%   ORIGINAL: h4/inftree/inftree__ind
% 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/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/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__ALL: !Q P. (!x. P x ==> Q x) ==> (!x. P x) ==> (!x. Q x)
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/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/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/inftree/is__tree__def: h4/inftree/is__tree = (\a0. !is__tree_27. (!a00. (?a. a00 = (\p. h4/sum/INL a)) \/ (?f b. a00 = (\p. h4/bool/COND (p = h4/list/NIL) (h4/sum/INR b) (f (h4/list/HD p) (h4/list/TL p))) /\ (!d. is__tree_27 (f d))) ==> is__tree_27 a00) ==> is__tree_27 a0)
% Assm: h4/inftree/inftree__bijections_c0: !a. h4/inftree/to__inftree (h4/inftree/from__inftree a) = a
% Assm: h4/inftree/inftree__bijections_c1: !r. h4/inftree/is__tree r <=> h4/inftree/from__inftree (h4/inftree/to__inftree r) = r
% Assm: h4/inftree/iLf__def: !a. h4/inftree/iLf a = h4/inftree/to__inftree (\p. h4/sum/INL a)
% Assm: h4/inftree/iNd__def: !f b. h4/inftree/iNd b f = h4/inftree/to__inftree (\p. h4/bool/COND (p = h4/list/NIL) (h4/sum/INR b) (h4/inftree/from__inftree (f (h4/list/HD p)) (h4/list/TL p)))
% Goal: !P. (!a. P (h4/inftree/iLf a)) /\ (!b f. (!d. P (f d)) ==> P (h4/inftree/iNd b f)) ==> (!t. P t)
%   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_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_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_ALL]: !Q P. (!x. happ P x ==> happ Q x) ==> (!x. happ P x) ==> (!x. happ Q x)
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_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_combins_ou_u_THM]: !x g f. h4/combin/o f g x = happ f (happ g x)
% Assm [h4s_inftrees_isu_u_treeu_u_def]: !x. h4/inftree/is__tree x <=> (!is__tree_27. (!a00. (?a. !x. happ a00 x = h4/sum/INL a) \/ (?f b. (!x. ?v. (v <=> x = h4/list/NIL) /\ happ a00 x = h4/bool/COND v (h4/sum/INR b) (happ (happ f (h4/list/HD x)) (h4/list/TL x))) /\ (!d. happ is__tree_27 (happ f d))) ==> happ is__tree_27 a00) ==> happ is__tree_27 x)
% Assm [h4s_inftrees_inftreeu_u_bijectionsu_c0]: !a. h4/inftree/to__inftree (h4/inftree/from__inftree a) = a
% Assm [h4s_inftrees_inftreeu_u_bijectionsu_c1]: !r. h4/inftree/is__tree r <=> h4/inftree/from__inftree (h4/inftree/to__inftree r) = r
% Assm [h4s_inftrees_iLfu_u_def]: !_0. (!a p. happ (happ _0 a) p = h4/sum/INL a) ==> (!a. h4/inftree/iLf a = h4/inftree/to__inftree (happ _0 a))
% Assm [h4s_inftrees_iNdu_u_def]: !_0. (!b f p. ?v. (v <=> p = h4/list/NIL) /\ happ (happ (happ _0 b) f) p = h4/bool/COND v (h4/sum/INR b) (happ (h4/inftree/from__inftree (happ f (h4/list/HD p))) (h4/list/TL p))) ==> (!f b. h4/inftree/iNd b f = h4/inftree/to__inftree (happ (happ _0 b) f))
% Goal: !P. (!a. happ P (h4/inftree/iLf a)) /\ (!b f. (!d. happ P (happ f d)) ==> happ P (h4/inftree/iNd b f)) ==> (!t. happ P t)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q299321,TV_Q299317]: ![V_f, V_g]: (![V_x]: s(TV_Q299317,happ(s(t_fun(TV_Q299321,TV_Q299317),V_f),s(TV_Q299321,V_x))) = s(TV_Q299317,happ(s(t_fun(TV_Q299321,TV_Q299317),V_g),s(TV_Q299321,V_x))) => s(t_fun(TV_Q299321,TV_Q299317),V_f) = s(t_fun(TV_Q299321,TV_Q299317),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_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,t0)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) => 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,t0))) <=> p(s(t_bool,t0)))).
