%   ORIGINAL: 'h4/thm/while/OWHILE_EQ_NONE_'
% 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 f G. 'h4/const/while/OWHILE' G f s = 'h4/const/option/NONE' <=> (!n. G ('h4/const/arithmetic/FUNPOW' f n s))
%   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 f G. 'h4/const/while/OWHILE' G f s = 'h4/const/option/NONE' <=> (!n. happ G ('h4/const/arithmetic/FUNPOW' f n s))
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_3f41809,V_3f41805]: ![F, G]: (![X]: s(V_3f41805,happ(s(fun(V_3f41809,V_3f41805),F),s(V_3f41809,X))) = s(V_3f41805,happ(s(fun(V_3f41809,V_3f41805),G),s(V_3f41809,X))) => s(fun(V_3f41809,V_3f41805),F) = s(fun(V_3f41809,V_3f41805),G))).
fof('h4/thm/while/OWHILE_EQ_NONE_', conjecture, ![A]: ![S, F, G]: (s('h4/type/option/option'(A),'h4/const/while/OWHILE'(s(fun(A,bool),G),s(fun(A,A),F),s(A,S))) = s('h4/type/option/option'(A),'h4/const/option/NONE') <=> ![N]: p(s(bool,happ(s(fun(A,bool),G),s(A,'h4/const/arithmetic/FUNPOW'(s(fun(A,A),F),s('h4/type/num/num',N),s(A,S)))))))).
