%   ORIGINAL: h4/sum__num/SUM__EQUAL
% 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/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/sum__num/GSUM__EQUAL: !p n m f. h4/sum__num/GSUM (h4/pair/_2C p m) f = h4/sum__num/GSUM (h4/pair/_2C p n) f <=> h4/arithmetic/_3C_3D m n /\ (!q. h4/arithmetic/_3C_3D (h4/arithmetic/_2B p m) q /\ h4/prim__rec/_3C q (h4/arithmetic/_2B p n) ==> f q = h4/num/0) \/ h4/prim__rec/_3C n m /\ (!q. h4/arithmetic/_3C_3D (h4/arithmetic/_2B p n) q /\ h4/prim__rec/_3C q (h4/arithmetic/_2B p m) ==> f q = h4/num/0)
% Assm: h4/sum__num/SUM0: !m f. h4/sum__num/SUM m f = h4/sum__num/GSUM (h4/pair/_2C h4/num/0 m) f
% Goal: !n m f. h4/sum__num/SUM m f = h4/sum__num/SUM n f <=> h4/arithmetic/_3C_3D m n /\ (!q. h4/arithmetic/_3C_3D m q /\ h4/prim__rec/_3C q n ==> f q = h4/num/0) \/ h4/prim__rec/_3C n m /\ (!q. h4/arithmetic/_3C_3D n q /\ h4/prim__rec/_3C q m ==> f q = h4/num/0)
%   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_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_sumu_u_nums_GSUMu_u_EQUAL]: !p n m f. h4/sum__num/GSUM (h4/pair/_2C p m) f = h4/sum__num/GSUM (h4/pair/_2C p n) f <=> h4/arithmetic/_3C_3D m n /\ (!q. h4/arithmetic/_3C_3D (h4/arithmetic/_2B p m) q /\ h4/prim__rec/_3C q (h4/arithmetic/_2B p n) ==> happ f q = h4/num/0) \/ h4/prim__rec/_3C n m /\ (!q. h4/arithmetic/_3C_3D (h4/arithmetic/_2B p n) q /\ h4/prim__rec/_3C q (h4/arithmetic/_2B p m) ==> happ f q = h4/num/0)
% Assm [h4s_sumu_u_nums_SUM0]: !m f. h4/sum__num/SUM m f = h4/sum__num/GSUM (h4/pair/_2C h4/num/0 m) f
% Goal: !n m f. h4/sum__num/SUM m f = h4/sum__num/SUM n f <=> h4/arithmetic/_3C_3D m n /\ (!q. h4/arithmetic/_3C_3D m q /\ h4/prim__rec/_3C q n ==> happ f q = h4/num/0) \/ h4/prim__rec/_3C n m /\ (!q. h4/arithmetic/_3C_3D n q /\ h4/prim__rec/_3C q m ==> happ f q = h4/num/0)
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_Q13152,TV_Q13148]: ![V_f, V_g]: (![V_x]: s(TV_Q13148,happ(s(t_fun(TV_Q13152,TV_Q13148),V_f),s(TV_Q13152,V_x))) = s(TV_Q13148,happ(s(t_fun(TV_Q13152,TV_Q13148),V_g),s(TV_Q13152,V_x))) => s(t_fun(TV_Q13152,TV_Q13148),V_f) = s(t_fun(TV_Q13152,TV_Q13148),V_g))).
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_ANDu_u_IMPu_u_INTRO, 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_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_arithmetics_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_sumu_u_nums_GSUMu_u_EQUAL, axiom, ![V_p, V_n, V_m, V_f]: (s(t_h4s_nums_num,h4s_sumu_u_nums_gsum(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_m))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),V_f))) = s(t_h4s_nums_num,h4s_sumu_u_nums_gsum(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),V_f))) <=> ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & ![V_q]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_q)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_n))))))) => s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),V_f),s(t_h4s_nums_num,V_q))) = s(t_h4s_nums_num,h4s_nums_0))) | (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))) & ![V_q]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_q)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_m))))))) => s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),V_f),s(t_h4s_nums_num,V_q))) = s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_sumu_u_nums_SUM0, axiom, ![V_m, V_f]: s(t_h4s_nums_num,h4s_sumu_u_nums_sum(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),V_f))) = s(t_h4s_nums_num,h4s_sumu_u_nums_gsum(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_m))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),V_f)))).
fof(ch4s_sumu_u_nums_SUMu_u_EQUAL, conjecture, ![V_n, V_m, V_f]: (s(t_h4s_nums_num,h4s_sumu_u_nums_sum(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),V_f))) = s(t_h4s_nums_num,h4s_sumu_u_nums_sum(s(t_h4s_nums_num,V_n),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),V_f))) <=> ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & ![V_q]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_q)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_n))))) => s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),V_f),s(t_h4s_nums_num,V_q))) = s(t_h4s_nums_num,h4s_nums_0))) | (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))) & ![V_q]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_q)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_m))))) => s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),V_f),s(t_h4s_nums_num,V_q))) = s(t_h4s_nums_num,h4s_nums_0)))))).
