%% apply the NP semantics to the VP semantics
s('@'(NpS,VpS)) --> np(NpS), vp(VpS).

%% apply the determiner semantics to the noun phrase semantics
np('@'(DetS,NS)) --> det(DetS), n(NS).

%% just take the proper name semantics
np(PnSem) --> pn(PnSem).

%% just take the semantics if the intransitive verb
vp(l(X,'@'(ViSem,X))) --> vi(ViSem).

%% Apply the semantics of the transitive verb to Y and X,
%% such that the first argument of kill is filled by the
%% variable Y and the second by X. Then lambda abstract over X and
%% apply the semantics of the object NP to the result. Finally, lambda
%% abstract over Y.
vp(l(Y,'@'(NpS,l(X,'@'('@'(VtS,Y),X))))) --> vt(VtS), np(NpS).


%% Lexicon

det(l(R, l(S, exists(X,'@'(R,X),'@'(S,X))))) --> [a].
det(l(R, l(S, exists(X,'@'(R,X),'@'(S,X))))) --> [the].

n(l(X,bride(X))) --> [bride].
n(l(X,nurse(X))) --> [nurse].
n(l(X,yakuza(X))) --> [yakuza].
n(l(X,whiskey(X))) --> [whiskey].

pn(l(P,'@'(P,bill))) --> [bill].
pn(l(P,'@'(P,gogo))) --> [gogo].

vi(l(X,whistle(X))) --> [whistles].
vi(l(X,fight(X))) --> [fights].

vt(l(X,l(Y,drink(Y,X)))) --> [drinks].
vt(l(X,l(Y,kill(X,Y)))) --> [kills].


%%%
%%% Predicate to do beta reduction
%%%
%%% Call beta_reduce(Sem,R) (with Sem being a semantic representation
%%% constructed by the grammar) to get a beta-reduced and therefore
%% more readable version of the semantics.
%%%

beta_reduce(F,Reduced) :- atom(F), !, Reduced = F.
beta_reduce(F,Reduced) :- var(F), !, Reduced = F.

beta_reduce('@'(l(X,F),Y),Reduced) :- !, beta_reduce(Y,YR), YR=X, beta_reduce(F,Reduced).
beta_reduce('@'(F,X),Reduced) :- !, beta_reduce(F,FR), beta_reduce(X,XR), beta_reduce('@'(FR,XR),Reduced).

beta_reduce(Term,Reduced) :- 
	Term =.. [Functor|Args],
	reduce_args(Args,RArgs),
	Reduced =.. [Functor|RArgs].
	
reduce_args([],[]).
reduce_args([Head|Rest],[RHead|RRest]) :- 
	beta_reduce(Head,RHead), 
	reduce_args(Rest,RRest).
    

Back to the exercise.