%   ORIGINAL: h4/transc/DIVISION__GT
% 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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/arithmetic/LESS__EQ__REFL: !m. h4/arithmetic/_3C_3D m m
% Assm: h4/transc/DIVISION__LT__GEN: !n m b a D. h4/transc/division (h4/pair/_2C a b) D /\ h4/prim__rec/_3C m n /\ h4/arithmetic/_3C_3D n (h4/transc/dsize D) ==> h4/realax/real__lt (D m) (D n)
% Goal: !b a D. h4/transc/division (h4/pair/_2C a b) D ==> (!n. h4/prim__rec/_3C n (h4/transc/dsize D) ==> h4/realax/real__lt (D n) (D (h4/transc/dsize D)))
%   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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_arithmetics_LESSu_u_EQu_u_REFL]: !m. h4/arithmetic/_3C_3D m m
% Assm [h4s_transcs_DIVISIONu_u_LTu_u_GEN]: !n m b a D. h4/transc/division (h4/pair/_2C a b) D /\ h4/prim__rec/_3C m n /\ h4/arithmetic/_3C_3D n (h4/transc/dsize D) ==> h4/realax/real__lt (happ D m) (happ D n)
% Goal: !b a D. h4/transc/division (h4/pair/_2C a b) D ==> (!n. h4/prim__rec/_3C n (h4/transc/dsize D) ==> h4/realax/real__lt (happ D n) (happ D (h4/transc/dsize D)))
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_Q196189,TV_Q196185]: ![V_f, V_g]: (![V_x]: s(TV_Q196185,happ(s(t_fun(TV_Q196189,TV_Q196185),V_f),s(TV_Q196189,V_x))) = s(TV_Q196185,happ(s(t_fun(TV_Q196189,TV_Q196185),V_g),s(TV_Q196189,V_x))) => s(t_fun(TV_Q196189,TV_Q196185),V_f) = s(t_fun(TV_Q196189,TV_Q196185),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_arithmetics_LESSu_u_EQu_u_REFL, axiom, ![V_m]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_m))))).
fof(ah4s_transcs_DIVISIONu_u_LTu_u_GEN, axiom, ![V_n, V_m, V_b, V_a, V_D]: ((p(s(t_bool,h4s_transcs_division(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D)))) & (p(s(t_bool,h4s_primu_u_recs_u_3c(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,V_n),s(t_h4s_nums_num,h4s_transcs_dsize(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D)))))))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D),s(t_h4s_nums_num,V_m))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D),s(t_h4s_nums_num,V_n)))))))).
fof(ch4s_transcs_DIVISIONu_u_GT, conjecture, ![V_b, V_a, V_D]: (p(s(t_bool,h4s_transcs_division(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D)))) => ![V_n]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_transcs_dsize(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D)))))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D),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_D),s(t_h4s_nums_num,h4s_transcs_dsize(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D))))))))))).
