%   ORIGINAL: 'h4/thm/prim_rec/DC_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !P R a. P a /\ (!x. P x ==> (?y. P y /\ R x y)) ==> (?f. f 'h4/const/num/0' = a /\ (!n. P (f n) /\ R (f n) (f ('h4/const/num/SUC' n))))
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !P R a. happ P a /\ (!x. happ P x ==> (?y. happ P y /\ happ (happ R x) y)) ==> (?f. happ f 'h4/const/num/0' = a /\ (!n. happ P (happ f n) /\ happ (happ R (happ f n)) (happ f ('h4/const/num/SUC' n))))
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f87044,V_3f87040]: ![F, G]: (![X]: s(V_3f87040,happ(s(fun(V_3f87044,V_3f87040),F),s(V_3f87044,X))) = s(V_3f87040,happ(s(fun(V_3f87044,V_3f87040),G),s(V_3f87044,X))) => s(fun(V_3f87044,V_3f87040),F) = s(fun(V_3f87044,V_3f87040),G))).
fof('h4/thm/prim_rec/DC_', conjecture, ![A]: ![P, R, A0]: ((p(s(bool,happ(s(fun(A,bool),P),s(A,A0)))) & ![X]: (p(s(bool,happ(s(fun(A,bool),P),s(A,X)))) => ?[Y]: (p(s(bool,happ(s(fun(A,bool),P),s(A,Y)))) & p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,X))),s(A,Y))))))) => ?[F]: (s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num','h4/const/num/0'))) = s(A,A0) & ![N]: (p(s(bool,happ(s(fun(A,bool),P),s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num',N)))))) & p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num',N))))),s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',N)))))))))))).
