%   ORIGINAL: h4/sptree/wf__union
% 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/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/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/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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/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/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% 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/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/spt__case__def_c0: !v f2 f1 f. h4/sptree/spt__CASE h4/sptree/LN v f f1 f2 = v
% Assm: h4/sptree/spt__case__def_c1: !v f2 f1 f a. h4/sptree/spt__CASE (h4/sptree/LS a) v f f1 f2 = f a
% Assm: h4/sptree/spt__case__def_c2: !v f2 f1 f a1 a0. h4/sptree/spt__CASE (h4/sptree/BN a0 a1) v f f1 f2 = f1 a0 a1
% Assm: h4/sptree/spt__case__def_c3: !v f2 f1 f a2 a1 a0. h4/sptree/spt__CASE (h4/sptree/BS a0 a1 a2) v f f1 f2 = f2 a0 a1 a2
% Assm: h4/sptree/wf__def_c0: h4/sptree/wf h4/sptree/LN <=> T
% Assm: h4/sptree/wf__def_c1: !a. h4/sptree/wf (h4/sptree/LS a) <=> T
% Assm: h4/sptree/wf__def_c2: !t2 t1. h4/sptree/wf (h4/sptree/BN t1 t2) <=> h4/sptree/wf t1 /\ h4/sptree/wf t2 /\ ~(t1 = h4/sptree/LN /\ t2 = h4/sptree/LN)
% Assm: h4/sptree/wf__def_c3: !t2 t1 a. h4/sptree/wf (h4/sptree/BS t1 a t2) <=> h4/sptree/wf t1 /\ h4/sptree/wf t2 /\ ~(t1 = h4/sptree/LN /\ t2 = h4/sptree/LN)
% Assm: h4/sptree/union__def_c0: !t. h4/sptree/union h4/sptree/LN t = t
% Assm: h4/sptree/union__def_c1: !t a. h4/sptree/union (h4/sptree/LS a) t = h4/sptree/spt__CASE t (h4/sptree/LS a) (\b. h4/sptree/LS a) (\t1 t2. h4/sptree/BS t1 a t2) (\t1_27 v4 t2_27. h4/sptree/BS t1_27 a t2_27)
% Assm: h4/sptree/union__def_c2: !t2 t1 t. h4/sptree/union (h4/sptree/BN t1 t2) t = h4/sptree/spt__CASE t (h4/sptree/BN t1 t2) (\a. h4/sptree/BS t1 a t2) (\t1_27 t2_27. h4/sptree/BN (h4/sptree/union t1 t1_27) (h4/sptree/union t2 t2_27)) (\t1_27_27 a_27 t2_27_27. h4/sptree/BS (h4/sptree/union t1 t1_27_27) a_27 (h4/sptree/union t2 t2_27_27))
% Assm: h4/sptree/union__def_c3: !t2 t1 t a. h4/sptree/union (h4/sptree/BS t1 a t2) t = h4/sptree/spt__CASE t (h4/sptree/BS t1 a t2) (\a_27. h4/sptree/BS t1 a t2) (\t1_27 t2_27. h4/sptree/BS (h4/sptree/union t1 t1_27) a (h4/sptree/union t2 t2_27)) (\t1_27_27 a_27_27 t2_27_27. h4/sptree/BS (h4/sptree/union t1 t1_27_27) a (h4/sptree/union t2 t2_27_27))
% Assm: h4/sptree/isEmpty__union: !m2 m1. h4/sptree/union m1 m2 = h4/sptree/LN <=> m1 = h4/sptree/LN /\ m2 = h4/sptree/LN
% Goal: !m2 m1. h4/sptree/wf m1 /\ h4/sptree/wf m2 ==> h4/sptree/wf (h4/sptree/union m1 m2)
%   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_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_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_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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_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_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% 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_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_sptu_u_caseu_u_defu_c0]: !v f2 f1 f. h4/sptree/spt__CASE h4/sptree/LN v f f1 f2 = v
% Assm [h4s_sptrees_sptu_u_caseu_u_defu_c1]: !v f2 f1 f a. h4/sptree/spt__CASE (happ h4/sptree/LS a) v f f1 f2 = happ f a
% Assm [h4s_sptrees_sptu_u_caseu_u_defu_c2]: !v f2 f1 f a1 a0. h4/sptree/spt__CASE (happ (happ h4/sptree/BN a0) a1) v f f1 f2 = happ (happ f1 a0) a1
% Assm [h4s_sptrees_sptu_u_caseu_u_defu_c3]: !v f2 f1 f a2 a1 a0. h4/sptree/spt__CASE (happ (happ (happ h4/sptree/BS a0) a1) a2) v f f1 f2 = happ (happ (happ f2 a0) a1) a2
