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% test file for second-order unification
%    quantificational logic


set(lambda).
set(demod_inf).
set(hyper_res).
%set(binary_res).
%set(very_verbose).
assign(max_distinct_vars,6).
assign(max_weight,32).
assign(pick_given_ratio,4).
weight_list(pick_and_purge).
weight(junk,50).
end_of_list.

% typings: 
% The types are:  PF  (propositional function),  prop (proposition), var, and obj (object)
% PF lambda(var, prop)
% prop all(PF)
% bool P(prop)
% prop i(prop,prop)
% prop n(prop)
% prop Ap(PF, obj)
% bool notfree(var,prop)
% bool notfree2(var, PF)

list(usable).
x=x.

%P(i(n(x),i(x,y))).  % for the intuitionistic system.


%-P(i(x,ap(y,z))) | -notfree(z,x) | P(i(x,all(y))).    % y is a functional lambda(z,Ap(a,z)).
%P(i(all(z),ap(z,y))).
% to permit working backwards:
%-notfree(z,x) | z != gensym.
% to permit working forwards:
%notfree(z,i(x,y)) | -notfree(z,x) | -notfree(z,y).
%notfree(z,n(x)) | -notfree(z,x).
%notfree(z,Ap(y,x)) | -notfree(z,x) | -notfree2(z,y).
%notfree2(z,lambda(u,v)) |   u!=z | -notfree(z,v).
end_of_list.

list(demodulators).
n(n(n(x))) = junk.
n(junk)= junk.
i(x,junk) = junk.
i(junk,x)= junk.
P(junk) = junk.
end_of_list.

list(sos).
-P(x) | -P(i(x,y)) | P(y).    % condensed detachment 
% axioms from Kleene IM p. 82
P(i(x,i(y,x))).       
P(i(i(x,y),i(i(x,i(y,z)),i(x,z)))).
P(i(i(x,y),i(i(x,n(y)),n(x)))).
P(i(n(n(x)),x)).    %for the classical system
%-P(i(z,y)) | -notfree(x,z) | P(i(z,all(lambda(x,y)))).
P(i(u,u)).
P(i( i(n(x),n(y)), i(y,x))).
P(i( i(x,y),  i(n(y),n(x)))).
P(i(all(lambda(x,Ap(y,x))), Ap(y,z))).
P(Ap(a,c)).
-P(n(all(lambda(x,n(Ap(a,x)))))).
end_of_list.