%   ORIGINAL: 'h4/thm/pred_set/FINITE_WEAK_ENUMERATE_'
% 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: !s. 'h4/const/pred_set/FINITE' s <=> (?f b. !e. 'h4/const/bool/IN' e s <=> (?n. 'h4/const/prim_rec/<' n b /\ e = f 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: !s. 'h4/const/pred_set/FINITE' s <=> (?f b. !e. 'h4/const/bool/IN' e s <=> (?n. 'h4/const/prim_rec/<' n b /\ e = happ f 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_3f20346,V_3f20342]: ![F, G]: (![X]: s(V_3f20342,happ(s(fun(V_3f20346,V_3f20342),F),s(V_3f20346,X))) = s(V_3f20342,happ(s(fun(V_3f20346,V_3f20342),G),s(V_3f20346,X))) => s(fun(V_3f20346,V_3f20342),F) = s(fun(V_3f20346,V_3f20342),G))).
fof('h4/thm/pred_set/FINITE_WEAK_ENUMERATE_', conjecture, ![A]: ![S]: (p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),S)))) <=> ?[F, B]: ![E]: (p(s(bool,'h4/const/bool/IN'(s(A,E),s(fun(A,bool),S)))) <=> ?[N]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',N),s('h4/type/num/num',B)))) & s(A,E) = s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num',N))))))).
