%   ORIGINAL: h4/lbtree/mindepth__thm_c0
% 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/FALSITY: !t. F ==> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/lbtree/mem__thm_c0: !a. h4/lbtree/mem a h4/lbtree/Lf <=> F
% Assm: h4/lbtree/mindepth__def: !x t. h4/lbtree/mindepth x t = h4/bool/COND (h4/lbtree/mem x t) (h4/option/SOME (h4/while/LEAST (\n. h4/lbtree/depth x t n))) h4/option/NONE
% Goal: !x. h4/lbtree/mindepth x h4/lbtree/Lf = h4/option/NONE
%   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_FALSITY]: !t. F ==> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_lbtrees_memu_u_thmu_c0]: !a. h4/lbtree/mem a h4/lbtree/Lf <=> F
% Assm [h4s_lbtrees_mindepthu_u_def]: !_0. (!x t n. happ (happ (happ _0 x) t) n <=> h4/lbtree/depth x t n) ==> (!x t. h4/lbtree/mindepth x t = h4/bool/COND (h4/lbtree/mem x t) (h4/option/SOME (h4/while/LEAST (happ (happ _0 x) t))) h4/option/NONE)
% Goal: !x. h4/lbtree/mindepth x h4/lbtree/Lf = h4/option/NONE
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_Q198419,TV_Q198415]: ![V_f, V_g]: (![V_x]: s(TV_Q198415,happ(s(t_fun(TV_Q198419,TV_Q198415),V_f),s(TV_Q198419,V_x))) = s(TV_Q198415,happ(s(t_fun(TV_Q198419,TV_Q198415),V_g),s(TV_Q198419,V_x))) => s(t_fun(TV_Q198419,TV_Q198415),V_f) = s(t_fun(TV_Q198419,TV_Q198415),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,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_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_lbtrees_memu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_bool,h4s_lbtrees_mem(s(TV_u_27a,V_a),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf))) = s(t_bool,f)).
fof(ah4s_lbtrees_mindepthu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_t, V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_uu_0),s(TV_u_27a,V_x))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t))),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_lbtrees_depth(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t),s(t_h4s_nums_num,V_n))) => ![V_x, V_t]: s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_mindepth(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_bools_cond(s(t_bool,h4s_lbtrees_mem(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_uu_0),s(TV_u_27a,V_x))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t))))))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))))).
fof(ch4s_lbtrees_mindepthu_u_thmu_c0, conjecture, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_mindepth(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none)).
