%   ORIGINAL: h4/util__prob/IN__PAIR
% 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/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/util__prob/pair__def: !Y X. h4/util__prob/pair X Y = h4/pair/UNCURRY (\x y. h4/bool/IN x X /\ h4/bool/IN y Y)
% Goal: !x Y X. h4/bool/IN x (h4/util__prob/pair X Y) <=> h4/bool/IN (h4/pair/FST x) X /\ h4/bool/IN (h4/pair/SND x) Y
%   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_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_utilu_u_probs_pairu_u_def]: !_1. (!x X Y y. happ (happ (happ (happ _1 x) X) Y) y <=> h4/bool/IN x X /\ h4/bool/IN y Y) ==> (!_0. (!X Y x. happ (happ (happ _0 X) Y) x = happ (happ (happ _1 x) X) Y) ==> (!Y X. h4/util__prob/pair X Y = h4/pair/UNCURRY (happ (happ _0 X) Y)))
% Goal: !x Y X. h4/bool/IN x (h4/util__prob/pair X Y) <=> h4/bool/IN (h4/pair/FST x) X /\ h4/bool/IN (h4/pair/SND x) Y
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_Q183258,TV_Q183254]: ![V_f, V_g]: (![V_x]: s(TV_Q183254,happ(s(t_fun(TV_Q183258,TV_Q183254),V_f),s(TV_Q183258,V_x))) = s(TV_Q183254,happ(s(t_fun(TV_Q183258,TV_Q183254),V_g),s(TV_Q183258,V_x))) => s(t_fun(TV_Q183258,TV_Q183254),V_f) = s(t_fun(TV_Q183258,TV_Q183254),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_predu_u_sets_SPECIFICATION, axiom, ![TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_utilu_u_probs_pairu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_x, V_X, V_Y, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,t_bool),V_X))),s(t_fun(TV_u_27b,t_bool),V_Y))),s(TV_u_27b,V_y)))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_X)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),V_Y)))))) => ![V_uu_0]: (![V_X, V_Y, V_x]: s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_X))),s(t_fun(TV_u_27b,t_bool),V_Y))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,t_bool),V_X))),s(t_fun(TV_u_27b,t_bool),V_Y))) => ![V_Y, V_X]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_utilu_u_probs_pair(s(t_fun(TV_u_27a,t_bool),V_X),s(t_fun(TV_u_27b,t_bool),V_Y))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_X))),s(t_fun(TV_u_27b,t_bool),V_Y)))))))).
fof(ch4s_utilu_u_probs_INu_u_PAIR, conjecture, ![TV_u_27a,TV_u_27b]: ![V_x, V_Y, V_X]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_utilu_u_probs_pair(s(t_fun(TV_u_27a,t_bool),V_X),s(t_fun(TV_u_27b,t_bool),V_Y)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_fun(TV_u_27a,t_bool),V_X)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_fun(TV_u_27b,t_bool),V_Y))))))).
