Sindbad~EG File Manager

% test file for second-order unification
%    quantificational logic
% This file finds a proof of the existential quantifier law, with an extra double negation in the hypothesis,
% but without being given the contrapositive law.  To get a short proof, uncomment the contrapositive law.

set(lambda).
set(demod_inf).
set(hyper_res).
%set(binary_res).
%set(very_verbose).
assign(max_distinct_vars,4).
assign(max_weight,24).
assign(pick_given_ratio,4).
weight_list(pick_and_purge).
weight(junk,50).
weight(cases($(1),$(1),$(1),$(1)), 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).
%-P(i(n(n(Ap(a,z))),n(all(lambda(x,n(Ap(a,x))))))).   % negation of goal
-P(i((Ap(a,z)),n(all(lambda(x,n(Ap(a,x))))))).   % negation of goal
 P(i(i(x,y),i(n(y),n(x)))).   %contrapositive law
x=x.
-P(x) | -P(i(x,y)) | P(y).    % condensed detachment


%-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.
end_of_list.

list(sos).
% axioms from Kleene IM p. 82
%P(i(n(x),i(x,y))).  % for the intuitionistic system.
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(n(n(x)),x)).
P(i(all(lambda(x,Ap(y,x))), Ap(y,z))). 
%-P(i(Ap(a,z),n(all(lambda(x,n(Ap(a,x))))))).
%-P(i(z,y)) | -notfree(x,z) | P(i(z,all(lambda(x,y)))).
% -P(i(i(a,b),i(n(b),n(a)))).   %contrapositive law as a negated goal
end_of_list.