%   ORIGINAL: h4/lbtree/lbtree__case__thm_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/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% 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/SELECT__REFL__2: !x. h4/min/_40 (\y. x = 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/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/lbtree/Lf__NOT__Nd: !t2 t1 a. ~(h4/lbtree/Lf = h4/lbtree/Nd a t1 t2)
% Assm: h4/lbtree/Nd__11: !u2 u1 t2 t1 a2 a1. h4/lbtree/Nd a1 t1 u1 = h4/lbtree/Nd a2 t2 u2 <=> a1 = a2 /\ t1 = t2 /\ u1 = u2
% 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: !t2 t1 f e a. h4/lbtree/lbtree__case e f (h4/lbtree/Nd a t1 t2) = f a t1 t2
%   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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% 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_SELECTu_u_REFLu_u_2]: !_0. (!x y. happ (happ _0 x) y <=> x = y) ==> (!x. h4/min/_40 (happ _0 x) = 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_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_lbtrees_Lfu_u_NOTu_u_Nd]: !t2 t1 a. ~(h4/lbtree/Lf = h4/lbtree/Nd a t1 t2)
% Assm [h4s_lbtrees_Ndu_u_11]: !u2 u1 t2 t1 a2 a1. h4/lbtree/Nd a1 t1 u1 = h4/lbtree/Nd a2 t2 u2 <=> a1 = a2 /\ t1 = t2 /\ u1 = u2
% 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: !t2 t1 f e a. h4/lbtree/lbtree__case e f (h4/lbtree/Nd a t1 t2) = happ (happ (happ f a) t1) t2
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_Q197443,TV_Q197439]: ![V_f, V_g]: (![V_x]: s(TV_Q197439,happ(s(t_fun(TV_Q197443,TV_Q197439),V_f),s(TV_Q197443,V_x))) = s(TV_Q197439,happ(s(t_fun(TV_Q197443,TV_Q197439),V_g),s(TV_Q197443,V_x))) => s(t_fun(TV_Q197443,TV_Q197439),V_f) = s(t_fun(TV_Q197443,TV_Q197439),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => 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_SELECTu_u_REFLu_u_2, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y)) => ![V_x]: s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))))) = 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_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & 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,V_a)))))).
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,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_lbtrees_Lfu_u_NOTu_u_Nd, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_a]: ~ (s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_a),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))))).
fof(ah4s_lbtrees_Ndu_u_11, axiom, ![TV_u_27a]: ![V_u2, V_u1, V_t2, V_t1, V_a2, V_a1]: (s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_a1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u1))) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_a2),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u2))) <=> (s(TV_u_27a,V_a1) = s(TV_u_27a,V_a2) & (s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1) = s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2) & s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u1) = s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u2))))).
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_c1, conjecture, ![TV_u_27a,TV_u_27b]: ![V_t2, V_t1, V_f, V_e, V_a]: 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_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))))) = 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,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t1))),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t2)))).