% Assm [h4s_sptrees_wfu_u_defu_c0]: h4/sptree/wf h4/sptree/LN <=> T
% Assm [h4s_sptrees_wfu_u_defu_c1]: !a. h4/sptree/wf (happ h4/sptree/LS a) <=> T
% Assm [h4s_sptrees_wfu_u_defu_c2]: !t2 t1. h4/sptree/wf (happ (happ h4/sptree/BN t1) t2) <=> h4/sptree/wf t1 /\ h4/sptree/wf t2 /\ ~(t1 = h4/sptree/LN /\ t2 = h4/sptree/LN)
% Assm [h4s_sptrees_wfu_u_defu_c3]: !t2 t1 a. h4/sptree/wf (happ (happ (happ h4/sptree/BS t1) a) t2) <=> h4/sptree/wf t1 /\ h4/sptree/wf t2 /\ ~(t1 = h4/sptree/LN /\ t2 = h4/sptree/LN)
% Assm [h4s_sptrees_unionu_u_defu_c0]: !t. h4/sptree/union h4/sptree/LN t = t
% Assm [h4s_sptrees_unionu_u_defu_c1]: !_4. (!_2 t1_27 a v4. happ (happ (happ (happ _4 _2) t1_27) a) v4 = happ (happ _2 t1_27) a) ==> (!_3. (!_2 a t1_27. happ (happ (happ _3 _2) a) t1_27 = happ (happ (happ _4 _2) t1_27) a) ==> (!_2. (!t1 a t2. happ (happ (happ _2 t1) a) t2 = happ (happ (happ h4/sptree/BS t1) a) t2) ==> (!_1. (!a t1. happ (happ _1 a) t1 = happ (happ _2 t1) a) ==> (!_0. (!a b. happ (happ _0 a) b = happ h4/sptree/LS a) ==> (!t a. h4/sptree/union (happ h4/sptree/LS a) t = h4/sptree/spt__CASE t (happ h4/sptree/LS a) (happ _0 a) (happ _1 a) (happ (happ _3 _2) a))))))
% Assm [h4s_sptrees_unionu_u_defu_c2]: !_5. (!t1 t1_27_27 a_27 t2 t2_27_27. happ (happ (happ (happ (happ _5 t1) t1_27_27) a_27) t2) t2_27_27 = happ (happ (happ h4/sptree/BS (h4/sptree/union t1 t1_27_27)) a_27) (h4/sptree/union t2 t2_27_27)) ==> (!_4. (!t1 t1_27_27 t2 a_27. happ (happ (happ (happ _4 t1) t1_27_27) t2) a_27 = happ (happ (happ (happ _5 t1) t1_27_27) a_27) t2) ==> (!_3. (!t1 t2 t1_27_27. happ (happ (happ _3 t1) t2) t1_27_27 = happ (happ (happ _4 t1) t1_27_27) t2) ==> (!_2. (!t1 t1_27 t2 t2_27. happ (happ (happ (happ _2 t1) t1_27) t2) t2_27 = happ (happ h4/sptree/BN (h4/sptree/union t1 t1_27)) (h4/sptree/union t2 t2_27)) ==> (!_1. (!t1 t2 t1_27. happ (happ (happ _1 t1) t2) t1_27 = happ (happ (happ _2 t1) t1_27) t2) ==> (!_0. (!t1 t2 a. happ (happ (happ _0 t1) t2) a = happ (happ (happ h4/sptree/BS t1) a) t2) ==> (!t2 t1 t. h4/sptree/union (happ (happ h4/sptree/BN t1) t2) t = h4/sptree/spt__CASE t (happ (happ h4/sptree/BN t1) t2) (happ (happ _0 t1) t2) (happ (happ _1 t1) t2) (happ (happ _3 t1) t2)))))))
% Assm [h4s_sptrees_unionu_u_defu_c3]: !_4. (!_2 t1 t1_27_27 a t2 a_27_27. happ (happ (happ (happ (happ (happ _4 _2) t1) t1_27_27) a) t2) a_27_27 = happ (happ (happ (happ _2 t1) t1_27_27) a) t2) ==> (!_3. (!_2 t1 a t2 t1_27_27. happ (happ (happ (happ (happ _3 _2) t1) a) t2) t1_27_27 = happ (happ (happ (happ (happ _4 _2) t1) t1_27_27) a) t2) ==> (!_2. (!t1 t1_27 a t2 t2_27. happ (happ (happ (happ (happ _2 t1) t1_27) a) t2) t2_27 = happ (happ (happ h4/sptree/BS (h4/sptree/union t1 t1_27)) a) (h4/sptree/union t2 t2_27)) ==> (!_1. (!t1 a t2 t1_27. happ (happ (happ (happ _1 t1) a) t2) t1_27 = happ (happ (happ (happ _2 t1) t1_27) a) t2) ==> (!_0. (!t1 a t2 a_27. happ (happ (happ (happ _0 t1) a) t2) a_27 = happ (happ (happ h4/sptree/BS t1) a) t2) ==> (!t2 t1 t a. h4/sptree/union (happ (happ (happ h4/sptree/BS t1) a) t2) t = h4/sptree/spt__CASE t (happ (happ (happ h4/sptree/BS t1) a) t2) (happ (happ (happ _0 t1) a) t2) (happ (happ (happ _1 t1) a) t2) (happ (happ (happ (happ _3 _2) t1) a) t2))))))
% Assm [h4s_sptrees_isEmptyu_u_union]: !m2 m1. h4/sptree/union m1 m2 = h4/sptree/LN <=> m1 = h4/sptree/LN /\ m2 = h4/sptree/LN
% Goal: !m2 m1. h4/sptree/wf m1 /\ h4/sptree/wf m2 ==> h4/sptree/wf (h4/sptree/union m1 m2)
