%   ORIGINAL: 'h4/thm/pred_set/SET_MINIMUM_'
% 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 M. (?x. 'h4/const/bool/IN' x s) <=> (?x. 'h4/const/bool/IN' x s /\ (!y. 'h4/const/bool/IN' y s ==> 'h4/const/arithmetic/<=' (M x) (M y)))
%   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 M. (?x. 'h4/const/bool/IN' x s) <=> (?x. 'h4/const/bool/IN' x s /\ (!y. 'h4/const/bool/IN' y s ==> 'h4/const/arithmetic/<=' (happ M x) (happ M y)))
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_3f14794,V_3f14790]: ![F, G]: (![X]: s(V_3f14790,happ(s(fun(V_3f14794,V_3f14790),F),s(V_3f14794,X))) = s(V_3f14790,happ(s(fun(V_3f14794,V_3f14790),G),s(V_3f14794,X))) => s(fun(V_3f14794,V_3f14790),F) = s(fun(V_3f14794,V_3f14790),G))).
fof('h4/thm/pred_set/SET_MINIMUM_', conjecture, ![A]: ![S, M]: (?[X]: p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),S)))) <=> ?[X]: (p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),S)))) & ![Y]: (p(s(bool,'h4/const/bool/IN'(s(A,Y),s(fun(A,bool),S)))) => p(s(bool,'h4/const/arithmetic/<='(s('h4/type/num/num',happ(s(fun(A,'h4/type/num/num'),M),s(A,X))),s('h4/type/num/num',happ(s(fun(A,'h4/type/num/num'),M),s(A,Y)))))))))).