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,f0))) <=> ~ (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_LEFTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_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_ALL, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_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,f0))))).
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,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
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,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
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_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_inftrees_isu_u_treeu_u_def, axiom, ![TV_u_27d,TV_u_27a,TV_u_27b]: ![V_x]: (p(s(t_bool,h4s_inftrees_isu_u_tree(s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_x)))) <=> ![V_isu_u_treeu_27]: (![V_a00]: ((?[V_a]: ![V_x0]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_a00),s(t_h4s_lists_list(TV_u_27d),V_x0))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_a))) | ?[V_f, V_b]: (![V_x0]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_lists_list(TV_u_27d),V_x0) = s(t_h4s_lists_list(TV_u_27d),h4s_lists_nil)) & s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_a00),s(t_h4s_lists_list(TV_u_27d),V_x0))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_b))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27d,t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27d,h4s_lists_hd(s(t_h4s_lists_list(TV_u_27d),V_x0))))),s(t_h4s_lists_list(TV_u_27d),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27d),V_x0)))))))) & ![V_d]: p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool),V_isu_u_treeu_27),s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27d,t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27d,V_d)))))))) => p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool),V_isu_u_treeu_27),s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool),V_isu_u_treeu_27),s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_x))))))).
fof(ah4s_inftrees_inftreeu_u_bijectionsu_c0, axiom, ![TV_u_27a,TV_u_27b,TV_u_27d]: ![V_a]: s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27d),h4s_inftrees_tou_u_inftree(s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_inftrees_fromu_u_inftree(s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27d),V_a))))) = s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27d),V_a)).
fof(ah4s_inftrees_inftreeu_u_bijectionsu_c1, axiom, ![TV_u_27d,TV_u_27a,TV_u_27b]: ![V_r]: (p(s(t_bool,h4s_inftrees_isu_u_tree(s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_r)))) <=> s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_inftrees_fromu_u_inftree(s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27d),h4s_inftrees_tou_u_inftree(s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_r))))) = s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_r))).
fof(ah4s_inftrees_iLfu_u_def, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_a, V_p]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),V_uu_0),s(TV_u_27a,V_a))),s(t_h4s_lists_list(TV_u_27c),V_p))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_a))) => ![V_a]: s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_ilf(s(TV_u_27a,V_a))) = s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_tou_u_inftree(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),V_uu_0),s(TV_u_27a,V_a))))))).
fof(ah4s_inftrees_iNdu_u_def, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_b, V_f, V_p]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_lists_list(TV_u_27c),V_p) = s(t_h4s_lists_list(TV_u_27c),h4s_lists_nil)) & s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),V_uu_0),s(TV_u_27b,V_b))),s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_lists_list(TV_u_27c),V_p))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_b))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_inftrees_fromu_u_inftree(s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f),s(TV_u_27c,h4s_lists_hd(s(t_h4s_lists_list(TV_u_27c),V_p))))))),s(t_h4s_lists_list(TV_u_27c),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27c),V_p)))))))) => ![V_f, V_b]: s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_ind(s(TV_u_27b,V_b),s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f))) = s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_tou_u_inftree(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),V_uu_0),s(TV_u_27b,V_b))),s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f))))))).
fof(ch4s_inftrees_inftreeu_u_ind, conjecture, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_P]: ((![V_a]: p(s(t_bool,happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),t_bool),V_P),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_ilf(s(TV_u_27a,V_a)))))) & ![V_b, V_f]: (![V_d]: p(s(t_bool,happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),t_bool),V_P),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f),s(TV_u_27c,V_d)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),t_bool),V_P),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_ind(s(TV_u_27b,V_b),s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f)))))))) => ![V_t]: p(s(t_bool,happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),t_bool),V_P),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),V_t)))))).
