%   ORIGINAL: h4/container/FINITE__BAG__OF__FMAP
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> 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/finite__map/fmap__INDUCT: !P. P h4/finite__map/FEMPTY /\ (!f. P f ==> (!x y. ~h4/bool/IN x (h4/finite__map/FDOM f) ==> P (h4/finite__map/FUPDATE f (h4/pair/_2C x y)))) ==> (!f. P f)
% Assm: h4/finite__map/DOMSUB__NOT__IN__DOM: !k fm. ~h4/bool/IN k (h4/finite__map/FDOM fm) ==> h4/finite__map/fdomsub fm k = fm
% Assm: h4/bag/FINITE__EMPTY__BAG: h4/bag/FINITE__BAG h4/bag/EMPTY__BAG
% Assm: h4/bag/FINITE__BAG__INSERT: !b. h4/bag/FINITE__BAG b ==> (!e. h4/bag/FINITE__BAG (h4/bag/BAG__INSERT e b))
% Assm: h4/container/BAG__OF__FMAP__THM_c0: !f. h4/container/BAG__OF__FMAP f h4/finite__map/FEMPTY = h4/bag/EMPTY__BAG
% Assm: h4/container/BAG__OF__FMAP__THM_c1: !v k f b. h4/container/BAG__OF__FMAP f (h4/finite__map/FUPDATE b (h4/pair/_2C k v)) = h4/bag/BAG__INSERT (f k v) (h4/container/BAG__OF__FMAP f (h4/finite__map/fdomsub b k))
% Goal: !f b. h4/bag/FINITE__BAG (h4/container/BAG__OF__FMAP f b)
%   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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> 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_finiteu_u_maps_fmapu_u_INDUCT]: !P. happ P h4/finite__map/FEMPTY /\ (!f. happ P f ==> (!x y. ~h4/bool/IN x (h4/finite__map/FDOM f) ==> happ P (h4/finite__map/FUPDATE f (h4/pair/_2C x y)))) ==> (!f. happ P f)
% Assm [h4s_finiteu_u_maps_DOMSUBu_u_NOTu_u_INu_u_DOM]: !k fm. ~h4/bool/IN k (h4/finite__map/FDOM fm) ==> h4/finite__map/fdomsub fm k = fm
% Assm [h4s_bags_FINITEu_u_EMPTYu_u_BAG]: h4/bag/FINITE__BAG h4/bag/EMPTY__BAG
% Assm [h4s_bags_FINITEu_u_BAGu_u_INSERT]: !b. h4/bag/FINITE__BAG b ==> (!e. h4/bag/FINITE__BAG (h4/bag/BAG__INSERT e b))
% Assm [h4s_containers_BAGu_u_OFu_u_FMAPu_u_THMu_c0]: !f. h4/container/BAG__OF__FMAP f h4/finite__map/FEMPTY = h4/bag/EMPTY__BAG
% Assm [h4s_containers_BAGu_u_OFu_u_FMAPu_u_THMu_c1]: !v k f b. h4/container/BAG__OF__FMAP f (h4/finite__map/FUPDATE b (h4/pair/_2C k v)) = h4/bag/BAG__INSERT (happ (happ f k) v) (h4/container/BAG__OF__FMAP f (h4/finite__map/fdomsub b k))
% Goal: !f b. h4/bag/FINITE__BAG (h4/container/BAG__OF__FMAP f b)
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_Q300319,TV_Q300315]: ![V_f, V_g]: (![V_x]: s(TV_Q300315,happ(s(t_fun(TV_Q300319,TV_Q300315),V_f),s(TV_Q300319,V_x))) = s(TV_Q300315,happ(s(t_fun(TV_Q300319,TV_Q300315),V_g),s(TV_Q300319,V_x))) => s(t_fun(TV_Q300319,TV_Q300315),V_f) = s(t_fun(TV_Q300319,TV_Q300315),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_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_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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> 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_finiteu_u_maps_fmapu_u_INDUCT, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fempty)))) & ![V_f]: (p(s(t_bool,happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))) => ![V_x, V_y]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),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))))))))))) => ![V_f]: p(s(t_bool,happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))).
fof(ah4s_finiteu_u_maps_DOMSUBu_u_NOTu_u_INu_u_DOM, axiom, ![TV_u_27a,TV_u_27b]: ![V_k, V_fm]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_k),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm))))))) => s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fdomsub(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(TV_u_27a,V_k))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm))).
fof(ah4s_bags_FINITEu_u_EMPTYu_u_BAG, axiom, ![TV_u_27a]: p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))))).
fof(ah4s_bags_FINITEu_u_BAGu_u_INSERT, axiom, ![TV_u_27a]: ![V_b]: (p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) => ![V_e]: p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))))))).
fof(ah4s_containers_BAGu_u_OFu_u_FMAPu_u_THMu_c0, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_f]: s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_containers_bagu_u_ofu_u_fmap(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fempty))) = s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_bags_emptyu_u_bag)).
fof(ah4s_containers_BAGu_u_OFu_u_FMAPu_u_THMu_c1, axiom, ![TV_u_27f,TV_u_27e,TV_u_27d]: ![V_v, V_k, V_f, V_b]: s(t_fun(TV_u_27f,t_h4s_nums_num),h4s_containers_bagu_u_ofu_u_fmap(s(t_fun(TV_u_27d,t_fun(TV_u_27e,TV_u_27f)),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27d,TV_u_27e),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27d,TV_u_27e),V_b),s(t_h4s_pairs_prod(TV_u_27d,TV_u_27e),h4s_pairs_u_2c(s(TV_u_27d,V_k),s(TV_u_27e,V_v))))))) = s(t_fun(TV_u_27f,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27f,happ(s(t_fun(TV_u_27e,TV_u_27f),happ(s(t_fun(TV_u_27d,t_fun(TV_u_27e,TV_u_27f)),V_f),s(TV_u_27d,V_k))),s(TV_u_27e,V_v))),s(t_fun(TV_u_27f,t_h4s_nums_num),h4s_containers_bagu_u_ofu_u_fmap(s(t_fun(TV_u_27d,t_fun(TV_u_27e,TV_u_27f)),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27d,TV_u_27e),h4s_finiteu_u_maps_fdomsub(s(t_h4s_finiteu_u_maps_fmap(TV_u_27d,TV_u_27e),V_b),s(TV_u_27d,V_k)))))))).
fof(ch4s_containers_FINITEu_u_BAGu_u_OFu_u_FMAP, conjecture, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_f, V_b]: p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_containers_bagu_u_ofu_u_fmap(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_b))))))).
