{\{ {{ s}[{\mathtt {p}}]} : {!}\langle {\mathtt {q}},{{{S}}}\rangle \}}){*}({\varDelta }_1',{{ s}[{\mathtt {p}}]}:{T})$ and the consistency of $({\varDelta }_2{*}({\varDelta }_1',{{ s}[{\mathtt {p}}]}:{!}\langle {\mathtt {q}},{{{S}}}\rangle .{T})){*}{\varDelta }_0$ implies the consistency of $(({\varDelta }_2{;}{\{ {{ s}[{\mathtt {p}}]} : {!}\langle {\mathtt {q}},{{{S}}}\rangle \}}){*}({\varDelta }_1',{{ s}[{\mathtt {p}}]}:{T})){*}{\varDelta }_0$.

•  [Rcv] ${{{ s}[{\mathtt {p}}]}?( {\mathtt {q}},{x}).{ P}} {\ |\ }{{ s} : {{{({\mathtt {q}},{\{{\mathtt {p}}\}},{ v})} \cdot {{h}}}}} {\longrightarrow }{ P}{\{{{{ v}}} / {{{x}}}\}} {\ |\ }{ s}:{h}$.

  By hypothesis, ${\varGamma \vdash _{\varSigma } {{{ s}[{\mathtt {p}}]}?( {\mathtt {q}},{x}).{ P}} {\ |\ }{{ s} : {{{({\mathtt {q}},{\{{\mathtt {p}}\}},{ v})} \cdot {{h}}}}} \triangleright {\varDelta }}.$ By Lemma 3(7), (1), and (2) we have $\varSigma ={\{{ s}\}}$ and

  $$\begin{aligned}&{\varGamma \vdash {{{ s}[{\mathtt {p}}]}?( {\mathtt {q}},{x}).{ P}}\triangleright {\varDelta }_1}\end{aligned}$$

  (15)
  $$\begin{aligned}&{\varGamma \vdash _{\{{ s}\}} {{ s} : {{{({\mathtt {q}},{\{{\mathtt {p}}\}},{ v})} \cdot {{h}}}}} \triangleright {\varDelta }_2} \end{aligned}$$

  (16)

  where ${\varDelta }={\varDelta }_2{*}{\varDelta }_1$. Using Lemma 2(8) on (15) we have

  $$\begin{aligned}&{\varDelta }_1={\varDelta }_1',{{ s}[{\mathtt {p}}]}:{?( {\mathtt {q}},{{{S}}} )}.{T}\nonumber \\&{\varGamma ,{{{x}}}:{{{S}}}\vdash { P}\triangleright {\varDelta }_1',{{ s}[{\mathtt {p}}]}:{T}} \end{aligned}$$

  (17)

  Using Lemma 4(1) on (16) we have

  $$\begin{aligned}&{\varDelta }_2={\{{{ s}[{\mathtt {q}}]} :\;{!}\langle {\{{\mathtt {p}}\}},{{{S}}}'\rangle \}}{*}{\varDelta }_2'\nonumber \\&\quad \quad \ \ {\varGamma \vdash _{{\{{ s}\}}} {{ s} : {{h}}} \triangleright {\varDelta }_2'}\end{aligned}$$

  (18)
  $$\begin{aligned}&{\varGamma \vdash v:{{{S}}}'}. \end{aligned}$$

  (19)

  The consistency of ${\varDelta }{*}{\varDelta }_0$ implies ${{{S}}}={{{S}}}'$. Using Lemma 5(1) from (17) and (19) we get

  ${\varGamma \vdash { P}{\{{{{ v}}} / {{{x}}}\}}\triangleright {\varDelta }_1',{{ s}[{\mathtt {p}}]}:{T}}$, which implies by rule (GInit)

  $$\begin{aligned} {\varGamma \vdash _{\emptyset } { P}{\{{{{ v}}} / {{{x}}}\}} \triangleright {\varDelta }_1',{{ s}[{\mathtt {p}}]}:{T}}. \end{aligned}$$

  (20)

  Using rule (GPar) on (20) and (18) we conclude

  $$\begin{aligned} {\varGamma \vdash _{\{{ s}\}} { P}{\{{{{ v}}} / {{{x}}}\}}{\ |\ }{ s}:{h} \triangleright {\varDelta }_2'{*}({\varDelta }_1',{s[{\mathtt {p}}]}:{T}}).\end{aligned}$$

