%   ORIGINAL: h4/seq/SER__LE
% 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/TRUTH: T
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/GREATER__EQ: !n m. h4/arithmetic/_3E_3D n m <=> h4/arithmetic/_3C_3D m n
% Assm: h4/real/SUM__LE: !n m g f. (!r. h4/arithmetic/_3C_3D m r /\ h4/prim__rec/_3C r (h4/arithmetic/_2B n m) ==> h4/real/real__lte (f r) (g r)) ==> h4/real/real__lte (h4/real/sum (h4/pair/_2C m n) f) (h4/real/sum (h4/pair/_2C m n) g)
% Assm: h4/seq/SEQ__LE: !m l g f. h4/seq/_2D_2D_3E f l /\ h4/seq/_2D_2D_3E g m /\ (?N. !n. h4/arithmetic/_3E_3D n N ==> h4/real/real__lte (f n) (g n)) ==> h4/real/real__lte l m
% Assm: h4/seq/sums0: !s f. h4/seq/sums f s <=> h4/seq/_2D_2D_3E (\n. h4/real/sum (h4/pair/_2C h4/num/0 n) f) s
% Assm: h4/seq/SUMMABLE__SUM: !f. h4/seq/summable f ==> h4/seq/sums f (h4/seq/suminf f)
% Goal: !g f. (!n. h4/real/real__lte (f n) (g n)) /\ h4/seq/summable f /\ h4/seq/summable g ==> h4/real/real__lte (h4/seq/suminf f) (h4/seq/suminf g)
%   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_TRUTH]: T
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_GREATERu_u_EQ]: !n m. h4/arithmetic/_3E_3D n m <=> h4/arithmetic/_3C_3D m n
% Assm [h4s_reals_SUMu_u_LE]: !n m g f. (!r. h4/arithmetic/_3C_3D m r /\ h4/prim__rec/_3C r (h4/arithmetic/_2B n m) ==> h4/real/real__lte (happ f r) (happ g r)) ==> h4/real/real__lte (h4/real/sum (h4/pair/_2C m n) f) (h4/real/sum (h4/pair/_2C m n) g)
% Assm [h4s_seqs_SEQu_u_LE]: !m l g f. h4/seq/_2D_2D_3E f l /\ h4/seq/_2D_2D_3E g m /\ (?N. !n. h4/arithmetic/_3E_3D n N ==> h4/real/real__lte (happ f n) (happ g n)) ==> h4/real/real__lte l m
% Assm [h4s_seqs_sums0]: !_0. (!f n. happ (happ _0 f) n = h4/real/sum (h4/pair/_2C h4/num/0 n) f) ==> (!s f. h4/seq/sums f s <=> h4/seq/_2D_2D_3E (happ _0 f) s)
% Assm [h4s_seqs_SUMMABLEu_u_SUM]: !f. h4/seq/summable f ==> h4/seq/sums f (h4/seq/suminf f)
% Goal: !g f. (!n. h4/real/real__lte (happ f n) (happ g n)) /\ h4/seq/summable f /\ h4/seq/summable g ==> h4/real/real__lte (h4/seq/suminf f) (h4/seq/suminf g)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q134283,TV_Q134279]: ![V_f, V_g]: (![V_x]: s(TV_Q134279,happ(s(t_fun(TV_Q134283,TV_Q134279),V_f),s(TV_Q134283,V_x))) = s(TV_Q134279,happ(s(t_fun(TV_Q134283,TV_Q134279),V_g),s(TV_Q134283,V_x))) => s(t_fun(TV_Q134283,TV_Q134279),V_f) = s(t_fun(TV_Q134283,TV_Q134279),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_CONJu_u_ASSOC, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) & (p(s(t_bool,V_t2)) & p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) & p(s(t_bool,V_t3))))).
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_IMPu_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_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
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_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_GREATERu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3eu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
fof(ah4s_reals_SUMu_u_LE, axiom, ![V_n, V_m, V_g, V_f]: (![V_r]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_r)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_r),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_h4s_nums_num,V_r))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_g),s(t_h4s_nums_num,V_r))))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_sum(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,h4s_reals_sum(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_g)))))))).
fof(ah4s_seqs_SEQu_u_LE, axiom, ![V_m, V_l, V_g, V_f]: ((p(s(t_bool,h4s_seqs_u_2du_2du_3e(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l)))) & (p(s(t_bool,h4s_seqs_u_2du_2du_3e(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_m)))) & ?[V_N]: ![V_n]: (p(s(t_bool,h4s_arithmetics_u_3eu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_N)))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_g),s(t_h4s_nums_num,V_n))))))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_m)))))).
fof(ah4s_seqs_sums0, axiom, ![V_uu_0]: (![V_f, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))),s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_reals_sum(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))) => ![V_s, V_f]: s(t_bool,h4s_seqs_sums(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_s))) = s(t_bool,h4s_seqs_u_2du_2du_3e(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_s))))).
fof(ah4s_seqs_SUMMABLEu_u_SUM, axiom, ![V_f]: (p(s(t_bool,h4s_seqs_summable(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f)))) => p(s(t_bool,h4s_seqs_sums(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,h4s_seqs_suminf(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f)))))))).
fof(ch4s_seqs_SERu_u_LE, conjecture, ![V_g, V_f]: ((![V_n]: p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_g),s(t_h4s_nums_num,V_n)))))) & (p(s(t_bool,h4s_seqs_summable(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f)))) & p(s(t_bool,h4s_seqs_summable(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_g)))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_seqs_suminf(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,h4s_seqs_suminf(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_g)))))))).
