%   ORIGINAL: h4/sptree/toListA__append
% 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/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/EQ__REFL: !x. x = x
% 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/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% 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__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~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/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/APPEND__ASSOC: !l3 l2 l1. h4/list/APPEND l1 (h4/list/APPEND l2 l3) = h4/list/APPEND (h4/list/APPEND l1 l2) l3
% Assm: h4/sptree/spt__repfns_c0: !a. h4/sptree/_20_40ind__typesptree4 (h4/sptree/_20_40ind__typesptree5 a) = a
% Assm: h4/sptree/spt__repfns_c1: !r. (\a0_27. !_27spt_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a0 (\n. h4/ind__type/BOTTOM)) a) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27spt_27 a0 /\ _27spt_27 a1) \/ (?a0 a1 a2. a0_270 = (\a00 a10 a20. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) a10 (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a20 (\n. h4/ind__type/BOTTOM)))) a0 a1 a2 /\ _27spt_27 a0 /\ _27spt_27 a2) ==> _27spt_27 a0_270) ==> _27spt_27 a0_27) r <=> h4/sptree/_20_40ind__typesptree5 (h4/sptree/_20_40ind__typesptree4 r) = r
% Assm: h4/sptree/hidden____20__40ind____typesptree0____def: h4/sptree/_20_40ind__typesptree0 = h4/sptree/_20_40ind__typesptree4 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/sptree/hidden____20__40ind____typesptree1____def: h4/sptree/_20_40ind__typesptree1 = (\a. h4/sptree/_20_40ind__typesptree4 ((\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a0 (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/sptree/hidden____20__40ind____typesptree2____def: h4/sptree/_20_40ind__typesptree2 = (\a0 a1. h4/sptree/_20_40ind__typesptree4 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) (h4/sptree/_20_40ind__typesptree5 a0) (h4/sptree/_20_40ind__typesptree5 a1)))
% Assm: h4/sptree/hidden____20__40ind____typesptree3____def: h4/sptree/_20_40ind__typesptree3 = (\a0 a1 a2. h4/sptree/_20_40ind__typesptree4 ((\a00 a10 a20. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) a10 (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a20 (\n. h4/ind__type/BOTTOM)))) (h4/sptree/_20_40ind__typesptree5 a0) a1 (h4/sptree/_20_40ind__typesptree5 a2)))
% Assm: h4/sptree/LN0: h4/sptree/LN = h4/sptree/_20_40ind__typesptree0
% Assm: h4/sptree/LS0: h4/sptree/LS = h4/sptree/_20_40ind__typesptree1
% Assm: h4/sptree/BN0: h4/sptree/BN = h4/sptree/_20_40ind__typesptree2
% Assm: h4/sptree/BS0: h4/sptree/BS = h4/sptree/_20_40ind__typesptree3
% Assm: h4/sptree/toListA__def_c0: !acc. h4/sptree/toListA acc h4/sptree/LN = acc
% Assm: h4/sptree/toListA__def_c1: !acc a. h4/sptree/toListA acc (h4/sptree/LS a) = h4/list/CONS a acc
% Assm: h4/sptree/toListA__def_c2: !t2 t1 acc. h4/sptree/toListA acc (h4/sptree/BN t1 t2) = h4/sptree/toListA (h4/sptree/toListA acc t2) t1
% Assm: h4/sptree/toListA__def_c3: !t2 t1 acc a. h4/sptree/toListA acc (h4/sptree/BS t1 a t2) = h4/sptree/toListA (h4/list/CONS a (h4/sptree/toListA acc t2)) t1
% Goal: !t acc. h4/sptree/toListA acc t = h4/list/APPEND (h4/sptree/toListA h4/list/NIL t) acc
%   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_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_EQu_u_REFL]: !x. x = x
% 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_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% 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_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% 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_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_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~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_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_APPENDu_u_ASSOC]: !l3 l2 l1. h4/list/APPEND l1 (h4/list/APPEND l2 l3) = h4/list/APPEND (h4/list/APPEND l1 l2) l3
% Assm [h4s_sptrees_sptu_u_repfnsu_c0]: !a. h4/sptree/_20_40ind__typesptree4 (h4/sptree/_20_40ind__typesptree5 a) = a
