%   ORIGINAL: 'h4/thm/relation/RC_lifts_monotonicities_'
% 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: !f R. (!x y. R x y ==> R (f x) (f y)) ==> (!x y. 'h4/const/relation/RC' R x y ==> 'h4/const/relation/RC' R (f x) (f 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: !f R. (!x y. happ (happ R x) y ==> happ (happ R (happ f x)) (happ f y)) ==> (!x y. 'h4/const/relation/RC' R x y ==> 'h4/const/relation/RC' R (happ f x) (happ f 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_3f25953,V_3f25949]: ![F, G]: (![X]: s(V_3f25949,happ(s(fun(V_3f25953,V_3f25949),F),s(V_3f25953,X))) = s(V_3f25949,happ(s(fun(V_3f25953,V_3f25949),G),s(V_3f25953,X))) => s(fun(V_3f25953,V_3f25949),F) = s(fun(V_3f25953,V_3f25949),G))).
fof('h4/thm/relation/RC_lifts_monotonicities_', conjecture, ![A]: ![F, R]: (![X, Y]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,X))),s(A,Y)))) => p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,happ(s(fun(A,A),F),s(A,X))))),s(A,happ(s(fun(A,A),F),s(A,Y))))))) => ![X, Y]: (p(s(bool,'h4/const/relation/RC'(s(fun(A,fun(A,bool)),R),s(A,X),s(A,Y)))) => p(s(bool,'h4/const/relation/RC'(s(fun(A,fun(A,bool)),R),s(A,happ(s(fun(A,A),F),s(A,X))),s(A,happ(s(fun(A,A),F),s(A,Y))))))))).