  Note that $({\{{{ s}[{\mathtt {q}}]} :\;{!}\langle {\{{\mathtt {p}}\}},{{{S}}}\rangle \}}{*}{\varDelta }_2')\,{*}\,({\varDelta }_1',{ s}[{\mathtt {p}}]:{?( {\mathtt {q}},{{{S}}} )};{T})~\Rightarrow {\varDelta }_2'\,{*}\,({\varDelta }_1',{{ s}[{\mathtt {p}}]}:{T})$ and the consistency of $(({\{{{ s}[{\mathtt {q}}]} :\;{!}\langle {\{{\mathtt {p}}\}},{{{S}}}\rangle \}}\,{*}\,{\varDelta }_2')\,{*}\,({\varDelta }_1',{ s}[{\mathtt {p}}]:{?( {\mathtt {q}},{{{S}}} )};{T}))\,{*}\,{\varDelta }_0$ implies the consistency of $({\varDelta }_2'\,{*}\,({\varDelta }_1',{{ s}[{\mathtt {p}}]}:{T}))\,{*}\,{\varDelta }_0$.

•  [Sel] ${{{{ s}[{\mathtt {p}}]} \oplus {\langle {\mathtt {p}}, l\rangle }.{ P}} {\ |\ }{{ s} : {{h}}}{\longrightarrow }{ P}{\ |\ }{{ s} : {{{{h}} \cdot {({\mathtt {p}},{\mathtt {q}},l)}}}}}$.

  By hypothesis, ${\varGamma \vdash _{\varSigma } {{{ s}[{\mathtt {p}}]} \oplus {\langle {\mathtt {q}}, l\rangle }.{ P}} {\ |\ }{{ s} : {{h}}} \triangleright {\varDelta }}$. Using Lemma 3(7), (1), and (2) we have $\varSigma ={\{{ s}\}}$ and

  $$\begin{aligned}&{\varGamma \vdash {{{ s}[{\mathtt {p}}]} \oplus {\langle {\mathtt {q}}, l\rangle }.{ P}}\triangleright {\varDelta }_1}\end{aligned}$$

  (21)
  $$\begin{aligned}&{\varGamma \vdash _{\{{ s}\}} {{ s} : {{h}}} \triangleright {\varDelta }_2} \end{aligned}$$

  (22)

  where ${\varDelta }={\varDelta }_2\,{*}\,{\varDelta }_1$. Using Lemma 2(2) on (21) we have for $l=l_j$ $(j\in I)$:

  $$\begin{aligned}&{\varDelta }_1={\varDelta }_1',{ s}[{\mathtt {p}}]:\oplus \langle {\mathtt {q}},\{l_i:{T}_i\}_{i\in I} \rangle \nonumber \\&\qquad \ \ {\varGamma \vdash { P}\triangleright {\varDelta }_1',{ s}[{\mathtt {p}}]:T_j}. \end{aligned}$$

  (23)

  Using rule (QSel) on (22) we derive

  $$\begin{aligned} {\varGamma \vdash _{\{s\}} {{ s} : {{{{h}} \cdot {({\mathtt {p}},{\mathtt {q}},l)}}}} \triangleright {\varDelta }_2{;}{\{ {{ s}[{\mathtt {p}}]} : \oplus \langle {\mathtt {q}},l\rangle \}}}. \end{aligned}$$

  (24)

  Using (GPar) on (23) and (24) we conclude

  $$\begin{aligned} {\varGamma \vdash _{\{{ s}\}} { P}{\ |\ }{{ s} : {{{{h}} \cdot {({\mathtt {p}},{\mathtt {q}},l)}}}} \triangleright ({\varDelta }_2{;}{\{ {{ s}[{\mathtt {p}}]} : \oplus \langle {\mathtt {q}},l\rangle \}}){*}({\varDelta }_1', {{ s}[{\mathtt {p}}]} : {T}_j )}. \end{aligned}$$

