%   ORIGINAL: 'h4/thm/arithmetic/transitive_monotone_'
% 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 f. 'h4/const/relation/transitive' R /\ (!n. R (f n) (f ('h4/const/num/SUC' n))) ==> (!m n. 'h4/const/prim_rec/<' m n ==> R (f m) (f 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: !R f. 'h4/const/relation/transitive' R /\ (!n. happ (happ R (happ f n)) (happ f ('h4/const/num/SUC' n))) ==> (!m n. 'h4/const/prim_rec/<' m n ==> happ (happ R (happ f m)) (happ f 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_3f84567,V_3f84563]: ![F, G]: (![X]: s(V_3f84563,happ(s(fun(V_3f84567,V_3f84563),F),s(V_3f84567,X))) = s(V_3f84563,happ(s(fun(V_3f84567,V_3f84563),G),s(V_3f84567,X))) => s(fun(V_3f84567,V_3f84563),F) = s(fun(V_3f84567,V_3f84563),G))).
fof('h4/thm/arithmetic/transitive_monotone_', conjecture, ![A]: ![R, F]: ((p(s(bool,'h4/const/relation/transitive'(s(fun(A,fun(A,bool)),R)))) & ![N]: p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num',N))))),s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',N))))))))) => ![M, N]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',M),s('h4/type/num/num',N)))) => p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num',M))))),s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num',N))))))))).
