%   ORIGINAL: h4/arithmetic/EXP__BASE__LEQ__MONO__SUC__IMP
% 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
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/arithmetic/EXP__BASE__LEQ__MONO__IMP: !n m b. h4/prim__rec/_3C h4/num/0 b /\ h4/arithmetic/_3C_3D m n ==> h4/arithmetic/_3C_3D (h4/arithmetic/EXP b m) (h4/arithmetic/EXP b n)
% Goal: !n m b. h4/arithmetic/_3C_3D m n ==> h4/arithmetic/_3C_3D (h4/arithmetic/EXP (h4/num/SUC b) m) (h4/arithmetic/EXP (h4/num/SUC b) n)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm [h4s_arithmetics_EXPu_u_BASEu_u_LEQu_u_MONOu_u_IMP]: !n m b. h4/prim__rec/_3C h4/num/0 b /\ h4/arithmetic/_3C_3D m n ==> h4/arithmetic/_3C_3D (h4/arithmetic/EXP b m) (h4/arithmetic/EXP b n)
% Goal: !n m b. h4/arithmetic/_3C_3D m n ==> h4/arithmetic/_3C_3D (h4/arithmetic/EXP (h4/num/SUC b) m) (h4/arithmetic/EXP (h4/num/SUC b) n)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q311598,TV_Q311594]: ![V_f, V_g]: (![V_x]: s(TV_Q311594,happ(s(t_fun(TV_Q311598,TV_Q311594),V_f),s(TV_Q311598,V_x))) = s(TV_Q311594,happ(s(t_fun(TV_Q311598,TV_Q311594),V_g),s(TV_Q311598,V_x))) => s(t_fun(TV_Q311598,TV_Q311594),V_f) = s(t_fun(TV_Q311598,TV_Q311594),V_g))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_primu_u_recs_LESSu_u_0, axiom, ![V_n]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_EXPu_u_BASEu_u_LEQu_u_MONOu_u_IMP, axiom, ![V_n, V_m, V_b]: ((p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_b)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_b),s(t_h4s_nums_num,V_n)))))))).
fof(ch4s_arithmetics_EXPu_u_BASEu_u_LEQu_u_MONOu_u_SUCu_u_IMP, conjecture, ![V_n, V_m, V_b]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_b))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_b))),s(t_h4s_nums_num,V_n)))))))).
