%   ORIGINAL: 'h4/thm/relation/RTC_INDUCT_'
% 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: !R P. (!x. P x x) /\ (!x y z. R x y /\ P y z ==> P x z) ==> (!x y. 'h4/const/relation/RTC' R x y ==> P x 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: !R P. (!x. happ (happ P x) x) /\ (!x y z. happ (happ R x) y /\ happ (happ P y) z ==> happ (happ P x) z) ==> (!x y. 'h4/const/relation/RTC' R x y ==> happ (happ P x) 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_3f25578,V_3f25574]: ![F, G]: (![X]: s(V_3f25574,happ(s(fun(V_3f25578,V_3f25574),F),s(V_3f25578,X))) = s(V_3f25574,happ(s(fun(V_3f25578,V_3f25574),G),s(V_3f25578,X))) => s(fun(V_3f25578,V_3f25574),F) = s(fun(V_3f25578,V_3f25574),G))).
fof('h4/thm/relation/RTC_INDUCT_', conjecture, ![A]: ![R, P]: ((![X]: p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),P),s(A,X))),s(A,X)))) & ![X, Y, Z]: ((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)),P),s(A,Y))),s(A,Z))))) => p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),P),s(A,X))),s(A,Z)))))) => ![X, Y]: (p(s(bool,'h4/const/relation/RTC'(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)),P),s(A,X))),s(A,Y))))))).
