%   ORIGINAL: h4/patricia/IS__EMPTY__def_c1
% 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/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/relation/WF__EMPTY__REL: h4/relation/WF h4/relation/EMPTY__REL
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/patricia/ptree__case__def_c0: !v f1 f. h4/patricia/ptree__CASE h4/patricia/Empty v f f1 = v
% Assm: h4/patricia/ptree__case__def_c1: !v f1 f a1 a0. h4/patricia/ptree__CASE (h4/patricia/Leaf a0 a1) v f f1 = f a0 a1
% Assm: h4/patricia/ptree__case__def_c2: !v f1 f a3 a2 a1 a0. h4/patricia/ptree__CASE (h4/patricia/Branch a0 a1 a2 a3) v f f1 = f1 a0 a1 a2 a3
% Assm: h4/patricia/IS__EMPTY__primitive__def: h4/patricia/IS__EMPTY = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R)) (\IS__EMPTY a. h4/patricia/ptree__CASE a (h4/combin/I T) (\v6 v7. h4/combin/I F) (\v8 v9 v10 v11. h4/combin/I F))
% Goal: !v1 v. h4/patricia/IS__EMPTY (h4/patricia/Leaf v v1) <=> F
%   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_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_relations_WFu_u_EMPTYu_u_REL]: h4/relation/WF h4/relation/EMPTY__REL
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_patricias_ptreeu_u_caseu_u_defu_c0]: !v f1 f. h4/patricia/ptree__CASE h4/patricia/Empty v f f1 = v
% Assm [h4s_patricias_ptreeu_u_caseu_u_defu_c1]: !v f1 f a1 a0. h4/patricia/ptree__CASE (h4/patricia/Leaf a0 a1) v f f1 = happ (happ f a0) a1
% Assm [h4s_patricias_ptreeu_u_caseu_u_defu_c2]: !v f1 f a3 a2 a1 a0. h4/patricia/ptree__CASE (h4/patricia/Branch a0 a1 a2 a3) v f f1 = happ (happ (happ (happ f1 a0) a1) a2) a3
% Assm [h4s_patricias_ISu_u_EMPTYu_u_primitiveu_u_def]: !_8. (!v11. happ _8 v11 <=> h4/combin/I F) ==> (!_7. (!v10. happ _7 v10 = _8) ==> (!_6. (!v9. happ _6 v9 = _7) ==> (!_5. (!v8. happ _5 v8 = _6) ==> (!_4. (!v7. happ _4 v7 <=> h4/combin/I F) ==> (!_3. (!v6. happ _3 v6 = _4) ==> (!_2. (!a. happ _2 a <=> h4/patricia/ptree__CASE a (h4/combin/I T) _3 _5) ==> (!_1. (!IS__EMPTY. happ _1 IS__EMPTY = _2) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R) ==> h4/patricia/IS__EMPTY = h4/relation/WFREC (h4/min/_40 _0) _1))))))))
% Goal: !v1 v. happ h4/patricia/IS__EMPTY (h4/patricia/Leaf v v1) <=> F
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_Q285759,TV_Q285755]: ![V_f, V_g]: (![V_x]: s(TV_Q285755,happ(s(t_fun(TV_Q285759,TV_Q285755),V_f),s(TV_Q285759,V_x))) = s(TV_Q285755,happ(s(t_fun(TV_Q285759,TV_Q285755),V_g),s(TV_Q285759,V_x))) => s(t_fun(TV_Q285759,TV_Q285755),V_f) = s(t_fun(TV_Q285759,TV_Q285755),V_g))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (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_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_relations_WFu_u_EMPTYu_u_REL, axiom, ![TV_u_27a]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_emptyu_u_rel))))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_patricias_ptreeu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f1, V_f]: s(TV_u_27b,h4s_patricias_ptreeu_u_case(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty),s(TV_u_27b,V_v),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_f),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1))) = s(TV_u_27b,V_v)).
fof(ah4s_patricias_ptreeu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f1, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_patricias_ptreeu_u_case(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_leaf(s(t_h4s_nums_num,V_a0),s(TV_u_27a,V_a1))),s(TV_u_27b,V_v),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_f),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_f),s(t_h4s_nums_num,V_a0))),s(TV_u_27a,V_a1)))).
fof(ah4s_patricias_ptreeu_u_caseu_u_defu_c2, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f1, V_f, V_a3, V_a2, V_a1, V_a0]: s(TV_u_27b,h4s_patricias_ptreeu_u_case(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_branch(s(t_h4s_nums_num,V_a0),s(t_h4s_nums_num,V_a1),s(t_h4s_patricias_ptree(TV_u_27a),V_a2),s(t_h4s_patricias_ptree(TV_u_27a),V_a3))),s(TV_u_27b,V_v),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_f),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1))) = s(TV_u_27b,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1),s(t_h4s_nums_num,V_a0))),s(t_h4s_nums_num,V_a1))),s(t_h4s_patricias_ptree(TV_u_27a),V_a2))),s(t_h4s_patricias_ptree(TV_u_27a),V_a3)))).
fof(ah4s_patricias_ISu_u_EMPTYu_u_primitiveu_u_def, axiom, ![TV_u_27a]: ![V_uu_8]: (![V_v11]: s(t_bool,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),V_uu_8),s(t_h4s_patricias_ptree(TV_u_27a),V_v11))) = s(t_bool,h4s_combins_i(s(t_bool,f))) => ![V_uu_7]: (![V_v10]: s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool)),V_uu_7),s(t_h4s_patricias_ptree(TV_u_27a),V_v10))) = s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),V_uu_8) => ![V_uu_6]: (![V_v9]: s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool))),V_uu_6),s(t_h4s_nums_num,V_v9))) = s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool)),V_uu_7) => ![V_uu_5]: (![V_v8]: s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool)))),V_uu_5),s(t_h4s_nums_num,V_v8))) = s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool))),V_uu_6) => ![V_uu_4]: (![V_v7]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_uu_4),s(TV_u_27a,V_v7))) = s(t_bool,h4s_combins_i(s(t_bool,f))) => ![V_uu_3]: (![V_v6]: s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_uu_3),s(t_h4s_nums_num,V_v6))) = s(t_fun(TV_u_27a,t_bool),V_uu_4) => ![V_uu_2]: (![V_a]: s(t_bool,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),V_uu_2),s(t_h4s_patricias_ptree(TV_u_27a),V_a))) = s(t_bool,h4s_patricias_ptreeu_u_case(s(t_h4s_patricias_ptree(TV_u_27a),V_a),s(t_bool,h4s_combins_i(s(t_bool,t))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_uu_3),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool)))),V_uu_5))) => ![V_uu_1]: (![V_ISu_u_EMPTY]: s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),happ(s(t_fun(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool)),V_uu_1),s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),V_ISu_u_EMPTY))) = s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),V_uu_2) => ![V_uu_0]: (![V_R]: s(t_bool,happ(s(t_fun(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool)),V_R))) = s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool)),V_R))) => s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),h4s_patricias_isu_u_empty) = s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),h4s_relations_wfrec(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool)),V_uu_1))))))))))))).
fof(ch4s_patricias_ISu_u_EMPTYu_u_defu_c1, conjecture, ![TV_u_27a]: ![V_v1, V_v]: s(t_bool,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),h4s_patricias_isu_u_empty),s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_leaf(s(t_h4s_nums_num,V_v),s(TV_u_27a,V_v1))))) = s(t_bool,f)).
