%   ORIGINAL: 'h4/thm/arithmetic/SUC_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: !P. (!n. P ('h4/const/num/SUC' n) n) <=> (!n. 'h4/const/prim_rec/<' 'h4/const/num/0' n ==> P n ('h4/const/arithmetic/-' n ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO'))))
%   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. (!n. happ (happ P ('h4/const/num/SUC' n)) n) <=> (!n. 'h4/const/prim_rec/<' 'h4/const/num/0' n ==> happ (happ P n) ('h4/const/arithmetic/-' n ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO'))))
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_3f83191,V_3f83187]: ![F, G]: (![X]: s(V_3f83187,happ(s(fun(V_3f83191,V_3f83187),F),s(V_3f83191,X))) = s(V_3f83187,happ(s(fun(V_3f83191,V_3f83187),G),s(V_3f83191,X))) => s(fun(V_3f83191,V_3f83187),F) = s(fun(V_3f83191,V_3f83187),G))).
fof('h4/thm/arithmetic/SUC_ELIM_THM_', conjecture, ![P]: (![N]: p(s(bool,happ(s(fun('h4/type/num/num',bool),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',bool)),P),s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',N))))),s('h4/type/num/num',N)))) <=> ![N]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num','h4/const/num/0'),s('h4/type/num/num',N)))) => p(s(bool,happ(s(fun('h4/type/num/num',bool),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',bool)),P),s('h4/type/num/num',N))),s('h4/type/num/num','h4/const/arithmetic/-'(s('h4/type/num/num',N),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'))))))))))))).
