%   ORIGINAL: h4/lbtree/lbtree__case__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/NOT__CLAUSES_c1: ~T <=> F
% 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_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/lbtree/lbtree__case__def: !t f e. h4/lbtree/lbtree__case e f t = h4/bool/COND (t = h4/lbtree/Lf) e (f (h4/min/_40 (\a. ?t1 t2. t = h4/lbtree/Nd a t1 t2)) (h4/min/_40 (\t1. ?a t2. t = h4/lbtree/Nd a t1 t2)) (h4/min/_40 (\t2. ?a t1. t = h4/lbtree/Nd a t1 t2)))
% Goal: !f e. h4/lbtree/lbtree__case e f h4/lbtree/Lf = e
%   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_NOTu_u_CLAUSESu_c1]: ~T <=> F
% 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_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_lbtrees_lbtreeu_u_caseu_u_def]: !_2. (!t t2. happ (happ _2 t) t2 <=> (?a t1. t = h4/lbtree/Nd a t1 t2)) ==> (!_1. (!t t1. happ (happ _1 t) t1 <=> (?a t2. t = h4/lbtree/Nd a t1 t2)) ==> (!_0. (!t a. happ (happ _0 t) a <=> (?t1 t2. t = h4/lbtree/Nd a t1 t2)) ==> (!t f e. ?v. (v <=> t = h4/lbtree/Lf) /\ h4/lbtree/lbtree__case e f t = h4/bool/COND v e (happ (happ (happ f (h4/min/_40 (happ _0 t))) (h4/min/_40 (happ _1 t))) (h4/min/_40 (happ _2 t))))))
% Goal: !f e. h4/lbtree/lbtree__case e f h4/lbtree/Lf = e
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_Q197417,TV_Q197413]: ![V_f, V_g]: (![V_x]: s(TV_Q197413,happ(s(t_fun(TV_Q197417,TV_Q197413),V_f),s(TV_Q197417,V_x))) = s(TV_Q197413,happ(s(t_fun(TV_Q197417,TV_Q197413),V_g),s(TV_Q197417,V_x))) => s(t_fun(TV_Q197417,TV_Q197413),V_f) = s(t_fun(TV_Q197417,TV_Q197413),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f0)))).
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_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_lbtrees_lbtreeu_u_caseu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_2]: (![V_t, V_t2]: (p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool)),V_uu_2),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t))),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t2)))) <=> ?[V_a, V_t1]: s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_nd(s(TV_u_27b,V_a),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t2)))) => ![V_uu_1]: (![V_t, V_t1]: (p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool)),V_uu_1),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t))),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t1)))) <=> ?[V_a, V_t2]: s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_nd(s(TV_u_27b,V_a),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t2)))) => ![V_uu_0]: (![V_t, V_a]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t))),s(TV_u_27b,V_a)))) <=> ?[V_t1, V_t2]: s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_nd(s(TV_u_27b,V_a),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t2)))) => ![V_t, V_f, V_e]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_lf)) & s(TV_u_27a,h4s_lbtrees_lbtreeu_u_case(s(TV_u_27a,V_e),s(t_fun(TV_u_27b,t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a))),V_f),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27a,V_e),s(TV_u_27a,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a))),V_f),s(TV_u_27b,h4s_mins_u_40(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t))))))),s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_mins_u_40(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool)),V_uu_1),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t))))))),s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_mins_u_40(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool)),V_uu_2),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t)))))))))))))).
fof(ch4s_lbtrees_lbtreeu_u_caseu_u_thmu_c0, conjecture, ![TV_u_27b,TV_u_27a]: ![V_f, V_e]: s(TV_u_27a,h4s_lbtrees_lbtreeu_u_case(s(TV_u_27a,V_e),s(t_fun(TV_u_27b,t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a))),V_f),s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_lf))) = s(TV_u_27a,V_e)).