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_Q81651,TV_Q81647]: ![V_f, V_g]: (![V_x]: s(TV_Q81647,happ(s(t_fun(TV_Q81651,TV_Q81647),V_f),s(TV_Q81651,V_x))) = s(TV_Q81647,happ(s(t_fun(TV_Q81651,TV_Q81647),V_g),s(TV_Q81651,V_x))) => s(t_fun(TV_Q81651,TV_Q81647),V_f) = s(t_fun(TV_Q81651,TV_Q81647),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_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_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,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
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_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_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_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => 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,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
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_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_sptu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f2, V_f1, V_f]: s(TV_u_27b,h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),V_f1),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b))),V_f2))) = s(TV_u_27b,V_v)).
fof(ah4s_sptrees_sptu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f2, V_f1, V_f, V_a]: s(TV_u_27b,h4s_sptrees_sptu_u_case(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(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),V_f1),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b))),V_f2))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_a)))).
fof(ah4s_sptrees_sptu_u_caseu_u_defu_c2, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f2, V_f1, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_sptrees_sptu_u_case(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_a0))),s(t_h4s_sptrees_spt(TV_u_27a),V_a1))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),V_f1),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b))),V_f2))) = s(TV_u_27b,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),V_f1),s(t_h4s_sptrees_spt(TV_u_27a),V_a0))),s(t_h4s_sptrees_spt(TV_u_27a),V_a1)))).
fof(ah4s_sptrees_sptu_u_caseu_u_defu_c3, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f2, V_f1, V_f, V_a2, V_a1, V_a0]: s(TV_u_27b,h4s_sptrees_sptu_u_case(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_a0))),s(TV_u_27a,V_a1))),s(t_h4s_sptrees_spt(TV_u_27a),V_a2))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),V_f1),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b))),V_f2))) = s(TV_u_27b,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),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),TV_u_27b))),V_f2),s(t_h4s_sptrees_spt(TV_u_27a),V_a0))),s(TV_u_27a,V_a1))),s(t_h4s_sptrees_spt(TV_u_27a),V_a2)))).
fof(ah4s_sptrees_wfu_u_defu_c0, axiom, ![TV_u_27a]: s(t_bool,h4s_sptrees_wf(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))) = s(t_bool,t)).
fof(ah4s_sptrees_wfu_u_defu_c1, axiom, ![TV_u_27a]: ![V_a]: s(t_bool,h4s_sptrees_wf(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_bool,t)).
fof(ah4s_sptrees_wfu_u_defu_c2, axiom, ![TV_u_27a]: ![V_t2, V_t1]: (p(s(t_bool,h4s_sptrees_wf(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)))))) <=> (p(s(t_bool,h4s_sptrees_wf(s(t_h4s_sptrees_spt(TV_u_27a),V_t1)))) & (p(s(t_bool,h4s_sptrees_wf(s(t_h4s_sptrees_spt(TV_u_27a),V_t2)))) & ~ ((s(t_h4s_sptrees_spt(TV_u_27a),V_t1) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln) & s(t_h4s_sptrees_spt(TV_u_27a),V_t2) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))))))).
