%   ORIGINAL: h4/alist/alist__to__fmap__MAP__values
% 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/AND__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/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/pred__set/INJ__DEF: !t s f. h4/pred__set/INJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (f x) t) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> f x = f y ==> x = y)
% Assm: h4/alist/alist__to__fmap__MAP__matchable: !v mal f2 f1 al. h4/pred__set/INJ f1 (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST al)) h4/pred__set/UNIV /\ mal = h4/list/MAP (h4/pair/UNCURRY (\x y. h4/pair/_2C (f1 x) (f2 y))) al /\ v = h4/finite__map/MAP__KEYS f1 (h4/finite__map/o__f f2 (h4/alist/alist__to__fmap al)) ==> h4/alist/alist__to__fmap mal = v
% Assm: h4/alist/MAP__KEYS__I: !fm. h4/finite__map/MAP__KEYS h4/combin/I fm = fm
% Goal: !f al. h4/alist/alist__to__fmap (h4/list/MAP (h4/pair/UNCURRY (\k v. h4/pair/_2C k (f v))) al) = h4/finite__map/o__f f (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_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_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_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_INJu_u_DEF]: !t s f. h4/pred__set/INJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (happ f x) t) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> happ f x = happ f y ==> x = y)
% Assm [h4s_alists_alistu_u_tou_u_fmapu_u_MAPu_u_matchable]: !_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) ==> (!v mal f2 f1 al. h4/pred__set/INJ f1 (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST al)) h4/pred__set/UNIV /\ mal = h4/list/MAP (h4/pair/UNCURRY (happ (happ _0 f1) f2)) al /\ v = h4/finite__map/MAP__KEYS f1 (h4/finite__map/o__f f2 (h4/alist/alist__to__fmap al)) ==> h4/alist/alist__to__fmap mal = v))
% Assm [h4s_alists_MAPu_u_KEYSu_u_I]: !fm. h4/finite__map/MAP__KEYS h4/combin/I fm = fm
% Goal: !_1. (!k f v. happ (happ (happ _1 k) f) v = h4/pair/_2C k (happ f v)) ==> (!_0. (!f k. happ (happ _0 f) k = happ (happ _1 k) f) ==> (!f al. h4/alist/alist__to__fmap (h4/list/MAP (h4/pair/UNCURRY (happ _0 f)) al) = h4/finite__map/o__f f (h4/alist/alist__to__fmap al)))
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_Q271250,TV_Q271246]: ![V_f, V_g]: (![V_x]: s(TV_Q271246,happ(s(t_fun(TV_Q271250,TV_Q271246),V_f),s(TV_Q271250,V_x))) = s(TV_Q271246,happ(s(t_fun(TV_Q271250,TV_Q271246),V_g),s(TV_Q271250,V_x))) => s(t_fun(TV_Q271250,TV_Q271246),V_f) = s(t_fun(TV_Q271250,TV_Q271246),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_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_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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
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_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![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),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) <=> (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => 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_x, V_y]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & 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_alists_alistu_u_tou_u_fmapu_u_MAPu_u_matchable, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b,TV_u_27d]: ![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_v, V_mal, 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_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27d)),V_mal) = 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),V_v) = 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))))))))) => 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)),V_mal))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27d),V_v))))).
fof(ah4s_alists_MAPu_u_KEYSu_u_I, axiom, ![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_mapu_u_keys(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),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),V_fm)).
fof(ch4s_alists_alistu_u_tou_u_fmapu_u_MAPu_u_values, conjecture, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_1]: (![V_k, V_f, V_v]: s(t_h4s_pairs_prod(TV_u_27c,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),V_uu_1),s(TV_u_27c,V_k))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_v))) = s(t_h4s_pairs_prod(TV_u_27c,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27c,V_k),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_v))))) => ![V_uu_0]: (![V_f, V_k]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27c,V_k))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),V_uu_1),s(TV_u_27c,V_k))),s(t_fun(TV_u_27a,TV_u_27b),V_f))) => ![V_f, V_al]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),h4s_alists_alistu_u_tou_u_fmap(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),h4s_pairs_uncurry(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f))))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),V_al))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),h4s_alists_alistu_u_tou_u_fmap(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),V_al)))))))).
