%   ORIGINAL: h4/list/LIST__REL__cases
% 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__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/list/LIST__REL__def0: h4/list/LIST__REL = (\R a0 a1. !LIST__REL_27. (!a00 a10. a00 = h4/list/NIL /\ a10 = h4/list/NIL \/ (?h1 h2 t1 t2. a00 = h4/list/CONS h1 t1 /\ a10 = h4/list/CONS h2 t2 /\ R h1 h2 /\ LIST__REL_27 t1 t2) ==> LIST__REL_27 a00 a10) ==> LIST__REL_27 a0 a1)
% Goal: !a1 a0 R. h4/list/LIST__REL R a0 a1 <=> a0 = h4/list/NIL /\ a1 = h4/list/NIL \/ (?h1 h2 t1 t2. a0 = h4/list/CONS h1 t1 /\ a1 = h4/list/CONS h2 t2 /\ R h1 h2 /\ h4/list/LIST__REL R t1 t2)
%   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_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_lists_LISTu_u_RELu_u_def0]: !x x x. h4/list/LIST__REL x x x <=> (!LIST__REL_27. (!a00 a10. a00 = h4/list/NIL /\ a10 = h4/list/NIL \/ (?h1 h2 t1 t2. a00 = h4/list/CONS h1 t1 /\ a10 = h4/list/CONS h2 t2 /\ happ (happ x h1) h2 /\ happ (happ LIST__REL_27 t1) t2) ==> happ (happ LIST__REL_27 a00) a10) ==> happ (happ LIST__REL_27 x) x)
% Goal: !a1 a0 R. h4/list/LIST__REL R a0 a1 <=> a0 = h4/list/NIL /\ a1 = h4/list/NIL \/ (?h1 h2 t1 t2. a0 = h4/list/CONS h1 t1 /\ a1 = h4/list/CONS h2 t2 /\ happ (happ R h1) h2 /\ h4/list/LIST__REL R t1 t2)
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_Q242129,TV_Q242125]: ![V_f, V_g]: (![V_x]: s(TV_Q242125,happ(s(t_fun(TV_Q242129,TV_Q242125),V_f),s(TV_Q242129,V_x))) = s(TV_Q242125,happ(s(t_fun(TV_Q242129,TV_Q242125),V_g),s(TV_Q242129,V_x))) => s(t_fun(TV_Q242129,TV_Q242125),V_f) = s(t_fun(TV_Q242129,TV_Q242125),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_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_lists_LISTu_u_RELu_u_def0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_x0, V_x1]: (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_x),s(t_h4s_lists_list(TV_u_27a),V_x0),s(t_h4s_lists_list(TV_u_27b),V_x1)))) <=> ![V_LISTu_u_RELu_27]: (![V_a00, V_a10]: (((s(t_h4s_lists_list(TV_u_27a),V_a00) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) & s(t_h4s_lists_list(TV_u_27b),V_a10) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)) | ?[V_h1, V_h2, V_t1, V_t2]: (s(t_h4s_lists_list(TV_u_27a),V_a00) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h1),s(t_h4s_lists_list(TV_u_27a),V_t1))) & (s(t_h4s_lists_list(TV_u_27b),V_a10) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_h2),s(t_h4s_lists_list(TV_u_27b),V_t2))) & (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_x),s(TV_u_27a,V_h1))),s(TV_u_27b,V_h2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool)),V_LISTu_u_RELu_27),s(t_h4s_lists_list(TV_u_27a),V_t1))),s(t_h4s_lists_list(TV_u_27b),V_t2)))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool)),V_LISTu_u_RELu_27),s(t_h4s_lists_list(TV_u_27a),V_a00))),s(t_h4s_lists_list(TV_u_27b),V_a10))))) => p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool)),V_LISTu_u_RELu_27),s(t_h4s_lists_list(TV_u_27a),V_x0))),s(t_h4s_lists_list(TV_u_27b),V_x1))))))).
fof(ch4s_lists_LISTu_u_RELu_u_cases, conjecture, ![TV_u_27a,TV_u_27b]: ![V_a1, V_a0, V_R]: (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_a0),s(t_h4s_lists_list(TV_u_27b),V_a1)))) <=> ((s(t_h4s_lists_list(TV_u_27a),V_a0) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) & s(t_h4s_lists_list(TV_u_27b),V_a1) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)) | ?[V_h1, V_h2, V_t1, V_t2]: (s(t_h4s_lists_list(TV_u_27a),V_a0) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h1),s(t_h4s_lists_list(TV_u_27a),V_t1))) & (s(t_h4s_lists_list(TV_u_27b),V_a1) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_h2),s(t_h4s_lists_list(TV_u_27b),V_t2))) & (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_h1))),s(TV_u_27b,V_h2)))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_t1),s(t_h4s_lists_list(TV_u_27b),V_t2)))))))))).
