%   ORIGINAL: 'h4/thm/while/LEAST_EXISTS_'
% 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 n) <=> p ('h4/const/while/LEAST' p) /\ (!n. 'h4/const/prim_rec/<' n ('h4/const/while/LEAST' p) ==> ~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. (?n. happ p n) <=> happ p ('h4/const/while/LEAST' p) /\ (!n. 'h4/const/prim_rec/<' n ('h4/const/while/LEAST' p) ==> ~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_3f41534,V_3f41530]: ![F, G]: (![X]: s(V_3f41530,happ(s(fun(V_3f41534,V_3f41530),F),s(V_3f41534,X))) = s(V_3f41530,happ(s(fun(V_3f41534,V_3f41530),G),s(V_3f41534,X))) => s(fun(V_3f41534,V_3f41530),F) = s(fun(V_3f41534,V_3f41530),G))).
fof('h4/thm/while/LEAST_EXISTS_', conjecture, ![P]: (?[N]: 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/while/LEAST'(s(fun('h4/type/num/num',bool),P)))))) & ![N]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',N),s('h4/type/num/num','h4/const/while/LEAST'(s(fun('h4/type/num/num',bool),P)))))) => ~ (p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num',N))))))))).
