%   ORIGINAL: 'h4/thm/arithmetic/BOUNDED_EXISTS_THM_'
% 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 c. 'h4/const/prim_rec/<' 'h4/const/num/0' c ==> ((?n. 'h4/const/prim_rec/<' n c /\ P n) <=> P ('h4/const/arithmetic/-' c ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO'))) \/ (?n. 'h4/const/prim_rec/<' n ('h4/const/arithmetic/-' c ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO'))) /\ P 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 c. 'h4/const/prim_rec/<' 'h4/const/num/0' c ==> ((?n. 'h4/const/prim_rec/<' n c /\ happ P n) <=> happ P ('h4/const/arithmetic/-' c ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO'))) \/ (?n. 'h4/const/prim_rec/<' n ('h4/const/arithmetic/-' c ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO'))) /\ happ P 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_3f84542,V_3f84538]: ![F, G]: (![X]: s(V_3f84538,happ(s(fun(V_3f84542,V_3f84538),F),s(V_3f84542,X))) = s(V_3f84538,happ(s(fun(V_3f84542,V_3f84538),G),s(V_3f84542,X))) => s(fun(V_3f84542,V_3f84538),F) = s(fun(V_3f84542,V_3f84538),G))).
fof('h4/thm/arithmetic/BOUNDED_EXISTS_THM_', conjecture, ![P, C]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num','h4/const/num/0'),s('h4/type/num/num',C)))) => (?[N]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',N),s('h4/type/num/num',C)))) & p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num',N))))) <=> (p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num','h4/const/arithmetic/-'(s('h4/type/num/num',C),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num','h4/const/arithmetic/ZERO')))))))))) | ?[N]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',N),s('h4/type/num/num','h4/const/arithmetic/-'(s('h4/type/num/num',C),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num','h4/const/arithmetic/ZERO')))))))))) & p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num',N))))))))).
