set - range restriction/domain restriction in Isabelle -

i trying input schema isabelle when add range restriction or domain restriction theorem prover doesn't want parse. have following schema in latex:

\begin{schema}{videoshop} members: \power person \\ rented: person \rel title \\ stocklevel: title \pfun \nat \where \dom rented \subseteq members \\ \ran rented \subseteq \dom stocklevel \\ \forall t: \ran rented @ \# (rented \rres \{t\}) \leq stocklevel~t \end{schema} 

when inputting isabelle following:

locale videoshop =  fixes members :: "person set" , rented :: "(person * title) set" , stocklevel :: "(title * nat) set" assumes "domain rented \<subseteq> members"   , "range rented \<subseteq> domain stocklevel"   , "(\<forall> t. (t \<in> range rented) \<and> (card (rented \<rhd> {t}) \<le> stocklevel t))" begin ..... 

it parses except last expression \<forall> t.....

i don't understand how add range restriction isabelle.

there multiple problems input.

1. the ⊳ symbol using in expression

   (rented ⊳ {t}) 

   is not associated operator, can't parsed. i'm not quite sure it's supposed mean. high-level idea of specification i'm guessing along lines of "all persons rented specific title". can expressed set comprehension:

   {p. (p, t) ∈ rented} 

2. you translated bounded universal quantifier quantifier containing conjunction. not want, because says "for t, t in range of rented and else". isabelle has notation bounded quantifiers.

   ∀t ∈ range rented. ... 

3. you trying use stocklevel function, isn't. latex input gather it's supposed partial function. isabelle calls these maps. appropriate type is:

   title ⇀ nat 

   note "harpoon" symbol instead of function arrow. domain function maps called dom. second locale assumption can expressed as:

   range rented ⊆ dom stocklevel 

   given that, can use stocklevel function title nat option.