%   ORIGINAL: h4/sorting/PERM__lifts__monotonicities
% 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/TRUTH: T
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/list/APPEND__ASSOC: !l3 l2 l1. h4/list/APPEND l1 (h4/list/APPEND l2 l3) = h4/list/APPEND (h4/list/APPEND l1 l2) l3
% Assm: h4/sorting/PERM__transitive: h4/relation/transitive h4/sorting/PERM
% Assm: h4/sorting/PERM__APPEND: !l2 l1. h4/sorting/PERM (h4/list/APPEND l1 l2) (h4/list/APPEND l2 l1)
% Assm: h4/sorting/PERM__APPEND__IFF_c0: !l2 l1 l. h4/sorting/PERM (h4/list/APPEND l l1) (h4/list/APPEND l l2) <=> h4/sorting/PERM l1 l2
% Assm: h4/sorting/PERM__lifts__transitive__relations: !f Q. (!x1 x2 x3. Q (f (h4/list/APPEND (h4/list/APPEND x1 x2) x3)) (f (h4/list/APPEND (h4/list/APPEND x1 x3) x2))) /\ h4/relation/transitive Q ==> (!x y. h4/sorting/PERM x y ==> Q (f x) (f y))
% Goal: !f. (!x1 x2 x3. ?x1_27 x2_27 x3_27. f (h4/list/APPEND (h4/list/APPEND x1 x2) x3) = h4/list/APPEND (h4/list/APPEND x1_27 x2_27) x3_27 /\ f (h4/list/APPEND (h4/list/APPEND x1 x3) x2) = h4/list/APPEND (h4/list/APPEND x1_27 x3_27) x2_27) ==> (!x y. h4/sorting/PERM x y ==> h4/sorting/PERM (f x) (f y))
%   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_TRUTH]: T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% 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_lists_APPENDu_u_ASSOC]: !l3 l2 l1. h4/list/APPEND l1 (h4/list/APPEND l2 l3) = h4/list/APPEND (h4/list/APPEND l1 l2) l3
% Assm [h4s_sortings_PERMu_u_transitive]: h4/relation/transitive h4/sorting/PERM
% Assm [h4s_sortings_PERMu_u_APPEND]: !l2 l1. happ (happ h4/sorting/PERM (h4/list/APPEND l1 l2)) (h4/list/APPEND l2 l1)
% Assm [h4s_sortings_PERMu_u_APPENDu_u_IFFu_c0]: !l2 l1 l. happ (happ h4/sorting/PERM (h4/list/APPEND l l1)) (h4/list/APPEND l l2) <=> happ (happ h4/sorting/PERM l1) l2
% Assm [h4s_sortings_PERMu_u_liftsu_u_transitiveu_u_relations]: !f Q. (!x1 x2 x3. happ (happ Q (happ f (h4/list/APPEND (h4/list/APPEND x1 x2) x3))) (happ f (h4/list/APPEND (h4/list/APPEND x1 x3) x2))) /\ h4/relation/transitive Q ==> (!x y. happ (happ h4/sorting/PERM x) y ==> happ (happ Q (happ f x)) (happ f y))
% Goal: !f. (!x1 x2 x3. ?x1_27 x2_27 x3_27. happ f (h4/list/APPEND (h4/list/APPEND x1 x2) x3) = h4/list/APPEND (h4/list/APPEND x1_27 x2_27) x3_27 /\ happ f (h4/list/APPEND (h4/list/APPEND x1 x3) x2) = h4/list/APPEND (h4/list/APPEND x1_27 x3_27) x2_27) ==> (!x y. happ (happ h4/sorting/PERM x) y ==> happ (happ h4/sorting/PERM (happ f x)) (happ f y))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q276031,TV_Q276027]: ![V_f, V_g]: (![V_x]: s(TV_Q276027,happ(s(t_fun(TV_Q276031,TV_Q276027),V_f),s(TV_Q276031,V_x))) = s(TV_Q276027,happ(s(t_fun(TV_Q276031,TV_Q276027),V_g),s(TV_Q276031,V_x))) => s(t_fun(TV_Q276031,TV_Q276027),V_f) = s(t_fun(TV_Q276031,TV_Q276027),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_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_lists_APPENDu_u_ASSOC, axiom, ![TV_u_27a]: ![V_l3, V_l2, V_l1]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l2),s(t_h4s_lists_list(TV_u_27a),V_l3))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2))),s(t_h4s_lists_list(TV_u_27a),V_l3)))).
fof(ah4s_sortings_PERMu_u_transitive, axiom, ![TV_u_27a]: p(s(t_bool,h4s_relations_transitive(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_bool)),h4s_sortings_perm))))).
fof(ah4s_sortings_PERMu_u_APPEND, axiom, ![TV_u_27a]: ![V_l2, V_l1]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_bool)),h4s_sortings_perm),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2))))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l2),s(t_h4s_lists_list(TV_u_27a),V_l1))))))).
fof(ah4s_sortings_PERMu_u_APPENDu_u_IFFu_c0, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_l]: s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_bool)),h4s_sortings_perm),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l),s(t_h4s_lists_list(TV_u_27a),V_l1))))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l),s(t_h4s_lists_list(TV_u_27a),V_l2))))) = s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_bool)),h4s_sortings_perm),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27a),V_l2)))).
fof(ah4s_sortings_PERMu_u_liftsu_u_transitiveu_u_relations, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_Q]: ((![V_x1, V_x2, V_x3]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_Q),s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_x1),s(t_h4s_lists_list(TV_u_27a),V_x2))),s(t_h4s_lists_list(TV_u_27a),V_x3))))))),s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_x1),s(t_h4s_lists_list(TV_u_27a),V_x3))),s(t_h4s_lists_list(TV_u_27a),V_x2)))))))) & p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_Q))))) => ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_bool)),h4s_sortings_perm),s(t_h4s_lists_list(TV_u_27a),V_x))),s(t_h4s_lists_list(TV_u_27a),V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_Q),s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_x))))),s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_y))))))))).
fof(ch4s_sortings_PERMu_u_liftsu_u_monotonicities, conjecture, ![TV_u_27b,TV_u_27a]: ![V_f]: (![V_x1, V_x2, V_x3]: ?[V_x1u_27, V_x2u_27, V_x3u_27]: (s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_x1),s(t_h4s_lists_list(TV_u_27a),V_x2))),s(t_h4s_lists_list(TV_u_27a),V_x3))))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_append(s(t_h4s_lists_list(TV_u_27b),h4s_lists_append(s(t_h4s_lists_list(TV_u_27b),V_x1u_27),s(t_h4s_lists_list(TV_u_27b),V_x2u_27))),s(t_h4s_lists_list(TV_u_27b),V_x3u_27))) & s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_x1),s(t_h4s_lists_list(TV_u_27a),V_x3))),s(t_h4s_lists_list(TV_u_27a),V_x2))))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_append(s(t_h4s_lists_list(TV_u_27b),h4s_lists_append(s(t_h4s_lists_list(TV_u_27b),V_x1u_27),s(t_h4s_lists_list(TV_u_27b),V_x3u_27))),s(t_h4s_lists_list(TV_u_27b),V_x2u_27)))) => ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_bool)),h4s_sortings_perm),s(t_h4s_lists_list(TV_u_27a),V_x))),s(t_h4s_lists_list(TV_u_27a),V_y)))) => 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_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_bool)),h4s_sortings_perm),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),V_f),s(t_h4s_lists_list(TV_u_27a),V_x))))),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),V_f),s(t_h4s_lists_list(TV_u_27a),V_y))))))))).
