%   ORIGINAL: h4/finite__map/DOMSUB__COMMUTES
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> 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/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/pred__set/DELETE__COMM: !y x s. h4/pred__set/DELETE (h4/pred__set/DELETE s x) y = h4/pred__set/DELETE (h4/pred__set/DELETE s y) x
% Assm: h4/finite__map/fmap__EQ: !g f. h4/finite__map/FDOM f = h4/finite__map/FDOM g /\ h4/finite__map/FAPPLY f = h4/finite__map/FAPPLY g <=> f = g
% Assm: h4/finite__map/FDOM__DOMSUB: !k fm. h4/finite__map/FDOM (h4/finite__map/fdomsub fm k) = h4/pred__set/DELETE (h4/finite__map/FDOM fm) k
% Assm: h4/finite__map/DOMSUB__FAPPLY: !k fm. h4/finite__map/FAPPLY (h4/finite__map/fdomsub fm k) k = h4/finite__map/FAPPLY h4/finite__map/FEMPTY k
% Assm: h4/finite__map/DOMSUB__FAPPLY__NEQ: !k2 k1 fm. ~(k1 = k2) ==> h4/finite__map/FAPPLY (h4/finite__map/fdomsub fm k1) k2 = h4/finite__map/FAPPLY fm k2
% Assm: h4/finite__map/DOMSUB__FAPPLY__THM: !k2 k1 fm. h4/finite__map/FAPPLY (h4/finite__map/fdomsub fm k1) k2 = h4/bool/COND (k1 = k2) (h4/finite__map/FAPPLY h4/finite__map/FEMPTY k2) (h4/finite__map/FAPPLY fm k2)
% Goal: !k2 k1 fm. h4/finite__map/fdomsub (h4/finite__map/fdomsub fm k1) k2 = h4/finite__map/fdomsub (h4/finite__map/fdomsub fm k2) k1
%   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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> 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_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_predu_u_sets_DELETEu_u_COMM]: !y x s. h4/pred__set/DELETE (h4/pred__set/DELETE s x) y = h4/pred__set/DELETE (h4/pred__set/DELETE s y) x
% Assm [h4s_finiteu_u_maps_fmapu_u_EQ]: !g f. h4/finite__map/FDOM f = h4/finite__map/FDOM g /\ h4/finite__map/FAPPLY f = h4/finite__map/FAPPLY g <=> f = g
% Assm [h4s_finiteu_u_maps_FDOMu_u_DOMSUB]: !k fm. h4/finite__map/FDOM (h4/finite__map/fdomsub fm k) = h4/pred__set/DELETE (h4/finite__map/FDOM fm) k
% Assm [h4s_finiteu_u_maps_DOMSUBu_u_FAPPLY]: !k fm. happ (h4/finite__map/FAPPLY (h4/finite__map/fdomsub fm k)) k = happ (h4/finite__map/FAPPLY h4/finite__map/FEMPTY) k
% Assm [h4s_finiteu_u_maps_DOMSUBu_u_FAPPLYu_u_NEQ]: !k2 k1 fm. ~(k1 = k2) ==> happ (h4/finite__map/FAPPLY (h4/finite__map/fdomsub fm k1)) k2 = happ (h4/finite__map/FAPPLY fm) k2
% Assm [h4s_finiteu_u_maps_DOMSUBu_u_FAPPLYu_u_THM]: !k2 k1 fm. ?v. (v <=> k1 = k2) /\ happ (h4/finite__map/FAPPLY (h4/finite__map/fdomsub fm k1)) k2 = h4/bool/COND v (happ (h4/finite__map/FAPPLY h4/finite__map/FEMPTY) k2) (happ (h4/finite__map/FAPPLY fm) k2)
% Goal: !k2 k1 fm. h4/finite__map/fdomsub (h4/finite__map/fdomsub fm k1) k2 = h4/finite__map/fdomsub (h4/finite__map/fdomsub fm k2) k1
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_Q75374,TV_Q75370]: ![V_f, V_g]: (![V_x]: s(TV_Q75370,happ(s(t_fun(TV_Q75374,TV_Q75370),V_f),s(TV_Q75374,V_x))) = s(TV_Q75370,happ(s(t_fun(TV_Q75374,TV_Q75370),V_g),s(TV_Q75374,V_x))) => s(t_fun(TV_Q75374,TV_Q75370),V_f) = s(t_fun(TV_Q75374,TV_Q75370),V_g))).
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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> 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_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: 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_g),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_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
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_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_predu_u_sets_DELETEu_u_COMM, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_y))),s(TV_u_27a,V_x)))).
fof(ah4s_finiteu_u_maps_fmapu_u_EQ, axiom, ![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_27b),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_27b),V_g))) & s(t_fun(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g)))) <=> s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))).
fof(ah4s_finiteu_u_maps_FDOMu_u_DOMSUB, axiom, ![TV_u_27b,TV_u_27a]: ![V_k, 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_fdomsub(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(TV_u_27a,V_k))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(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(TV_u_27a,V_k)))).
fof(ah4s_finiteu_u_maps_DOMSUBu_u_FAPPLY, axiom, ![TV_u_27b,TV_u_27a]: ![V_k, V_fm]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,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_fdomsub(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(TV_u_27a,V_k))))),s(TV_u_27a,V_k))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,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_fempty))),s(TV_u_27a,V_k)))).
fof(ah4s_finiteu_u_maps_DOMSUBu_u_FAPPLYu_u_NEQ, axiom, ![TV_u_27b,TV_u_27a]: ![V_k2, V_k1, V_fm]: (~ (s(TV_u_27a,V_k1) = s(TV_u_27a,V_k2)) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,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_fdomsub(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(TV_u_27a,V_k1))))),s(TV_u_27a,V_k2))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm))),s(TV_u_27a,V_k2))))).
fof(ah4s_finiteu_u_maps_DOMSUBu_u_FAPPLYu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_k2, V_k1, V_fm]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_k1) = s(TV_u_27a,V_k2)) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,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_fdomsub(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(TV_u_27a,V_k1))))),s(TV_u_27a,V_k2))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,happ(s(t_fun(TV_u_27a,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_fempty))),s(TV_u_27a,V_k2))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm))),s(TV_u_27a,V_k2))))))).
fof(ch4s_finiteu_u_maps_DOMSUBu_u_COMMUTES, conjecture, ![TV_u_27b,TV_u_27a]: ![V_k2, V_k1, 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_fdomsub(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(TV_u_27a,V_k1))),s(TV_u_27a,V_k2))) = 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_fdomsub(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(TV_u_27a,V_k2))),s(TV_u_27a,V_k1)))).