  Note that ${\varDelta }_2\,{*}\,({\varDelta }_1',{ s}[{\mathtt {p}}]:\oplus \langle {\mathtt {q}},\{l_i:{T}_i\}_{i\in I} \rangle )~\Rightarrow ({\varDelta }_2{;}{\{ {{ s}[{\mathtt {p}}]} : \oplus \langle {\mathtt {q}},l\rangle \}})\,{*}\,({\varDelta }_1', {{ s}[{\mathtt {p}}]} : {T}_j )$ and the consistency of $({\varDelta }_2\,{*}\,({\varDelta }_1',{ s}[{\mathtt {p}}]:\oplus \langle {\mathtt {q}},\{l_i:{T}_i\}_{i\in I} \rangle ))\,{*}\,{\varDelta }_0$ implies the consistency of $(({\varDelta }_2{;}{\{ {{ s}[{\mathtt {p}}]} :\oplus \langle {\mathtt {q}},l\rangle \}})\,{*}\,({\varDelta }_1', {{ s}[{\mathtt {p}}]} : {T}_j ))\,{*}\,{\varDelta }_0$.

•  [Branch] ${{{ s}[{\mathtt {p}}]} \& ({{\mathtt {q}}},{\{L_i : { P}_i\}_{i \in I}})} {\ |\ }{{ s} : {{{({\mathtt {q}},{\{{\mathtt {p}}\}},l_{j})} \cdot {{h}}}}} {\longrightarrow }{ P}_{j} {\ |\ }{ s}:{h}$.

  By hypothesis, ${\varGamma \vdash _{\varSigma } {{{ s}[{\mathtt {p}}]} \& ({{\mathtt {q}}},{\{L_i : { P}_i\}_{i \in I}})} {\ |\ }{{ s} : {{{({\mathtt {q}},{\{{\mathtt {p}}\}},l_{j})} \cdot {{h}}}}} \triangleright {\varDelta }}$. Using Lemma 3(7), (1), and (2) we have $\varSigma ={\{{ s}\}}$ and

  $$ \begin{aligned}&{\varGamma \vdash {{{ s}[{\mathtt {p}}]} \& ({{\mathtt {q}}},{\{L_i : { P}_i\}_{i \in I}})}\triangleright {\varDelta }_1}\end{aligned}$$

  (25)
  $$\begin{aligned}&{\varGamma \vdash _{\{{ s}\}} {{ s} : {{{({\mathtt {q}},{\{{\mathtt {p}}\}},l_{j})} \cdot {{h}}}}} \triangleright {\varDelta }_2} \end{aligned}$$

  (26)

  where ${\varDelta }={\varDelta }_2\,{*}\,{\varDelta }_1$. Using Lemma 2(12) on (25) we have

  $$ \begin{aligned}&{\varDelta }_1={\varDelta }_1', {{ s}[{\mathtt {p}}]}: \& ({\mathtt {q}},\{l_i:T_i\}_{i\in I})\nonumber \\&\ \ {\varGamma \vdash { P}_i\triangleright {\varDelta }_1',{{ s}[{\mathtt {p}}]}:{T}_i}\ \ \ \forall i\in I. \end{aligned}$$

  (27)

  Using Lemma 4(3) on (26) we have

  $$\begin{aligned}&{\varDelta }_2={\{{{ s}[{\mathtt {q}}]} :\oplus \langle {\mathtt {p}},l_j\rangle \}}\,{*}\,{\varDelta }_2'\nonumber \\&\quad \quad \ \, {\varGamma \vdash _{\{s\}} {{ s} : {h}} \triangleright {\varDelta }_2'}. \end{aligned}$$

  (28)

  Using (GPar) on (27) and (28) we conclude

  $$\begin{aligned} {\varGamma \vdash _{\{{ s}\}} { P}_{j} {\ |\ }{ s}:h \triangleright {\varDelta }_2'\,{*}\,({\varDelta }_1',{{ s}[{\mathtt {p}}]}:{T}_{j})}. \end{aligned}$$