fof(ah4s_sptrees_wfu_u_defu_c3, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_a]: (p(s(t_bool,h4s_sptrees_wf(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)))))) <=> (p(s(t_bool,h4s_sptrees_wf(s(t_h4s_sptrees_spt(TV_u_27a),V_t1)))) & (p(s(t_bool,h4s_sptrees_wf(s(t_h4s_sptrees_spt(TV_u_27a),V_t2)))) & ~ ((s(t_h4s_sptrees_spt(TV_u_27a),V_t1) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln) & s(t_h4s_sptrees_spt(TV_u_27a),V_t2) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))))))).
fof(ah4s_sptrees_unionu_u_defu_c0, axiom, ![TV_u_27a]: ![V_t]: s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_union(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln),s(t_h4s_sptrees_spt(TV_u_27a),V_t))) = s(t_h4s_sptrees_spt(TV_u_27a),V_t)).
fof(ah4s_sptrees_unionu_u_defu_c1, axiom, ![TV_u_27a]: ![V_uu_4]: (![V_uu_2, V_t1u_27, V_a, V_v4]: 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(TV_u_27a,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(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),happ(s(t_fun(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)))),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))))),V_uu_4),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)))),V_uu_2))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))),s(TV_u_27a,V_a))),s(TV_u_27a,V_v4))) = 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)))),V_uu_2),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))),s(TV_u_27a,V_a))) => ![V_uu_3]: (![V_uu_2, V_a, V_t1u_27]: 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)))),happ(s(t_fun(TV_u_27a,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))))),happ(s(t_fun(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)))),t_fun(TV_u_27a,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)))))),V_uu_3),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)))),V_uu_2))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))) = 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(TV_u_27a,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(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),happ(s(t_fun(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)))),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))))),V_uu_4),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)))),V_uu_2))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))),s(TV_u_27a,V_a))) => ![V_uu_2]: (![V_t1, V_a, V_t2]: 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)))),V_uu_2),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_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))) => ![V_uu_1]: (![V_a, V_t1]: 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))),happ(s(t_fun(TV_u_27a,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)))),V_uu_1),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))) = 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)))),V_uu_2),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(TV_u_27a,V_a))) => ![V_uu_0]: (![V_a, V_b]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),V_uu_0),s(TV_u_27a,V_a))),s(TV_u_27a,V_b))) = 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))) => ![V_t, V_a]: s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_union(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_sptrees_spt(TV_u_27a),V_t))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),V_t),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_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),V_uu_0),s(TV_u_27a,V_a))),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))),happ(s(t_fun(TV_u_27a,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)))),V_uu_1),s(TV_u_27a,V_a))),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)))),happ(s(t_fun(TV_u_27a,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))))),happ(s(t_fun(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)))),t_fun(TV_u_27a,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)))))),V_uu_3),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)))),V_uu_2))),s(TV_u_27a,V_a))))))))))).
fof(ah4s_sptrees_unionu_u_defu_c2, axiom, ![TV_u_27a]: ![V_uu_5]: (![V_t1, V_t1u_27u_27, V_au_27, V_t2, V_t2u_27u_27]: 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))),happ(s(t_fun(TV_u_27a,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)))),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_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_fun(TV_u_27a,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)))))),V_uu_5),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27u_27))),s(TV_u_27a,V_au_27))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2u_27u_27))) = 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),h4s_sptrees_union(s(t_h4s_sptrees_spt(TV_u_27a),V_t1),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27u_27))))),s(TV_u_27a,V_au_27))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_union(s(t_h4s_sptrees_spt(TV_u_27a),V_t2),s(t_h4s_sptrees_spt(TV_u_27a),V_t2u_27u_27))))) => ![V_uu_4]: (![V_t1, V_t1u_27u_27, V_t2, V_au_27]: 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)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),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))))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),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)))))),V_uu_4),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27u_27))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))),s(TV_u_27a,V_au_27))) = 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))),happ(s(t_fun(TV_u_27a,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)))),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_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_fun(TV_u_27a,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)))))),V_uu_5),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27u_27))),s(TV_u_27a,V_au_27))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))) => ![V_uu_3]: (![V_t1, V_t2, V_t1u_27u_27]: 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)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),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))))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),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)))))),V_uu_3),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27u_27))) = 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)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),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))))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),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)))))),V_uu_4),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27u_27))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))) => ![V_uu_2]: (![V_t1, V_t1u_27, V_t2, V_t2u_27]: 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))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),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)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),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))))),V_uu_2),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2u_27))) = 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),h4s_sptrees_union(s(t_h4s_sptrees_spt(TV_u_27a),V_t1),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_union(s(t_h4s_sptrees_spt(TV_u_27a),V_t2),s(t_h4s_sptrees_spt(TV_u_27a),V_t2u_27))))) => ![V_uu_1]: (![V_t1, V_t2, V_t1u_27]: 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))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),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)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),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))))),V_uu_1),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))) = 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))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),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)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),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))))),V_uu_2),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))) => ![V_uu_0]: (![V_t1, V_t2, V_a]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(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_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_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)))),V_uu_0),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))),s(TV_u_27a,V_a))) = 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))) => ![V_t2, V_t1, V_t]: s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_union(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_sptrees_spt(TV_u_27a),V_t))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),V_t),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_fun(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_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_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)))),V_uu_0),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))),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))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),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)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),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))))),V_uu_1),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))),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)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),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))))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),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)))))),V_uu_3),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2)))))))))))).
