%   ORIGINAL: 'h4/thm/arithmetic/PRE_ELIM_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: !n P. P ('h4/const/prim_rec/PRE' n) <=> (!m. (n = 'h4/const/num/0' ==> P 'h4/const/num/0') /\ (n = 'h4/const/num/SUC' m ==> P m))
%   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: !n P. happ P ('h4/const/prim_rec/PRE' n) <=> (!m. (n = 'h4/const/num/0' ==> happ P 'h4/const/num/0') /\ (n = 'h4/const/num/SUC' m ==> happ P m))
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_3f83266,V_3f83262]: ![F, G]: (![X]: s(V_3f83262,happ(s(fun(V_3f83266,V_3f83262),F),s(V_3f83266,X))) = s(V_3f83262,happ(s(fun(V_3f83266,V_3f83262),G),s(V_3f83266,X))) => s(fun(V_3f83266,V_3f83262),F) = s(fun(V_3f83266,V_3f83262),G))).
fof('h4/thm/arithmetic/PRE_ELIM_THM_', conjecture, ![N, P]: (p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num','h4/const/prim_rec/PRE'(s('h4/type/num/num',N)))))) <=> ![M]: ((s('h4/type/num/num',N) = s('h4/type/num/num','h4/const/num/0') => p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num','h4/const/num/0'))))) & (s('h4/type/num/num',N) = s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',M))) => p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num',M)))))))).
