%   ORIGINAL: h4/fmaptree/fmaptree__nchotomy
% 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/ETA__AX: !t. (\x. t x) = t
% 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/FALSITY: !t. F ==> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__REFL: !x. x = x
% 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/BOOL__EQ__DISTINCT_c0: ~(T <=> F)
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> 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/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% 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/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/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/MONO__IMP: !z y x w. (y ==> x) /\ (z ==> w) ==> (x ==> z) ==> y ==> w
% Assm: h4/bool/MONO__ALL: !Q P. (!x. P x ==> Q x) ==> (!x. P x) ==> (!x. Q x)
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% 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/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/list/list__case__def_c0: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm: h4/list/list__case__def_c1: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = f a0 a1
% Assm: h4/finite__map/fmap__EXT: !g f. f = g <=> h4/finite__map/FDOM f = h4/finite__map/FDOM g /\ (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> h4/finite__map/FAPPLY f x = h4/finite__map/FAPPLY g x)
% Assm: h4/finite__map/FDOM__o__f: !g f. h4/finite__map/FDOM (h4/finite__map/o__f f g) = h4/finite__map/FDOM g
% Assm: h4/finite__map/o__f__FAPPLY: !x g f. h4/bool/IN x (h4/finite__map/FDOM g) ==> h4/finite__map/FAPPLY (h4/finite__map/o__f f g) x = f (h4/finite__map/FAPPLY g x)
% Assm: h4/fmaptree/construct__def: !kl kfm a. h4/fmaptree/construct a kfm kl = h4/list/list__CASE kl (h4/option/SOME a) (\h t. h4/bool/COND (h4/bool/IN h (h4/finite__map/FDOM kfm)) (h4/finite__map/FAPPLY kfm h t) h4/option/NONE)
% Assm: h4/fmaptree/wf__def: h4/fmaptree/wf = (\a0. !wf_27. (!a00. (?a fm. a00 = h4/fmaptree/construct a fm /\ (!k. h4/bool/IN k (h4/finite__map/FDOM fm) ==> wf_27 (h4/finite__map/FAPPLY fm k))) ==> wf_27 a00) ==> wf_27 a0)
% Assm: h4/fmaptree/fmap__bij__thm_c0: !a. h4/fmaptree/fromF (h4/fmaptree/toF a) = a
% Assm: h4/fmaptree/fmap__bij__thm_c0: !a. h4/fmaptree/fromF (h4/fmaptree/toF a) = a
% Assm: h4/fmaptree/fmap__bij__thm_c1: !r. h4/fmaptree/wf r <=> h4/fmaptree/toF (h4/fmaptree/fromF r) = r
% Assm: h4/fmaptree/fmap__bij__thm_c1: !r. h4/fmaptree/wf r <=> h4/fmaptree/toF (h4/fmaptree/fromF r) = r
% Assm: h4/fmaptree/FTNode__def: !i fm. h4/fmaptree/FTNode i fm = h4/fmaptree/fromF (h4/fmaptree/construct i (h4/finite__map/o__f h4/fmaptree/toF fm))
% Goal: !ft. ?i fm. ft = h4/fmaptree/FTNode i fm
%   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_ETAu_u_AX]: !t x. happ t x = happ t 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_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~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_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% 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_BOOLu_u_EQu_u_DISTINCTu_c0]: ~(T <=> F)
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> 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_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% 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_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_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_MONOu_u_IMP]: !z y x w. (y ==> x) /\ (z ==> w) ==> (x ==> z) ==> y ==> w
% Assm [h4s_bools_MONOu_u_ALL]: !Q P. (!x. happ P x ==> happ Q x) ==> (!x. happ P x) ==> (!x. happ Q x)
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% 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_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_lists_listu_u_caseu_u_defu_c0]: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm [h4s_lists_listu_u_caseu_u_defu_c1]: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = happ (happ f a0) a1
% Assm [h4s_finiteu_u_maps_fmapu_u_EXT]: !g f. f = g <=> h4/finite__map/FDOM f = h4/finite__map/FDOM g /\ (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> h4/finite__map/FAPPLY f x = h4/finite__map/FAPPLY g x)