% Assm [h4s_sptrees_sptu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27spt_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0 \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a _0) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _0)) /\ happ _27spt_27 a0 /\ happ _27spt_27 a1) \/ (?a0 a1 a2. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) a1 (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a2 _0)) /\ happ _27spt_27 a0 /\ happ _27spt_27 a2) ==> happ _27spt_27 a0_270) ==> happ _27spt_27 r) <=> h4/sptree/_20_40ind__typesptree5 (h4/sptree/_20_40ind__typesptree4 r) = r)
% Assm [h4s_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/sptree/_20_40ind__typesptree0 = h4/sptree/_20_40ind__typesptree4 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0)
% Assm [h4s_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/sptree/_20_40ind__typesptree1 x = h4/sptree/_20_40ind__typesptree4 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) x _0))
% Assm [h4s_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree2u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x'. happ (happ h4/sptree/_20_40ind__typesptree2 x) x' = h4/sptree/_20_40ind__typesptree4 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS (h4/sptree/_20_40ind__typesptree5 x) (h4/ind__type/FCONS (h4/sptree/_20_40ind__typesptree5 x') _0))))
% Assm [h4s_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree3u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x x'. happ (happ (happ h4/sptree/_20_40ind__typesptree3 x) x) x' = h4/sptree/_20_40ind__typesptree4 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) x (h4/ind__type/FCONS (h4/sptree/_20_40ind__typesptree5 x) (h4/ind__type/FCONS (h4/sptree/_20_40ind__typesptree5 x') _0))))
% Assm [h4s_sptrees_LN0]: h4/sptree/LN = h4/sptree/_20_40ind__typesptree0
% Assm [h4s_sptrees_LS0]: h4/sptree/LS = h4/sptree/_20_40ind__typesptree1
% Assm [h4s_sptrees_BN0]: h4/sptree/BN = h4/sptree/_20_40ind__typesptree2
% Assm [h4s_sptrees_BS0]: h4/sptree/BS = h4/sptree/_20_40ind__typesptree3
% Assm [h4s_sptrees_toListAu_u_defu_c0]: !acc. h4/sptree/toListA acc h4/sptree/LN = acc
% Assm [h4s_sptrees_toListAu_u_defu_c1]: !acc a. h4/sptree/toListA acc (happ h4/sptree/LS a) = h4/list/CONS a acc
% Assm [h4s_sptrees_toListAu_u_defu_c2]: !t2 t1 acc. h4/sptree/toListA acc (happ (happ h4/sptree/BN t1) t2) = h4/sptree/toListA (h4/sptree/toListA acc t2) t1
% Assm [h4s_sptrees_toListAu_u_defu_c3]: !t2 t1 acc a. h4/sptree/toListA acc (happ (happ (happ h4/sptree/BS t1) a) t2) = h4/sptree/toListA (h4/list/CONS a (h4/sptree/toListA acc t2)) t1
% Goal: !t acc. h4/sptree/toListA acc t = h4/list/APPEND (h4/sptree/toListA h4/list/NIL t) acc
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q82398,TV_Q82394]: ![V_f, V_g]: (![V_x]: s(TV_Q82394,happ(s(t_fun(TV_Q82398,TV_Q82394),V_f),s(TV_Q82398,V_x))) = s(TV_Q82394,happ(s(t_fun(TV_Q82398,TV_Q82394),V_g),s(TV_Q82398,V_x))) => s(t_fun(TV_Q82398,TV_Q82394),V_f) = s(t_fun(TV_Q82398,TV_Q82394),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,t0))).
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_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t0)))).
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_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_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,t0) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_ANDu_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,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_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_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_OR, 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_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_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_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_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_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_APPENDu_u_ASSOC, axiom, ![TV_u_27a]: ![V_l3, V_l2, V_l1]: 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),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l2),s(t_h4s_lists_list(TV_u_27a),V_l3))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(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))),s(t_h4s_lists_list(TV_u_27a),V_l3)))).
fof(ah4s_sptrees_sptu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree4(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree5(s(t_h4s_sptrees_spt(TV_u_27a),V_a))))) = s(t_h4s_sptrees_spt(TV_u_27a),V_a)).