  Note that

  $$ \begin{aligned} ({\{{{ s}[{\mathtt {q}}]} :\oplus \langle {\mathtt {p}},l_j\rangle \}}\,{*}\,{\varDelta }_2')\,{*}\,({\varDelta }_1', {{ s}[{\mathtt {p}}]}: \& ({\mathtt {q}},\{l_{i}:T_i\}_{i\in I}))~\Rightarrow {\varDelta }_2'\,{*}\,({\varDelta }_1',{{ s}[{\mathtt {p}}]}:{T}_{j}). \end{aligned}$$

  and the consistency of $(({\{{{ s}[{\mathtt {q}}]} \!:\!\oplus \langle {\mathtt {p}},l_j\rangle \}}{*}{\varDelta }_2')\,{*}\,({\varDelta }_1', {{ s}[{\mathtt {p}}]}\!:\! \& ({\mathtt {q}},\{l_{i}\!:\!T_i\}_{i\in I})))\,{*}\,{\varDelta }_0$ implies the consistency of $({\varDelta }_2'\,{*}\,({\varDelta }_1',{{ s}[{\mathtt {p}}]}:{T}_{j}))\,{*}\,{\varDelta }_0$ for $j\in I$   $\square$

Theorem 1 (Subject Reduction). If ${\varGamma \vdash _{\varSigma } { P} \triangleright {\varDelta }}$ with ${\varDelta }$ consistent and ${{ P}{\longrightarrow }^*{ P}'}$, then ${\varGamma \vdash _{\varSigma } { P}' \triangleright {{\varDelta }'}}$ for some consistent ${{\varDelta }'}$ such that ${{{\varDelta }}~\Rightarrow ^*~{{{\varDelta }'}}}$.

Proof

Let ${ P}\equiv {\mathcal {E}}[{ P}_0]$ and ${ P}'\equiv {\mathcal {E}}[{ P}_0']$, where ${{ P}_0{\longrightarrow }{ P}_0'}$ by one of the rules considered in Lemma 1. By structural equivalence we can assume ${\mathcal {E}}{=}(\overrightarrow{\nu a})(\overrightarrow{{{\mathsf{def}}\ {D}\ {\mathsf{in}}\ }}(\overrightarrow{\nu { s}})([\;]{\ |\ }{ P}_1))$ without loss of generality. Theorem 2 and Lemma 3(9), (10) and (8) applied to ${\varGamma \vdash _{\varSigma } { P} \triangleright {\varDelta }}$ give ${\varGamma , \overrightarrow{a:\mathsf {{G}}}, \overrightarrow{X:S\,\mu \mathbf t .T} \vdash _{\varSigma _0} { P}_0 \triangleright {\varDelta }_0}$, and ${\varGamma , \overrightarrow{a:\mathsf {{G}}}, \overrightarrow{X:S\,\mu \mathbf t .T} \vdash _{\varSigma _1} { P}_1 \triangleright {\varDelta }_1}$ and $\overrightarrow{{\varGamma , \overrightarrow{a:\mathsf {{G}}}, X:S\;\mathbf t \vdash _{} {Q} \triangleright {\{y:T\}}}}$, where $\overrightarrow{D}=\overrightarrow{X(x,y)=Q}$, $\varSigma =(\varSigma _0\cup \varSigma _1)\setminus \overrightarrow{{ s}}$ and ${\varDelta }=({\varDelta }_0\,{*}\,{\varDelta }_1)\setminus \overrightarrow{{ s}}$. The consistency of ${\varDelta }$ implies the consistency of ${\varDelta }_0\,{*}\,{\varDelta }_1$ by Lemma 3(8). By Lemma 1 there is ${\varDelta }_0'$ such that ${\varGamma , \overrightarrow{a:\mathsf {{G}}}, \overrightarrow{X:S\,\mu \mathbf t .T} \vdash _{\varSigma _0} { P}'_0 \triangleright {\varDelta }'_0}$ and ${{{\varDelta }_0}~\Rightarrow ^*~{{\varDelta }_0'}}$ and ${\varDelta }'_0\,{*}\,{\varDelta }_1$ is consistent. We derive ${\varGamma \vdash _{\varSigma } { P}' \triangleright {{\varDelta }'}}$, where ${\varDelta }'= ({\varDelta }_0\,{*}\,{\varDelta }'_1)\setminus \overrightarrow{{ s}}$ by applying typing rules (GPar), (GSRes), (GDef) and (GNRes). Observe that ${{{\varDelta }}~\Rightarrow ^*~{{\varDelta }'}}$ and ${\varDelta }'$ is consistent.   $\square$