%   ORIGINAL: h4/alist/alist__to__fmap__MAP
% 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/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: 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_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> 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/marker/Abbrev__def: !x. h4/marker/Abbrev x <=> x
% 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/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/pred__set/INJ__EMPTY_c0: !s f. h4/pred__set/INJ f h4/pred__set/EMPTY s
% Assm: h4/pred__set/INJ__INSERT: !x t s f. h4/pred__set/INJ f (h4/pred__set/INSERT x s) t <=> h4/pred__set/INJ f s t /\ h4/bool/IN (f x) t /\ (!y. h4/bool/IN y s /\ f x = f y ==> x = y)
% Assm: h4/list/MAP0_c0: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm: h4/list/MAP0_c1: !t h f. h4/list/MAP f (h4/list/CONS h t) = h4/list/CONS (f h) (h4/list/MAP f t)
% Assm: h4/list/LIST__TO__SET0_c0: h4/list/LIST__TO__SET h4/list/NIL = h4/pred__set/EMPTY
% Assm: h4/list/LIST__TO__SET0_c1: !t h. h4/list/LIST__TO__SET (h4/list/CONS h t) = h4/pred__set/INSERT h (h4/list/LIST__TO__SET t)
% Assm: h4/list/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/finite__map/FDOM__o__f: !g f. h4/finite__map/FDOM (h4/finite__map/o__f f g) = h4/finite__map/FDOM g
% Assm: h4/finite__map/o__f__FEMPTY: !f. h4/finite__map/o__f f h4/finite__map/FEMPTY = h4/finite__map/FEMPTY
% Assm: h4/finite__map/o__f__DOMSUB: !k g fm. h4/finite__map/fdomsub (h4/finite__map/o__f g fm) k = h4/finite__map/o__f g (h4/finite__map/fdomsub fm k)
% Assm: h4/finite__map/o__f__FUPDATE: !v k fm f. h4/finite__map/o__f f (h4/finite__map/FUPDATE fm (h4/pair/_2C k v)) = h4/finite__map/FUPDATE (h4/finite__map/o__f f (h4/finite__map/fdomsub fm k)) (h4/pair/_2C k (f v))
% Assm: h4/finite__map/FUPDATE__PURGE: !y x f. h4/finite__map/FUPDATE f (h4/pair/_2C x y) = h4/finite__map/FUPDATE (h4/finite__map/fdomsub f x) (h4/pair/_2C x y)
% Assm: h4/finite__map/MAP__KEYS__FEMPTY: !f. h4/finite__map/MAP__KEYS f h4/finite__map/FEMPTY = h4/finite__map/FEMPTY
% Assm: h4/finite__map/MAP__KEYS__FUPDATE: !v k fm f. h4/pred__set/INJ f (h4/pred__set/INSERT k (h4/finite__map/FDOM fm)) h4/pred__set/UNIV ==> h4/finite__map/MAP__KEYS f (h4/finite__map/FUPDATE fm (h4/pair/_2C k v)) = h4/finite__map/FUPDATE (h4/finite__map/MAP__KEYS f fm) (h4/pair/_2C (f k) v)
% Assm: h4/alist/alist__to__fmap__thm_c0: h4/alist/alist__to__fmap h4/list/NIL = h4/finite__map/FEMPTY
% Assm: h4/alist/alist__to__fmap__thm_c1: !v t k. h4/alist/alist__to__fmap (h4/list/CONS (h4/pair/_2C k v) t) = h4/finite__map/FUPDATE (h4/alist/alist__to__fmap t) (h4/pair/_2C k v)
% Assm: h4/alist/FDOM__alist__to__fmap: !al. h4/finite__map/FDOM (h4/alist/alist__to__fmap al) = h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST al)
% Goal: !f2 f1 al. h4/pred__set/INJ f1 (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST al)) h4/pred__set/UNIV ==> h4/alist/alist__to__fmap (h4/list/MAP (h4/pair/UNCURRY (\x y. h4/pair/_2C (f1 x) (f2 y))) al) = h4/finite__map/MAP__KEYS f1 (h4/finite__map/o__f f2 (h4/alist/alist__to__fmap al))
%   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_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: 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_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> 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_markers_Abbrevu_u_def]: !x. h4/marker/Abbrev x <=> x
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_FST0]: !y x. happ h4/pair/FST (h4/pair/_2C x y) = x
% 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_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_INJu_u_EMPTYu_c0]: !s f. h4/pred__set/INJ f h4/pred__set/EMPTY s