fof(ah4s_sptrees_sptu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27sptu_27]: (![V_a0u_270]: ((s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))) | (?[V_a]: s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))) | (?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27sptu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27sptu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1)))))) | ?[V_a0, V_a1, V_a2]: (s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(TV_u_27a,V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a2),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27sptu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27sptu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a2))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27sptu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27sptu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) <=> s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree5(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree4(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r)))).
fof(ah4s_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree0) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree4(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))).
fof(ah4s_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_u_20u_40indu_u_typesptree1),s(TV_u_27a,V_x))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree4(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_x),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))).
fof(ah4s_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree2u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_x, V_xi_]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_u_20u_40indu_u_typesptree2),s(t_h4s_sptrees_spt(TV_u_27a),V_x))),s(t_h4s_sptrees_spt(TV_u_27a),V_xi_))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree4(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree5(s(t_h4s_sptrees_spt(TV_u_27a),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree5(s(t_h4s_sptrees_spt(TV_u_27a),V_xi_))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))))).
fof(ah4s_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree3u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_x, V_x0, V_xi_]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_u_20u_40indu_u_typesptree3),s(t_h4s_sptrees_spt(TV_u_27a),V_x))),s(TV_u_27a,V_x0))),s(t_h4s_sptrees_spt(TV_u_27a),V_xi_))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree4(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(TV_u_27a,V_x0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree5(s(t_h4s_sptrees_spt(TV_u_27a),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree5(s(t_h4s_sptrees_spt(TV_u_27a),V_xi_))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))))).
fof(ah4s_sptrees_LN0, axiom, ![TV_u_27a]: s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree0)).
fof(ah4s_sptrees_LS0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_ls) = s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_u_20u_40indu_u_typesptree1)).
fof(ah4s_sptrees_BN0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_bn) = s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_u_20u_40indu_u_typesptree2)).
fof(ah4s_sptrees_BS0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_bs) = s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_u_20u_40indu_u_typesptree3)).
fof(ah4s_sptrees_toListAu_u_defu_c0, axiom, ![TV_u_27a]: ![V_acc]: s(t_h4s_lists_list(TV_u_27a),h4s_sptrees_tolista(s(t_h4s_lists_list(TV_u_27a),V_acc),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))) = s(t_h4s_lists_list(TV_u_27a),V_acc)).
fof(ah4s_sptrees_toListAu_u_defu_c1, axiom, ![TV_u_27a]: ![V_acc, V_a]: s(t_h4s_lists_list(TV_u_27a),h4s_sptrees_tolista(s(t_h4s_lists_list(TV_u_27a),V_acc),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_ls),s(TV_u_27a,V_a))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a),s(t_h4s_lists_list(TV_u_27a),V_acc)))).
fof(ah4s_sptrees_toListAu_u_defu_c2, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_acc]: s(t_h4s_lists_list(TV_u_27a),h4s_sptrees_tolista(s(t_h4s_lists_list(TV_u_27a),V_acc),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_bn),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))))) = s(t_h4s_lists_list(TV_u_27a),h4s_sptrees_tolista(s(t_h4s_lists_list(TV_u_27a),h4s_sptrees_tolista(s(t_h4s_lists_list(TV_u_27a),V_acc),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1)))).
fof(ah4s_sptrees_toListAu_u_defu_c3, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_acc, V_a]: s(t_h4s_lists_list(TV_u_27a),h4s_sptrees_tolista(s(t_h4s_lists_list(TV_u_27a),V_acc),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_bs),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))))) = s(t_h4s_lists_list(TV_u_27a),h4s_sptrees_tolista(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a),s(t_h4s_lists_list(TV_u_27a),h4s_sptrees_tolista(s(t_h4s_lists_list(TV_u_27a),V_acc),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1)))).
fof(ch4s_sptrees_toListAu_u_append, conjecture, ![TV_u_27a]: ![V_t, V_acc]: s(t_h4s_lists_list(TV_u_27a),h4s_sptrees_tolista(s(t_h4s_lists_list(TV_u_27a),V_acc),s(t_h4s_sptrees_spt(TV_u_27a),V_t))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_sptrees_tolista(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_sptrees_spt(TV_u_27a),V_t))),s(t_h4s_lists_list(TV_u_27a),V_acc)))).