% Assm [h4s_finiteu_u_maps_FDOMu_u_ou_u_f]: !g f. h4/finite__map/FDOM (h4/finite__map/o__f f g) = h4/finite__map/FDOM g
% Assm [h4s_finiteu_u_maps_ou_u_fu_u_FAPPLY]: !x g f. h4/bool/IN x (h4/finite__map/FDOM g) ==> h4/finite__map/FAPPLY (h4/finite__map/o__f f g) x = happ f (h4/finite__map/FAPPLY g x)
% Assm [h4s_fmaptrees_constructu_u_def]: !_1. (!kfm h t. happ (happ (happ _1 kfm) h) t = h4/bool/COND (h4/bool/IN h (h4/finite__map/FDOM kfm)) (happ (h4/finite__map/FAPPLY kfm h) t) h4/option/NONE) ==> (!_0. (!kfm h. happ (happ _0 kfm) h = happ (happ _1 kfm) h) ==> (!kl kfm a. happ (h4/fmaptree/construct a kfm) kl = h4/list/list__CASE kl (h4/option/SOME a) (happ _0 kfm)))
% Assm [h4s_fmaptrees_wfu_u_def]: !x. h4/fmaptree/wf x <=> (!wf_27. (!a00. (?a fm. a00 = h4/fmaptree/construct a fm /\ (!k. h4/bool/IN k (h4/finite__map/FDOM fm) ==> happ wf_27 (h4/finite__map/FAPPLY fm k))) ==> happ wf_27 a00) ==> happ wf_27 x)
% Assm [h4s_fmaptrees_fmapu_u_biju_u_thmu_c0]: !a. h4/fmaptree/fromF (happ h4/fmaptree/toF a) = a
% Assm [h4s_fmaptrees_fmapu_u_biju_u_thmu_c0]: !a. h4/fmaptree/fromF (happ h4/fmaptree/toF a) = a
% Assm [h4s_fmaptrees_fmapu_u_biju_u_thmu_c1]: !r. h4/fmaptree/wf r <=> happ h4/fmaptree/toF (h4/fmaptree/fromF r) = r
% Assm [h4s_fmaptrees_fmapu_u_biju_u_thmu_c1]: !r. h4/fmaptree/wf r <=> happ h4/fmaptree/toF (h4/fmaptree/fromF r) = r
% Assm [h4s_fmaptrees_FTNodeu_u_def]: !i fm. h4/fmaptree/FTNode i fm = h4/fmaptree/fromF (h4/fmaptree/construct i (h4/finite__map/o__f h4/fmaptree/toF fm))
% Goal: !ft. ?i fm. ft = h4/fmaptree/FTNode i fm
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_Q255231,TV_Q255227]: ![V_f, V_g]: (![V_x]: s(TV_Q255227,happ(s(t_fun(TV_Q255231,TV_Q255227),V_f),s(TV_Q255231,V_x))) = s(TV_Q255227,happ(s(t_fun(TV_Q255231,TV_Q255227),V_g),s(TV_Q255231,V_x))) => s(t_fun(TV_Q255231,TV_Q255227),V_f) = s(t_fun(TV_Q255231,TV_Q255227),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_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_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_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_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_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_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_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,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_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_BOOLu_u_EQu_u_DISTINCTu_c0, axiom, ~ (s(t_bool,t) = s(t_bool,f))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
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_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_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_LEFTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((![V_x]: 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,V_Q))) <=> ![V_x]: (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,V_Q))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
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_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
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_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_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_IMP, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_y)) => p(s(t_bool,V_x))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) => p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_ALL, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (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_Q),s(TV_u_27a,V_x))))) => (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (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_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_EXISTSu_u_REFL, axiom, ![TV_u_27a]: ![V_a]: ?[V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_a)).
fof(ah4s_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & 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,V_a)))))).
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_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,f))))).
