%   ORIGINAL: 'h4/thm/arithmetic/STRICTLY_INCREASING_UNBOUNDED_'
% 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. (!n. 'h4/const/prim_rec/<' (f n) (f ('h4/const/num/SUC' n))) ==> (!b. ?n. 'h4/const/prim_rec/<' b (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: !f. (!n. 'h4/const/prim_rec/<' (happ f n) (happ f ('h4/const/num/SUC' n))) ==> (!b. ?n. 'h4/const/prim_rec/<' b (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_3f84667,V_3f84663]: ![F, G]: (![X]: s(V_3f84663,happ(s(fun(V_3f84667,V_3f84663),F),s(V_3f84667,X))) = s(V_3f84663,happ(s(fun(V_3f84667,V_3f84663),G),s(V_3f84667,X))) => s(fun(V_3f84667,V_3f84663),F) = s(fun(V_3f84667,V_3f84663),G))).
fof('h4/thm/arithmetic/STRICTLY_INCREASING_UNBOUNDED_', conjecture, ![F]: (![N]: p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',happ(s(fun('h4/type/num/num','h4/type/num/num'),F),s('h4/type/num/num',N))),s('h4/type/num/num',happ(s(fun('h4/type/num/num','h4/type/num/num'),F),s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',N)))))))) => ![B]: ?[N]: p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',B),s('h4/type/num/num',happ(s(fun('h4/type/num/num','h4/type/num/num'),F),s('h4/type/num/num',N)))))))).
