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\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\axjustifies\trans{\StmPush~\texttt{"\textquotedblright{}u\textquotedblright{}p\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gh\textquotedblright{}v\textquotedblright{}g+\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gt\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}g\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}o\textquotedblright{}+\textquotedblright{}w\textquotedblright{}gq\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}u\textquotedblright{}gq++\textquotedblright{}u\textquotedblright{}g+\textquotedblright{}w\textquotedblright{}gp\textquotedblright{}\textquotedblright{}pgx"}:\StmPush~\texttt{"u"}:\StmPut:\StmPush~\texttt{"v"}:\StmPut:\StmPush~\texttt{"w"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmHead:\StmPush~\texttt{"v"}:\StmGet:\StmCat:\StmPush~\texttt{"v"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmTail:\StmPush~\texttt{"w"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmPush~\texttt{"v"}:\StmGet:\StmQuotify:\StmPush~\texttt{"o"}:\StmCat:\StmPush~\texttt{"w"}:\StmGet:\StmQuotify:\StmPush~\texttt{"v"}:\StmGet:\StmQuotify:\StmPush~\texttt{"u"}:\StmGet:\StmQuotify:\StmCat:\StmCat:\StmPush~\texttt{"u"}:\StmGet:\StmCat:\StmPush~\texttt{"w"}:\StmGet:\StmPut:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{""}:\texttt{"c$t$"}:\Nil,\emptyset\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rule{assumption}{}}\]
\justifies{}\trans{\StmPush~\texttt{""}:\StmPush~\texttt{"\textquotedblright{}u\textquotedblright{}p\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gh\textquotedblright{}v\textquotedblright{}g+\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gt\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}g\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}o\textquotedblright{}+\textquotedblright{}w\textquotedblright{}gq\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}u\textquotedblright{}gq++\textquotedblright{}u\textquotedblright{}g+\textquotedblright{}w\textquotedblright{}gp\textquotedblright{}\textquotedblright{}pgx"}:\StmPush~\texttt{"u"}:\StmPut:\StmPush~\texttt{"v"}:\StmPut:\StmPush~\texttt{"w"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmHead:\StmPush~\texttt{"v"}:\StmGet:\StmCat:\StmPush~\texttt{"v"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmTail:\StmPush~\texttt{"w"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmPush~\texttt{"v"}:\StmGet:\StmQuotify:\StmPush~\texttt{"o"}:\StmCat:\StmPush~\texttt{"w"}:\StmGet:\StmQuotify:\StmPush~\texttt{"v"}:\StmGet:\StmQuotify:\StmPush~\texttt{"u"}:\StmGet:\StmQuotify:\StmCat:\StmCat:\StmPush~\texttt{"u"}:\StmGet:\StmCat:\StmPush~\texttt{"w"}:\StmGet:\StmPut:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"c$t$"}:\Nil,\emptyset\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rpushns}\]
\justifies{}\trans{\StmPush~\texttt{"c$t$"}:\StmPush~\texttt{""}:\StmPush~\texttt{"\textquotedblright{}u\textquotedblright{}p\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gh\textquotedblright{}v\textquotedblright{}g+\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gt\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}g\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}o\textquotedblright{}+\textquotedblright{}w\textquotedblright{}gq\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}u\textquotedblright{}gq++\textquotedblright{}u\textquotedblright{}g+\textquotedblright{}w\textquotedblright{}gp\textquotedblright{}\textquotedblright{}pgx"}:\StmPush~\texttt{"u"}:\StmPut:\StmPush~\texttt{"v"}:\StmPut:\StmPush~\texttt{"w"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmHead:\StmPush~\texttt{"v"}:\StmGet:\StmCat:\StmPush~\texttt{"v"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmTail:\StmPush~\texttt{"w"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmPush~\texttt{"v"}:\StmGet:\StmQuotify:\StmPush~\texttt{"o"}:\StmCat:\StmPush~\texttt{"w"}:\StmGet:\StmQuotify:\StmPush~\texttt{"v"}:\StmGet:\StmQuotify:\StmPush~\texttt{"u"}:\StmGet:\StmQuotify:\StmCat:\StmCat:\StmPush~\texttt{"u"}:\StmGet:\StmCat:\StmPush~\texttt{"w"}:\StmGet:\StmPut:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\Nil,\emptyset\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rpushns}\]
