%   ORIGINAL: h4/fmaptree/apply__path__SNOC
% 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/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/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/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/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/list/APPEND0_c0: !l. h4/list/APPEND h4/list/NIL l = l
% Assm: h4/list/APPEND0_c1: !l2 l1 h. h4/list/APPEND (h4/list/CONS h l1) l2 = h4/list/CONS h (h4/list/APPEND l1 l2)
% 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/FLOOKUP__DEF: !x f. h4/finite__map/FLOOKUP f x = h4/bool/COND (h4/bool/IN x (h4/finite__map/FDOM f)) (h4/option/SOME (h4/finite__map/FAPPLY f x)) h4/option/NONE
% Assm: h4/fmaptree/apply__path__def_c0: !ft. h4/fmaptree/apply__path h4/list/NIL ft = h4/option/SOME ft
% Assm: h4/fmaptree/apply__path__def_c1: !t h ft. h4/fmaptree/apply__path (h4/list/CONS h t) ft = h4/bool/COND (h4/bool/IN h (h4/finite__map/FDOM (h4/fmaptree/map ft))) (h4/fmaptree/apply__path t (h4/finite__map/FAPPLY (h4/fmaptree/map ft) h)) h4/option/NONE
% Goal: !x p ft. h4/fmaptree/apply__path (h4/list/APPEND p (h4/list/CONS x h4/list/NIL)) ft = h4/option/option__CASE (h4/fmaptree/apply__path p ft) h4/option/NONE (\ft_27. h4/finite__map/FLOOKUP (h4/fmaptree/map ft_27) x)
%   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_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_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_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_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = happ f x
% Assm [h4s_lists_APPEND0u_c0]: !l. h4/list/APPEND h4/list/NIL l = l
% Assm [h4s_lists_APPEND0u_c1]: !l2 l1 h. h4/list/APPEND (h4/list/CONS h l1) l2 = h4/list/CONS h (h4/list/APPEND l1 l2)
% 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_FLOOKUPu_u_DEF]: !x f. h4/finite__map/FLOOKUP f x = h4/bool/COND (h4/bool/IN x (h4/finite__map/FDOM f)) (h4/option/SOME (h4/finite__map/FAPPLY f x)) h4/option/NONE
% Assm [h4s_fmaptrees_applyu_u_pathu_u_defu_c0]: !ft. h4/fmaptree/apply__path h4/list/NIL ft = h4/option/SOME ft
% Assm [h4s_fmaptrees_applyu_u_pathu_u_defu_c1]: !t h ft. h4/fmaptree/apply__path (h4/list/CONS h t) ft = h4/bool/COND (h4/bool/IN h (h4/finite__map/FDOM (h4/fmaptree/map ft))) (h4/fmaptree/apply__path t (h4/finite__map/FAPPLY (h4/fmaptree/map ft) h)) h4/option/NONE
% Goal: !_0. (!x ft_27. happ (happ _0 x) ft_27 = h4/finite__map/FLOOKUP (h4/fmaptree/map ft_27) x) ==> (!x p ft. h4/fmaptree/apply__path (h4/list/APPEND p (h4/list/CONS x h4/list/NIL)) ft = h4/option/option__CASE (h4/fmaptree/apply__path p ft) h4/option/NONE (happ _0 x))
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_Q255357,TV_Q255353]: ![V_f, V_g]: (![V_x]: s(TV_Q255353,happ(s(t_fun(TV_Q255357,TV_Q255353),V_f),s(TV_Q255357,V_x))) = s(TV_Q255353,happ(s(t_fun(TV_Q255357,TV_Q255353),V_g),s(TV_Q255357,V_x))) => s(t_fun(TV_Q255357,TV_Q255353),V_f) = s(t_fun(TV_Q255357,TV_Q255353),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_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_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_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_options_optionu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_options_optionu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_lists_APPEND0u_c0, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
fof(ah4s_lists_APPEND0u_c1, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(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_l1))),s(t_h4s_lists_list(TV_u_27a),V_l2))) = 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),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
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_FLOOKUPu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: 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_f),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27b),h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),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_h4s_options_option(TV_u_27b),h4s_options_some(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_x))))),s(t_h4s_options_option(TV_u_27b),h4s_options_none)))).
fof(ah4s_fmaptrees_applyu_u_pathu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_ft]: s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_applyu_u_path(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))) = s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_options_some(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft)))).
fof(ah4s_fmaptrees_applyu_u_pathu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_t, V_h, V_ft]: s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_applyu_u_path(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_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))) = s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_h),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_map(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))))))),s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_applyu_u_path(s(t_h4s_lists_list(TV_u_27a),V_t),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_map(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))),s(TV_u_27a,V_h))))),s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_options_none)))).
fof(ch4s_fmaptrees_applyu_u_pathu_u_SNOC, conjecture, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_x, V_ftu_27]: s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)))),V_uu_0),s(TV_u_27a,V_x))),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ftu_27))) = s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_map(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ftu_27))),s(TV_u_27a,V_x))) => ![V_x, V_p, V_ft]: s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_applyu_u_path(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_p),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))))),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))) = s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_applyu_u_path(s(t_h4s_lists_list(TV_u_27a),V_p),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))),s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_options_none),s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)))),V_uu_0),s(TV_u_27a,V_x))))))).