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,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
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,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
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_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_lists_listu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_lists_listu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1)))).
fof(ah4s_finiteu_u_maps_fmapu_u_EXT, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (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) <=> (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))) & ![V_x]: (p(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(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(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))))).
fof(ah4s_finiteu_u_maps_FDOMu_u_ou_u_f, axiom, ![TV_u_27c,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_27c),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))))) = 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)))).
fof(ah4s_finiteu_u_maps_ou_u_fu_u_FAPPLY, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: (p(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_g)))))) => s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))))).
fof(ah4s_fmaptrees_constructu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_1]: (![V_kfm, V_h, V_t]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)))),V_uu_1),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_kfm))),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_t))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27b,V_h),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_kfm))))),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_kfm),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_t))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) => ![V_uu_0]: (![V_kfm, V_h]: s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)))),V_uu_0),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_kfm))),s(TV_u_27b,V_h))) = s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)))),V_uu_1),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_kfm))),s(TV_u_27b,V_h))) => ![V_kl, V_kfm, V_a]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_fmaptrees_construct(s(TV_u_27a,V_a),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_kfm))),s(t_h4s_lists_list(TV_u_27b),V_kl))) = s(t_h4s_options_option(TV_u_27a),h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27b),V_kl),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_a))),s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)))),V_uu_0),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_kfm)))))))).
fof(ah4s_fmaptrees_wfu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_fmaptrees_wf(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),V_x)))) <=> ![V_wfu_27]: (![V_a00]: (?[V_a, V_fm]: (s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),V_a00) = s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_fmaptrees_construct(s(TV_u_27a,V_a),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_fm))) & ![V_k]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_k),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_fm)))))) => p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),t_bool),V_wfu_27),s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_fm),s(TV_u_27b,V_k)))))))) => p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),t_bool),V_wfu_27),s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),t_bool),V_wfu_27),s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),V_x))))))).
fof(ah4s_fmaptrees_fmapu_u_biju_u_thmu_c0, axiom, ![TV_u_27key,TV_u_27value]: ![V_a]: s(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),h4s_fmaptrees_fromf(s(t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value)),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value))),h4s_fmaptrees_tof),s(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),V_a))))) = s(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),V_a)).
fof(ah4s_fmaptrees_fmapu_u_biju_u_thmu_c0, axiom, ![TV_u_27key,TV_u_27value]: ![V_a]: s(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),h4s_fmaptrees_fromf(s(t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value)),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value))),h4s_fmaptrees_tof),s(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),V_a))))) = s(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),V_a)).
fof(ah4s_fmaptrees_fmapu_u_biju_u_thmu_c1, axiom, ![TV_u_27key,TV_u_27value]: ![V_r]: (p(s(t_bool,h4s_fmaptrees_wf(s(t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value)),V_r)))) <=> s(t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value)),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value))),h4s_fmaptrees_tof),s(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),h4s_fmaptrees_fromf(s(t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value)),V_r))))) = s(t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value)),V_r))).
fof(ah4s_fmaptrees_fmapu_u_biju_u_thmu_c1, axiom, ![TV_u_27key,TV_u_27value]: ![V_r]: (p(s(t_bool,h4s_fmaptrees_wf(s(t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value)),V_r)))) <=> s(t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value)),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value))),h4s_fmaptrees_tof),s(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),h4s_fmaptrees_fromf(s(t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value)),V_r))))) = s(t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value)),V_r))).
fof(ah4s_fmaptrees_FTNodeu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_i, V_fm]: s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),h4s_fmaptrees_ftnode(s(TV_u_27b,V_i),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),V_fm))) = s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),h4s_fmaptrees_fromf(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_fmaptrees_construct(s(TV_u_27b,V_i),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(TV_u_27b))),h4s_finiteu_u_maps_ou_u_f(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(TV_u_27b))),h4s_fmaptrees_tof),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),V_fm)))))))).
fof(ch4s_fmaptrees_fmaptreeu_u_nchotomy, conjecture, ![TV_u_27a,TV_u_27b]: ![V_ft]: ?[V_i, V_fm]: s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft) = s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),h4s_fmaptrees_ftnode(s(TV_u_27b,V_i),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),V_fm)))).
