%   ORIGINAL: 'h4/thm/prim_rec/num_Axiom_old_'
% 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: !e f. 'h4/const/bool/?!' (\fn1. fn1 'h4/const/num/0' = e /\ (!n. fn1 ('h4/const/num/SUC' n) = f (fn1 n) 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: !_0. (!e f fn1. happ (happ (happ _0 e) f) fn1 <=> happ fn1 'h4/const/num/0' = e /\ (!n. happ fn1 ('h4/const/num/SUC' n) = happ (happ f (happ fn1 n)) n)) ==> (!e f. 'h4/const/bool/?!' (happ (happ _0 e) f))
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_3f87069,V_3f87065]: ![F, G]: (![X]: s(V_3f87065,happ(s(fun(V_3f87069,V_3f87065),F),s(V_3f87069,X))) = s(V_3f87065,happ(s(fun(V_3f87069,V_3f87065),G),s(V_3f87069,X))) => s(fun(V_3f87069,V_3f87065),F) = s(fun(V_3f87069,V_3f87065),G))).
fof('h4/thm/prim_rec/num_Axiom_old_', conjecture, ![A]: ![V__0]: (![E, F, Fn1]: (p(s(bool,happ(s(fun(fun('h4/type/num/num',A),bool),happ(s(fun(fun(A,fun('h4/type/num/num',A)),fun(fun('h4/type/num/num',A),bool)),happ(s(fun(A,fun(fun(A,fun('h4/type/num/num',A)),fun(fun('h4/type/num/num',A),bool))),V__0),s(A,E))),s(fun(A,fun('h4/type/num/num',A)),F))),s(fun('h4/type/num/num',A),Fn1)))) <=> (s(A,happ(s(fun('h4/type/num/num',A),Fn1),s('h4/type/num/num','h4/const/num/0'))) = s(A,E) & ![N]: s(A,happ(s(fun('h4/type/num/num',A),Fn1),s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',N))))) = s(A,happ(s(fun('h4/type/num/num',A),happ(s(fun(A,fun('h4/type/num/num',A)),F),s(A,happ(s(fun('h4/type/num/num',A),Fn1),s('h4/type/num/num',N))))),s('h4/type/num/num',N))))) => ![E, F]: p(s(bool,'h4/const/bool/?!'(s(fun(fun('h4/type/num/num',A),bool),happ(s(fun(fun(A,fun('h4/type/num/num',A)),fun(fun('h4/type/num/num',A),bool)),happ(s(fun(A,fun(fun(A,fun('h4/type/num/num',A)),fun(fun('h4/type/num/num',A),bool))),V__0),s(A,E))),s(fun(A,fun('h4/type/num/num',A)),F)))))))).