% Assm [h4s_predu_u_sets_INJu_u_INSERT]: !x t s f. h4/pred__set/INJ f (h4/pred__set/INSERT x s) t <=> h4/pred__set/INJ f s t /\ h4/bool/IN (happ f x) t /\ (!y. h4/bool/IN y s /\ happ f x = happ f y ==> x = y)
% Assm [h4s_lists_MAP0u_c0]: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_MAP0u_c1]: !t h f. h4/list/MAP f (h4/list/CONS h t) = h4/list/CONS (happ f h) (h4/list/MAP f t)
% Assm [h4s_lists_LISTu_u_TOu_u_SET0u_c0]: h4/list/LIST__TO__SET h4/list/NIL = h4/pred__set/EMPTY
% Assm [h4s_lists_LISTu_u_TOu_u_SET0u_c1]: !t h. h4/list/LIST__TO__SET (h4/list/CONS h t) = h4/pred__set/INSERT h (h4/list/LIST__TO__SET t)
% Assm [h4s_lists_listu_u_induction]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_finiteu_u_maps_FDOMu_u_ou_u_f]: !g f. h4/finite__map/FDOM (h4/finite__map/o__f f g) = h4/finite__map/FDOM g
% Assm [h4s_finiteu_u_maps_ou_u_fu_u_FEMPTY]: !f. h4/finite__map/o__f f h4/finite__map/FEMPTY = h4/finite__map/FEMPTY
% Assm [h4s_finiteu_u_maps_ou_u_fu_u_DOMSUB]: !k g fm. h4/finite__map/fdomsub (h4/finite__map/o__f g fm) k = h4/finite__map/o__f g (h4/finite__map/fdomsub fm k)
% Assm [h4s_finiteu_u_maps_ou_u_fu_u_FUPDATE]: !v k fm f. h4/finite__map/o__f f (h4/finite__map/FUPDATE fm (h4/pair/_2C k v)) = h4/finite__map/FUPDATE (h4/finite__map/o__f f (h4/finite__map/fdomsub fm k)) (h4/pair/_2C k (happ f v))
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_PURGE]: !y x f. h4/finite__map/FUPDATE f (h4/pair/_2C x y) = h4/finite__map/FUPDATE (h4/finite__map/fdomsub f x) (h4/pair/_2C x y)
% Assm [h4s_finiteu_u_maps_MAPu_u_KEYSu_u_FEMPTY]: !f. h4/finite__map/MAP__KEYS f h4/finite__map/FEMPTY = h4/finite__map/FEMPTY
% Assm [h4s_finiteu_u_maps_MAPu_u_KEYSu_u_FUPDATE]: !v k fm f. h4/pred__set/INJ f (h4/pred__set/INSERT k (h4/finite__map/FDOM fm)) h4/pred__set/UNIV ==> h4/finite__map/MAP__KEYS f (h4/finite__map/FUPDATE fm (h4/pair/_2C k v)) = h4/finite__map/FUPDATE (h4/finite__map/MAP__KEYS f fm) (h4/pair/_2C (happ f k) v)
% Assm [h4s_alists_alistu_u_tou_u_fmapu_u_thmu_c0]: h4/alist/alist__to__fmap h4/list/NIL = h4/finite__map/FEMPTY
% Assm [h4s_alists_alistu_u_tou_u_fmapu_u_thmu_c1]: !v t k. h4/alist/alist__to__fmap (h4/list/CONS (h4/pair/_2C k v) t) = h4/finite__map/FUPDATE (h4/alist/alist__to__fmap t) (h4/pair/_2C k v)
% Assm [h4s_alists_FDOMu_u_alistu_u_tou_u_fmap]: !al. h4/finite__map/FDOM (h4/alist/alist__to__fmap al) = h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST al)
% Goal: !_1. (!f1 x f2 y. happ (happ (happ (happ _1 f1) x) f2) y = h4/pair/_2C (happ f1 x) (happ f2 y)) ==> (!_0. (!f1 f2 x. happ (happ (happ _0 f1) f2) x = happ (happ (happ _1 f1) x) f2) ==> (!f2 f1 al. h4/pred__set/INJ f1 (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST al)) h4/pred__set/UNIV ==> h4/alist/alist__to__fmap (h4/list/MAP (h4/pair/UNCURRY (happ (happ _0 f1) f2)) al) = h4/finite__map/MAP__KEYS f1 (h4/finite__map/o__f f2 (h4/alist/alist__to__fmap al))))
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_Q270740,TV_Q270736]: ![V_f, V_g]: (![V_x]: s(TV_Q270736,happ(s(t_fun(TV_Q270740,TV_Q270736),V_f),s(TV_Q270740,V_x))) = s(TV_Q270736,happ(s(t_fun(TV_Q270740,TV_Q270736),V_g),s(TV_Q270740,V_x))) => s(t_fun(TV_Q270740,TV_Q270736),V_f) = s(t_fun(TV_Q270740,TV_Q270736),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
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_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_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_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_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_markers_Abbrevu_u_def, axiom, ![V_x]: s(t_bool,h4s_markers_abbrev(s(t_bool,V_x))) = s(t_bool,V_x)).
