%   ORIGINAL: h4/alist/MEM__pair__fmap__to__alist__FLOOKUP
% 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/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/alist/MEM__fmap__to__alist__FLOOKUP: !p fm. h4/bool/IN p (h4/list/LIST__TO__SET (h4/alist/fmap__to__alist fm)) <=> h4/finite__map/FLOOKUP fm (h4/pair/FST p) = h4/option/SOME (h4/pair/SND p)
% Goal: !y x fm. h4/bool/IN (h4/pair/_2C x y) (h4/list/LIST__TO__SET (h4/alist/fmap__to__alist fm)) <=> h4/finite__map/FLOOKUP fm x = h4/option/SOME 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_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_alists_MEMu_u_fmapu_u_tou_u_alistu_u_FLOOKUP]: !p fm. h4/bool/IN p (h4/list/LIST__TO__SET (h4/alist/fmap__to__alist fm)) <=> h4/finite__map/FLOOKUP fm (h4/pair/FST p) = h4/option/SOME (h4/pair/SND p)
% Goal: !y x fm. h4/bool/IN (h4/pair/_2C x y) (h4/list/LIST__TO__SET (h4/alist/fmap__to__alist fm)) <=> h4/finite__map/FLOOKUP fm x = h4/option/SOME 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_Q270311,TV_Q270307]: ![V_f, V_g]: (![V_x]: s(TV_Q270307,happ(s(t_fun(TV_Q270311,TV_Q270307),V_f),s(TV_Q270311,V_x))) = s(TV_Q270307,happ(s(t_fun(TV_Q270311,TV_Q270307),V_g),s(TV_Q270311,V_x))) => s(t_fun(TV_Q270311,TV_Q270307),V_f) = s(t_fun(TV_Q270311,TV_Q270307),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_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_alists_MEMu_u_fmapu_u_tou_u_alistu_u_FLOOKUP, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_fm]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_alists_fmapu_u_tou_u_alist(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm)))))))) <=> s(t_h4s_options_option(TV_u_27b),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))))))).
fof(ch4s_alists_MEMu_u_pairu_u_fmapu_u_tou_u_alistu_u_FLOOKUP, conjecture, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_fm]: (p(s(t_bool,h4s_bools_in(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_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_alists_fmapu_u_tou_u_alist(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm)))))))) <=> s(t_h4s_options_option(TV_u_27b),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_y))))).