\justifies{}\trans{\StmExec}{\Nil}{\left(\texttt{"\textquotedblright{}c\textit{q}($t$)\textquotedblright{}
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"}:\Nil,\{\texttt{""}\mapsto\texttt{"\textquotedblright{}\textquotedblright{}o"},\texttt{"c$t$"}\mapsto\texttt{"\textquotedblright{}c\textit{q}($t$)\textquotedblright{}
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"},\texttt{"input"}\mapsto\texttt{"c$t$"}\}\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rexecns}\]
\justifies{}\trans{\StmGet:\StmExec}{\Nil}{\left(\texttt{"c$t$"}:\Nil,\{\texttt{""}\mapsto\texttt{"\textquotedblright{}\textquotedblright{}o"},\texttt{"c$t$"}\mapsto\texttt{"\textquotedblright{}c\textit{q}($t$)\textquotedblright{}
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"},\texttt{"input"}\mapsto\texttt{"c$t$"}\}\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rgetns}\]
\justifies{}\trans{\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{""}:\texttt{"\textquotedblright{}\textquotedblright{}o"}:\texttt{"c$t$"}:\Nil,\{\texttt{"c$t$"}\mapsto\texttt{"\textquotedblright{}c\textit{q}($t$)\textquotedblright{}
\textquotedblright{}\textquotedblright{}\textquotedblright{}\textbackslash{}\textquotedblright{}u\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gh\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}g+\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gt\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}g\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}gq\textbackslash{}\textquotedblright{}o\textbackslash{}\textquotedblright{}+\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gq\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}gq\textbackslash{}\textquotedblright{}u\textbackslash{}\textquotedblright{}gq++\textbackslash{}\textquotedblright{}u\textbackslash{}\textquotedblright{}g+\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gp\textbackslash{}\textquotedblright{}\textbackslash{}\textquotedblright{}pgx\textquotedblright{}\textquotedblright{}u\textquotedblright{}p\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gh\textquotedblright{}v\textquotedblright{}g+\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gt\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}g\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}o\textquotedblright{}+\textquotedblright{}w\textquotedblright{}gq\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}u\textquotedblright{}gq++\textquotedblright{}u\textquotedblright{}g+\textquotedblright{}w\textquotedblright{}gp\textquotedblright{}\textquotedblright{}pgx
"},\texttt{"input"}\mapsto\texttt{"c$t$"}\}\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rputns}\]
\justifies{}\trans{\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"\textquotedblright{}\textquotedblright{}o"}:\texttt{"c$t$"}:\Nil,\{\texttt{"c$t$"}\mapsto\texttt{"\textquotedblright{}c\textit{q}($t$)\textquotedblright{}
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"},\texttt{"input"}\mapsto\texttt{"c$t$"}\}\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rpushns}\]
\justifies{}\trans{\StmPush~\texttt{"\textquotedblright{}\textquotedblright{}o"}:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"c$t$"}:\Nil,\{\texttt{"c$t$"}\mapsto\texttt{"\textquotedblright{}c\textit{q}($t$)\textquotedblright{}
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"},\texttt{"input"}\mapsto\texttt{"c$t$"}\}\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rpushns}\]
\justifies{}\trans{\StmPut:\StmPush~\texttt{"\textquotedblright{}\textquotedblright{}o"}:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"c$t$"}:\texttt{"\textquotedblright{}c\textit{q}($t$)\textquotedblright{}
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"}:\texttt{"c$t$"}:\Nil,\{\texttt{"input"}\mapsto\texttt{"c$t$"}\}\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rputns}\]
\justifies{}\trans{\StmGet:\StmPut:\StmPush~\texttt{"\textquotedblright{}\textquotedblright{}o"}:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"input"}:\texttt{"\textquotedblright{}c\textit{q}($t$)\textquotedblright{}
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