%   ORIGINAL: h4/fmapal/fmap__FDOM
% 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/pred__set/UNION__EMPTY_c0: !s. h4/pred__set/UNION h4/pred__set/EMPTY s = s
% Assm: h4/list/LIST__TO__SET__REVERSE: !ls. h4/list/LIST__TO__SET (h4/list/REVERSE ls) = h4/list/LIST__TO__SET ls
% Assm: h4/rich__list/MAP__REVERSE: !l f. h4/list/MAP f (h4/list/REVERSE l) = h4/list/REVERSE (h4/list/MAP f l)
% Assm: h4/finite__map/FDOM__FEMPTY: h4/finite__map/FDOM h4/finite__map/FEMPTY = h4/pred__set/EMPTY
% Assm: h4/finite__map/FDOM__FUPDATE__LIST: !kvl fm. h4/finite__map/FDOM (h4/finite__map/FUPDATE__LIST fm kvl) = h4/pred__set/UNION (h4/finite__map/FDOM fm) (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl))
% Assm: h4/fmapal/fmap0: !l. h4/fmapal/fmap l = h4/finite__map/FUPDATE__LIST h4/finite__map/FEMPTY (h4/list/REVERSE l)
% Goal: !l. h4/finite__map/FDOM (h4/fmapal/fmap l) = h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST l)
%   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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_predu_u_sets_UNIONu_u_EMPTYu_c0]: !s. h4/pred__set/UNION h4/pred__set/EMPTY s = s
% Assm [h4s_lists_LISTu_u_TOu_u_SETu_u_REVERSE]: !ls. h4/list/LIST__TO__SET (h4/list/REVERSE ls) = h4/list/LIST__TO__SET ls
% Assm [h4s_richu_u_lists_MAPu_u_REVERSE]: !l f. h4/list/MAP f (h4/list/REVERSE l) = h4/list/REVERSE (h4/list/MAP f l)
% Assm [h4s_finiteu_u_maps_FDOMu_u_FEMPTY]: h4/finite__map/FDOM h4/finite__map/FEMPTY = h4/pred__set/EMPTY
% Assm [h4s_finiteu_u_maps_FDOMu_u_FUPDATEu_u_LIST]: !kvl fm. h4/finite__map/FDOM (h4/finite__map/FUPDATE__LIST fm kvl) = h4/pred__set/UNION (h4/finite__map/FDOM fm) (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl))
% Assm [h4s_fmapals_fmap0]: !l. h4/fmapal/fmap l = h4/finite__map/FUPDATE__LIST h4/finite__map/FEMPTY (h4/list/REVERSE l)
% Goal: !l. h4/finite__map/FDOM (h4/fmapal/fmap l) = h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST l)
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_Q220094,TV_Q220090]: ![V_f, V_g]: (![V_x]: s(TV_Q220090,happ(s(t_fun(TV_Q220094,TV_Q220090),V_f),s(TV_Q220094,V_x))) = s(TV_Q220090,happ(s(t_fun(TV_Q220094,TV_Q220090),V_g),s(TV_Q220094,V_x))) => s(t_fun(TV_Q220094,TV_Q220090),V_f) = s(t_fun(TV_Q220094,TV_Q220090),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_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_predu_u_sets_UNIONu_u_EMPTYu_c0, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),V_s)).
fof(ah4s_lists_LISTu_u_TOu_u_SETu_u_REVERSE, axiom, ![TV_u_27a]: ![V_ls]: s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),V_ls))))) = s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_ls)))).
fof(ah4s_richu_u_lists_MAPu_u_REVERSE, axiom, ![TV_u_27b,TV_u_27a]: ![V_l, V_f]: s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_finiteu_u_maps_FDOMu_u_FEMPTY, axiom, ![TV_u_27b,TV_u_27a]: 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),h4s_finiteu_u_maps_fempty))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_finiteu_u_maps_FDOMu_u_FUPDATEu_u_LIST, axiom, ![TV_u_27a,TV_u_27b]: ![V_kvl, V_fm]: 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),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(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_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl)))))))).
fof(ah4s_fmapals_fmap0, axiom, ![TV_u_27a,TV_u_27b]: ![V_l]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_fmapals_fmap(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fempty),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_reverse(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l)))))).
fof(ch4s_fmapals_fmapu_u_FDOM, conjecture, ![TV_u_27a,TV_u_27b]: ![V_l]: 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),h4s_fmapals_fmap(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))))) = s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l)))))).