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,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_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_INu_u_UNIV, axiom, ![TV_u_27a]: ![V_x]: p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))).
fof(ah4s_predu_u_sets_INJu_u_EMPTYu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_s, V_f]: p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(TV_u_27b,t_bool),V_s))))).
fof(ah4s_predu_u_sets_INJu_u_INSERT, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_t, V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27b,t_bool),V_t)))) <=> (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) & (p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),V_t)))) & ![V_y]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))))).
fof(ah4s_lists_MAP0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![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_nil))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)).
fof(ah4s_lists_MAP0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, 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_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_h))),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_t)))))).
fof(ah4s_lists_LISTu_u_TOu_u_SET0u_c0, axiom, ![TV_u_27a]: 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_nil))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_lists_LISTu_u_TOu_u_SET0u_c1, axiom, ![TV_u_27b]: ![V_t, V_h]: s(t_fun(TV_u_27b,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_h),s(t_h4s_lists_list(TV_u_27b),V_t))))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_insert(s(TV_u_27b,V_h),s(t_fun(TV_u_27b,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27b),V_t)))))).
fof(ah4s_lists_listu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_finiteu_u_maps_FDOMu_u_ou_u_f, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_g, V_f]: 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_27c),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))))) = 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_g)))).
fof(ah4s_finiteu_u_maps_ou_u_fu_u_FEMPTY, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fempty))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fempty)).
fof(ah4s_finiteu_u_maps_ou_u_fu_u_DOMSUB, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_k, V_g, 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),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27c,TV_u_27b),V_g),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))),s(TV_u_27a,V_k))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27c,TV_u_27b),V_g),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fdomsub(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm),s(TV_u_27a,V_k)))))).
fof(ah4s_finiteu_u_maps_ou_u_fu_u_FUPDATE, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_v, V_k, V_fm, V_f]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27a,V_k),s(TV_u_27c,V_v))))))) = 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),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fdomsub(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm),s(TV_u_27a,V_k))))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_k),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,V_v)))))))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_PURGE, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: 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))))) = 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),h4s_finiteu_u_maps_fdomsub(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),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)))))).
fof(ah4s_finiteu_u_maps_MAPu_u_KEYSu_u_FEMPTY, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_f]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_mapu_u_keys(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fempty))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_fempty)).
fof(ah4s_finiteu_u_maps_MAPu_u_KEYSu_u_FUPDATE, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_v, V_k, V_fm, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(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_27c),V_fm))))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_univ)))) => s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_mapu_u_keys(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27a,V_k),s(TV_u_27c,V_v))))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_mapu_u_keys(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))),s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_k))),s(TV_u_27c,V_v))))))).
fof(ah4s_alists_alistu_u_tou_u_fmapu_u_thmu_c0, axiom, ![TV_u_27a,TV_u_27b]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_alists_alistu_u_tou_u_fmap(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_nil))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fempty)).
fof(ah4s_alists_alistu_u_tou_u_fmapu_u_thmu_c1, axiom, ![TV_u_27c,TV_u_27d]: ![V_v, V_t, V_k]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27d),h4s_alists_alistu_u_tou_u_fmap(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27c,TV_u_27d)),h4s_lists_cons(s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),h4s_pairs_u_2c(s(TV_u_27c,V_k),s(TV_u_27d,V_v))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27c,TV_u_27d)),V_t))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27d),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27d),h4s_alists_alistu_u_tou_u_fmap(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27c,TV_u_27d)),V_t))),s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),h4s_pairs_u_2c(s(TV_u_27c,V_k),s(TV_u_27d,V_v)))))).
fof(ah4s_alists_FDOMu_u_alistu_u_tou_u_fmap, axiom, ![TV_u_27a,TV_u_27b]: ![V_al]: 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_alists_alistu_u_tou_u_fmap(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_al))))) = 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_al)))))).
fof(ch4s_alists_alistu_u_tou_u_fmapu_u_MAP, conjecture, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_uu_1]: (![V_f1, V_x, V_f2, V_y]: s(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),happ(s(t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27d))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27d)))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27d))))),V_uu_1),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(TV_u_27a,V_x))),s(t_fun(TV_u_27c,TV_u_27d),V_f2))),s(TV_u_27c,V_y))) = s(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),h4s_pairs_u_2c(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(TV_u_27a,V_x))),s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f2),s(TV_u_27c,V_y))))) => ![V_uu_0]: (![V_f1, V_f2, V_x]: s(t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27d)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27d)))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27d))))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(t_fun(TV_u_27c,TV_u_27d),V_f2))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27d))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27d)))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27d))))),V_uu_1),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(TV_u_27a,V_x))),s(t_fun(TV_u_27c,TV_u_27d),V_f2))) => ![V_f2, V_f1, V_al]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f1),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_27c),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),V_al))))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_univ)))) => s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27d),h4s_alists_alistu_u_tou_u_fmap(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27d)),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_h4s_pairs_prod(TV_u_27b,TV_u_27d)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27d)))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27d))))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(t_fun(TV_u_27c,TV_u_27d),V_f2))))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),V_al))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27d),h4s_finiteu_u_maps_mapu_u_keys(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27d),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27c,TV_u_27d),V_f2),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_alists_alistu_u_tou_u_fmap(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),V_al))))))))))).
