%   ORIGINAL: h4/sorting/QSORT3__SORTED
% 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/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/sorting/SORTS__DEF: !f R. h4/sorting/SORTS f R <=> (!l. h4/sorting/PERM l (f R l) /\ h4/sorting/SORTED R (f R l))
% Assm: h4/sorting/QSORT3__SORTS: !R. h4/relation/transitive R /\ h4/relation/total R ==> h4/sorting/SORTS h4/sorting/QSORT3 R
% Goal: !R L. h4/relation/transitive R /\ h4/relation/total R ==> h4/sorting/SORTED R (h4/sorting/QSORT3 R L)
%   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_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_sortings_SORTSu_u_DEF]: !f R. h4/sorting/SORTS f R <=> (!l. h4/sorting/PERM l (happ (happ f R) l) /\ h4/sorting/SORTED R (happ (happ f R) l))
% Assm [h4s_sortings_QSORT3u_u_SORTS]: !R. h4/relation/transitive R /\ h4/relation/total R ==> h4/sorting/SORTS h4/sorting/QSORT3 R
% Goal: !R L. h4/relation/transitive R /\ h4/relation/total R ==> h4/sorting/SORTED R (happ (happ h4/sorting/QSORT3 R) L)
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_Q277412,TV_Q277408]: ![V_f, V_g]: (![V_x]: s(TV_Q277408,happ(s(t_fun(TV_Q277412,TV_Q277408),V_f),s(TV_Q277412,V_x))) = s(TV_Q277408,happ(s(t_fun(TV_Q277412,TV_Q277408),V_g),s(TV_Q277412,V_x))) => s(t_fun(TV_Q277412,TV_Q277408),V_f) = s(t_fun(TV_Q277412,TV_Q277408),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_sortings_SORTSu_u_DEF, axiom, ![TV_u_27a]: ![V_f, V_R]: (p(s(t_bool,h4s_sortings_sorts(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_l]: (p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_l),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_h4s_lists_list(TV_u_27a),V_l)))))) & p(s(t_bool,h4s_sortings_sorted(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_h4s_lists_list(TV_u_27a),V_l))))))))).
fof(ah4s_sortings_QSORT3u_u_SORTS, axiom, ![TV_u_27a]: ![V_R]: ((p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & p(s(t_bool,h4s_relations_total(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) => p(s(t_bool,h4s_sortings_sorts(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_sortings_qsort3),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ch4s_sortings_QSORT3u_u_SORTED, conjecture, ![TV_u_27a]: ![V_R, V_L]: ((p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & p(s(t_bool,h4s_relations_total(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) => p(s(t_bool,h4s_sortings_sorted(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_sortings_qsort3),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_h4s_lists_list(TV_u_27a),V_L)))))))).
