%   ORIGINAL: h4/probability/joint__distribution__pos
% 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/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% 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/pred__set/INTER__SUBSET_c1: !t s. h4/pred__set/SUBSET (h4/pred__set/INTER t s) s
% Assm: h4/pred__set/IN__POW: !set e. h4/bool/IN e (h4/pred__set/POW set) <=> h4/pred__set/SUBSET e set
% Assm: h4/probability/joint__distribution__def: !p Y X. h4/probability/joint__distribution p X Y = (\a. h4/probability/prob p (h4/pred__set/INTER (h4/util__prob/PREIMAGE (\x. h4/pair/_2C (X x) (Y x)) a) (h4/probability/p__space p)))
% Assm: h4/probability/PROB__POSITIVE: !s p. h4/probability/prob__space p /\ h4/bool/IN s (h4/probability/events p) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/probability/prob p s)
% Goal: !p a Y X. h4/probability/prob__space p /\ h4/probability/events p = h4/pred__set/POW (h4/probability/p__space p) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/probability/joint__distribution p X Y a)
%   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_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% 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_predu_u_sets_INTERu_u_SUBSETu_c1]: !t s. h4/pred__set/SUBSET (h4/pred__set/INTER t s) s
% Assm [h4s_predu_u_sets_INu_u_POW]: !set e. h4/bool/IN e (h4/pred__set/POW set) <=> h4/pred__set/SUBSET e set
% Assm [h4s_probabilitys_jointu_u_distributionu_u_def]: !_0. (!X Y x. happ (happ (happ _0 X) Y) x = h4/pair/_2C (happ X x) (happ Y x)) ==> (!p Y X x. h4/probability/joint__distribution p X Y x = h4/probability/prob p (h4/pred__set/INTER (h4/util__prob/PREIMAGE (happ (happ _0 X) Y) x) (h4/probability/p__space p)))
% Assm [h4s_probabilitys_PROBu_u_POSITIVE]: !s p. h4/probability/prob__space p /\ h4/bool/IN s (h4/probability/events p) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/probability/prob p s)
% Goal: !p a Y X. h4/probability/prob__space p /\ h4/probability/events p = h4/pred__set/POW (h4/probability/p__space p) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/probability/joint__distribution p X Y a)
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_Q224108,TV_Q224104]: ![V_f, V_g]: (![V_x]: s(TV_Q224104,happ(s(t_fun(TV_Q224108,TV_Q224104),V_f),s(TV_Q224108,V_x))) = s(TV_Q224104,happ(s(t_fun(TV_Q224108,TV_Q224104),V_g),s(TV_Q224108,V_x))) => s(t_fun(TV_Q224108,TV_Q224104),V_f) = s(t_fun(TV_Q224108,TV_Q224104),V_g))).
fof(ah4s_bools_TRUTH, axiom, 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_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_predu_u_sets_INTERu_u_SUBSETu_c1, axiom, ![TV_u_27a]: ![V_t, V_s]: p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_s))))).
fof(ah4s_predu_u_sets_INu_u_POW, axiom, ![TV_u_27a]: ![V_set, V_e]: s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_e),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,t_bool),V_set))))) = s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_e),s(t_fun(TV_u_27a,t_bool),V_set)))).
fof(ah4s_probabilitys_jointu_u_distributionu_u_def, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_X, V_Y, V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27c,TV_u_27a),V_X))),s(t_fun(TV_u_27c,TV_u_27b),V_Y))),s(TV_u_27c,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_X),s(TV_u_27c,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_Y),s(TV_u_27c,V_x))))) => ![V_p, V_Y, V_X, V_x]: s(t_h4s_realaxs_real,h4s_probabilitys_jointu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27c,TV_u_27a),V_X),s(t_fun(TV_u_27c,TV_u_27b),V_Y),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_x))) = s(t_h4s_realaxs_real,h4s_probabilitys_prob(s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27c,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27c,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27c,TV_u_27a),V_X))),s(t_fun(TV_u_27c,TV_u_27b),V_Y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_x))),s(t_fun(TV_u_27c,t_bool),h4s_probabilitys_pu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_p))))))))).
fof(ah4s_probabilitys_PROBu_u_POSITIVE, axiom, ![TV_u_27a]: ![V_s, V_p]: ((p(s(t_bool,h4s_probabilitys_probu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_probabilitys_events(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_probabilitys_prob(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ch4s_probabilitys_jointu_u_distributionu_u_pos, conjecture, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_p, V_a, V_Y, V_X]: ((p(s(t_bool,h4s_probabilitys_probu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) & s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_probabilitys_events(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,t_bool),h4s_probabilitys_pu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_probabilitys_jointu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27a,TV_u_27c),V_Y),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),V_a)))))))).
