%   ORIGINAL: 'h4/thm/while/FULL_LEAST_INTRO_'
% 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 x. P x ==> P ('h4/const/while/LEAST' P) /\ 'h4/const/arithmetic/<=' ('h4/const/while/LEAST' P) x
%   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 x. happ P x ==> happ P ('h4/const/while/LEAST' P) /\ 'h4/const/arithmetic/<=' ('h4/const/while/LEAST' P) x
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_3f41484,V_3f41480]: ![F, G]: (![X]: s(V_3f41480,happ(s(fun(V_3f41484,V_3f41480),F),s(V_3f41484,X))) = s(V_3f41480,happ(s(fun(V_3f41484,V_3f41480),G),s(V_3f41484,X))) => s(fun(V_3f41484,V_3f41480),F) = s(fun(V_3f41484,V_3f41480),G))).
fof('h4/thm/while/FULL_LEAST_INTRO_', conjecture, ![P, X]: (p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num',X)))) => (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)))))) & p(s(bool,'h4/const/arithmetic/<='(s('h4/type/num/num','h4/const/while/LEAST'(s(fun('h4/type/num/num',bool),P))),s('h4/type/num/num',X))))))).
