%   ORIGINAL: h4/finite__map/FUPDATE__LIST__APPLY__HO__THM
% 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/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/finite__map/FUPDATE__LIST__APPLY__NOT__MEM: !kvl k f. ~h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl)) ==> h4/finite__map/FAPPLY (h4/finite__map/FUPDATE__LIST f kvl) k = h4/finite__map/FAPPLY f k
% Assm: h4/finite__map/FUPDATE__LIST__APPLY__MEM: !v n kvl k f. h4/prim__rec/_3C n (h4/list/LENGTH kvl) /\ k = h4/list/EL n (h4/list/MAP h4/pair/FST kvl) /\ v = h4/list/EL n (h4/list/MAP h4/pair/SND kvl) /\ (!m. h4/prim__rec/_3C n m /\ h4/prim__rec/_3C m (h4/list/LENGTH kvl) ==> ~(h4/list/EL m (h4/list/MAP h4/pair/FST kvl) = k)) ==> h4/finite__map/FAPPLY (h4/finite__map/FUPDATE__LIST f kvl) k = v
% Goal: !kvl k f P. (?n. h4/prim__rec/_3C n (h4/list/LENGTH kvl) /\ k = h4/list/EL n (h4/list/MAP h4/pair/FST kvl) /\ P (h4/list/EL n (h4/list/MAP h4/pair/SND kvl)) /\ (!m. h4/prim__rec/_3C n m /\ h4/prim__rec/_3C m (h4/list/LENGTH kvl) ==> ~(h4/list/EL m (h4/list/MAP h4/pair/FST kvl) = k))) \/ ~h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl)) /\ P (h4/finite__map/FAPPLY f k) ==> P (h4/finite__map/FAPPLY (h4/finite__map/FUPDATE__LIST f kvl) k)
%   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_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_APPLYu_u_NOTu_u_MEM]: !kvl k f. ~h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl)) ==> h4/finite__map/FAPPLY (h4/finite__map/FUPDATE__LIST f kvl) k = h4/finite__map/FAPPLY f k
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_APPLYu_u_MEM]: !v n kvl k f. h4/prim__rec/_3C n (h4/list/LENGTH kvl) /\ k = h4/list/EL n (h4/list/MAP h4/pair/FST kvl) /\ v = h4/list/EL n (h4/list/MAP h4/pair/SND kvl) /\ (!m. h4/prim__rec/_3C n m /\ h4/prim__rec/_3C m (h4/list/LENGTH kvl) ==> ~(h4/list/EL m (h4/list/MAP h4/pair/FST kvl) = k)) ==> h4/finite__map/FAPPLY (h4/finite__map/FUPDATE__LIST f kvl) k = v
% Goal: !kvl k f P. (?n. h4/prim__rec/_3C n (h4/list/LENGTH kvl) /\ k = h4/list/EL n (h4/list/MAP h4/pair/FST kvl) /\ happ P (h4/list/EL n (h4/list/MAP h4/pair/SND kvl)) /\ (!m. h4/prim__rec/_3C n m /\ h4/prim__rec/_3C m (h4/list/LENGTH kvl) ==> ~(h4/list/EL m (h4/list/MAP h4/pair/FST kvl) = k))) \/ ~h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl)) /\ happ P (h4/finite__map/FAPPLY f k) ==> happ P (h4/finite__map/FAPPLY (h4/finite__map/FUPDATE__LIST f kvl) k)
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_Q77802,TV_Q77798]: ![V_f, V_g]: (![V_x]: s(TV_Q77798,happ(s(t_fun(TV_Q77802,TV_Q77798),V_f),s(TV_Q77802,V_x))) = s(TV_Q77798,happ(s(t_fun(TV_Q77802,TV_Q77798),V_g),s(TV_Q77802,V_x))) => s(t_fun(TV_Q77802,TV_Q77798),V_f) = s(t_fun(TV_Q77802,TV_Q77798),V_g))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
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_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = 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_LEFTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) & p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) & (p(s(t_bool,V_A)) | p(s(t_bool,V_C)))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_APPLYu_u_NOTu_u_MEM, axiom, ![TV_u_27b,TV_u_27a]: ![V_kvl, V_k, V_f]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_k),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))))))))) => s(TV_u_27b,h4s_finiteu_u_maps_fapply(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_f),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))),s(TV_u_27a,V_k))) = s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_k))))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_APPLYu_u_MEM, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_n, V_kvl, V_k, V_f]: ((p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl)))))) & (s(TV_u_27a,V_k) = s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_n),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))))) & (s(TV_u_27b,V_v) = s(TV_u_27b,h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))))) & ![V_m]: ((p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))))))) => ~ (s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_m),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))))) = s(TV_u_27a,V_k)))))) => s(TV_u_27b,h4s_finiteu_u_maps_fapply(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_f),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))),s(TV_u_27a,V_k))) = s(TV_u_27b,V_v))).
fof(ch4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_APPLYu_u_HOu_u_THM, conjecture, ![TV_u_27a,TV_u_27b]: ![V_kvl, V_k, V_f, V_P]: ((?[V_n]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_kvl)))))) & (s(TV_u_27b,V_k) = s(TV_u_27b,h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),TV_u_27b),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_kvl))))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),TV_u_27a),h4s_pairs_snd),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_kvl)))))))) & ![V_m]: ((p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_kvl))))))) => ~ (s(TV_u_27b,h4s_lists_el(s(t_h4s_nums_num,V_m),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),TV_u_27b),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_kvl))))) = s(TV_u_27b,V_k)))))) | (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_k),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_map(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),TV_u_27b),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_kvl))))))))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f),s(TV_u_27b,V_k)))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_kvl))),s(TV_u_27b,V_k)))))))).