fof(ah4s_sptrees_unionu_u_defu_c3, axiom, ![TV_u_27a]: ![V_uu_4]: (![V_uu_2, V_t1, V_t1u_27u_27, V_a, V_t2, V_au_27u_27]: 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)))),happ(s(t_fun(TV_u_27a,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))))),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_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(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,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))))))),happ(s(t_fun(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,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)))))),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,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)))))))),V_uu_4),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,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)))))),V_uu_2))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27u_27))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))),s(TV_u_27a,V_au_27u_27))) = 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))),happ(s(t_fun(TV_u_27a,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)))),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_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_fun(TV_u_27a,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)))))),V_uu_2),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27u_27))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))) => ![V_uu_3]: (![V_uu_2, V_t1, V_a, V_t2, V_t1u_27u_27]: 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)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),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))))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),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)))))),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_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))))))),happ(s(t_fun(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,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)))))),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),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)))))))),V_uu_3),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,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)))))),V_uu_2))),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_sptrees_spt(TV_u_27a),V_t1u_27u_27))) = 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)))),happ(s(t_fun(TV_u_27a,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))))),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_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(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,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))))))),happ(s(t_fun(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,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)))))),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,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)))))))),V_uu_4),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,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)))))),V_uu_2))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27u_27))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))) => ![V_uu_2]: (![V_t1, V_t1u_27, V_a, V_t2, V_t2u_27]: 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))),happ(s(t_fun(TV_u_27a,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)))),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_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_fun(TV_u_27a,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)))))),V_uu_2),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2u_27))) = 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),h4s_sptrees_union(s(t_h4s_sptrees_spt(TV_u_27a),V_t1),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_union(s(t_h4s_sptrees_spt(TV_u_27a),V_t2),s(t_h4s_sptrees_spt(TV_u_27a),V_t2u_27))))) => ![V_uu_1]: (![V_t1, V_a, V_t2, V_t1u_27]: 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))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),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)))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),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))))),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_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))))),V_uu_1),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_sptrees_spt(TV_u_27a),V_t1u_27))) = 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))),happ(s(t_fun(TV_u_27a,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)))),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_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_fun(TV_u_27a,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)))))),V_uu_2),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))) => ![V_uu_0]: (![V_t1, V_a, V_t2, V_au_27]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(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_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(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_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))))),V_uu_0),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(TV_u_27a,V_au_27))) = 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))) => ![V_t2, V_t1, V_t, V_a]: s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_union(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_sptrees_spt(TV_u_27a),V_t))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),V_t),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_fun(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_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(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_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))))),V_uu_0),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_fun(t_h4s_sptrees_spt(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(t_h4s_sptrees_spt(TV_u_27a),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_fun(t_h4s_sptrees_spt(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_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))))),V_uu_1),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_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)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),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))))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),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)))))),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_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))))))),happ(s(t_fun(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,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)))))),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),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)))))))),V_uu_3),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,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)))))),V_uu_2))),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))))))))))).
fof(ah4s_sptrees_isEmptyu_u_union, axiom, ![TV_u_27a]: ![V_m2, V_m1]: (s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_union(s(t_h4s_sptrees_spt(TV_u_27a),V_m1),s(t_h4s_sptrees_spt(TV_u_27a),V_m2))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln) <=> (s(t_h4s_sptrees_spt(TV_u_27a),V_m1) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln) & s(t_h4s_sptrees_spt(TV_u_27a),V_m2) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln)))).
fof(ch4s_sptrees_wfu_u_union, conjecture, ![TV_u_27a]: ![V_m2, V_m1]: ((p(s(t_bool,h4s_sptrees_wf(s(t_h4s_sptrees_spt(TV_u_27a),V_m1)))) & p(s(t_bool,h4s_sptrees_wf(s(t_h4s_sptrees_spt(TV_u_27a),V_m2))))) => p(s(t_bool,h4s_sptrees_wf(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_union(s(t_h4s_sptrees_spt(TV_u_27a),V_m1),s(t_h4s_sptrees_spt(TV_u_27a),V_m2)))))))).
