%   ORIGINAL: h4/inftree/relrec__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/IMP__CLAUSES_c3: !t. t ==> t <=> T
% 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/inftree/relrec__def: h4/inftree/relrec = (\a0 a1 a2 a3. !relrec_27. (!a00 a10 a20 a30. (?a. a20 = h4/inftree/iLf a /\ a30 = a00 a) \/ (?b df g. a20 = h4/inftree/iNd b df /\ a30 = a10 b g /\ (!d. relrec_27 a00 a10 (df d) (g d))) ==> relrec_27 a00 a10 a20 a30) ==> relrec_27 a0 a1 a2 a3)
% Goal: !relrec_27. (!lf nd a. relrec_27 lf nd (h4/inftree/iLf a) (lf a)) /\ (!lf nd b df g. (!d. relrec_27 lf nd (df d) (g d)) ==> relrec_27 lf nd (h4/inftree/iNd b df) (nd b g)) ==> (!a0 a1 a2 a3. h4/inftree/relrec a0 a1 a2 a3 ==> relrec_27 a0 a1 a2 a3)
%   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_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_inftrees_relrecu_u_def]: !x x x x. h4/inftree/relrec x x x x <=> (!relrec_27. (!a00 a10 a20 a30. (?a. a20 = h4/inftree/iLf a /\ a30 = happ a00 a) \/ (?b df g. a20 = h4/inftree/iNd b df /\ a30 = happ (happ a10 b) g /\ (!d. happ (happ (happ (happ relrec_27 a00) a10) (happ df d)) (happ g d))) ==> happ (happ (happ (happ relrec_27 a00) a10) a20) a30) ==> happ (happ (happ (happ relrec_27 x) x) x) x)
% Goal: !relrec_27. (!lf nd a. happ (happ (happ (happ relrec_27 lf) nd) (h4/inftree/iLf a)) (happ lf a)) /\ (!lf nd b df g. (!d. happ (happ (happ (happ relrec_27 lf) nd) (happ df d)) (happ g d)) ==> happ (happ (happ (happ relrec_27 lf) nd) (h4/inftree/iNd b df)) (happ (happ nd b) g)) ==> (!a0 a1 a2 a3. h4/inftree/relrec a0 a1 a2 a3 ==> happ (happ (happ (happ relrec_27 a0) a1) a2) a3)
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_Q299398,TV_Q299394]: ![V_f, V_g]: (![V_x]: s(TV_Q299394,happ(s(t_fun(TV_Q299398,TV_Q299394),V_f),s(TV_Q299398,V_x))) = s(TV_Q299394,happ(s(t_fun(TV_Q299398,TV_Q299394),V_g),s(TV_Q299398,V_x))) => s(t_fun(TV_Q299398,TV_Q299394),V_f) = s(t_fun(TV_Q299398,TV_Q299394),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_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_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_inftrees_relrecu_u_def, axiom, ![TV_u_27a,TV_u_27c,TV_u_27d,TV_u_27b]: ![V_x, V_x0, V_x1, V_x2]: (p(s(t_bool,h4s_inftrees_relrec(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_x0),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),V_x1),s(TV_u_27b,V_x2)))) <=> ![V_relrecu_27]: (![V_a00, V_a10, V_a20, V_a30]: ((?[V_a]: (s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),V_a20) = s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),h4s_inftrees_ilf(s(TV_u_27a,V_a))) & s(TV_u_27b,V_a30) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_a00),s(TV_u_27a,V_a)))) | ?[V_b, V_df, V_g]: (s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),V_a20) = s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),h4s_inftrees_ind(s(TV_u_27c,V_b),s(t_fun(TV_u_27d,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d)),V_df))) & (s(TV_u_27b,V_a30) = s(TV_u_27b,happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b),happ(s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_a10),s(TV_u_27c,V_b))),s(t_fun(TV_u_27d,TV_u_27b),V_g))) & ![V_d]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)))),V_relrecu_27),s(t_fun(TV_u_27a,TV_u_27b),V_a00))),s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_a10))),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),happ(s(t_fun(TV_u_27d,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d)),V_df),s(TV_u_27d,V_d))))),s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_g),s(TV_u_27d,V_d))))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)))),V_relrecu_27),s(t_fun(TV_u_27a,TV_u_27b),V_a00))),s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_a10))),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),V_a20))),s(TV_u_27b,V_a30))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)))),V_relrecu_27),s(t_fun(TV_u_27a,TV_u_27b),V_x))),s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_x0))),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),V_x1))),s(TV_u_27b,V_x2))))))).
fof(ch4s_inftrees_relrecu_u_ind, conjecture, ![TV_u_27a,TV_u_27c,TV_u_27d,TV_u_27b]: ![V_relrecu_27]: ((![V_lf, V_nd, V_a]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)))),V_relrecu_27),s(t_fun(TV_u_27a,TV_u_27b),V_lf))),s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_nd))),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),h4s_inftrees_ilf(s(TV_u_27a,V_a))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_lf),s(TV_u_27a,V_a)))))) & ![V_lf, V_nd, V_b, V_df, V_g]: (![V_d]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)))),V_relrecu_27),s(t_fun(TV_u_27a,TV_u_27b),V_lf))),s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_nd))),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),happ(s(t_fun(TV_u_27d,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d)),V_df),s(TV_u_27d,V_d))))),s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_g),s(TV_u_27d,V_d)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)))),V_relrecu_27),s(t_fun(TV_u_27a,TV_u_27b),V_lf))),s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_nd))),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),h4s_inftrees_ind(s(TV_u_27c,V_b),s(t_fun(TV_u_27d,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d)),V_df))))),s(TV_u_27b,happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b),happ(s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_nd),s(TV_u_27c,V_b))),s(t_fun(TV_u_27d,TV_u_27b),V_g)))))))) => ![V_a0, V_a1, V_a2, V_a3]: (p(s(t_bool,h4s_inftrees_relrec(s(t_fun(TV_u_27a,TV_u_27b),V_a0),s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_a1),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),V_a2),s(TV_u_27b,V_a3)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)))),V_relrecu_27),s(t_fun(TV_u_27a,TV_u_27b),V_a0))),s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_a1))),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),V_a2))),s(TV_u_27b,V_a3))))))).
