%   ORIGINAL: h4/lbtree/finite__cases
% 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/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/lbtree/finite__def: h4/lbtree/finite = (\a0. !finite_27. (!a00. a00 = h4/lbtree/Lf \/ (?a t1 t2. a00 = h4/lbtree/Nd a t1 t2 /\ finite_27 t1 /\ finite_27 t2) ==> finite_27 a00) ==> finite_27 a0)
% Goal: !a0. h4/lbtree/finite a0 <=> a0 = h4/lbtree/Lf \/ (?a t1 t2. a0 = h4/lbtree/Nd a t1 t2 /\ h4/lbtree/finite t1 /\ h4/lbtree/finite 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_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_lbtrees_finiteu_u_def]: !x. h4/lbtree/finite x <=> (!finite_27. (!a00. a00 = h4/lbtree/Lf \/ (?a t1 t2. a00 = h4/lbtree/Nd a t1 t2 /\ happ finite_27 t1 /\ happ finite_27 t2) ==> happ finite_27 a00) ==> happ finite_27 x)
% Goal: !a0. h4/lbtree/finite a0 <=> a0 = h4/lbtree/Lf \/ (?a t1 t2. a0 = h4/lbtree/Nd a t1 t2 /\ h4/lbtree/finite t1 /\ h4/lbtree/finite t2)
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_Q197894,TV_Q197890]: ![V_f, V_g]: (![V_x]: s(TV_Q197890,happ(s(t_fun(TV_Q197894,TV_Q197890),V_f),s(TV_Q197894,V_x))) = s(TV_Q197890,happ(s(t_fun(TV_Q197894,TV_Q197890),V_g),s(TV_Q197894,V_x))) => s(t_fun(TV_Q197894,TV_Q197890),V_f) = s(t_fun(TV_Q197894,TV_Q197890),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
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_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (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_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_lbtrees_finiteu_u_def, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_lbtrees_finite(s(t_h4s_lbtrees_lbtree(TV_u_27a),V_x)))) <=> ![V_finiteu_27]: (![V_a00]: ((s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a00) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf) | ?[V_a, V_t1, V_t2]: (s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a00) = 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))) & (p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),V_finiteu_27),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1)))) & p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),V_finiteu_27),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),V_finiteu_27),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),V_finiteu_27),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_x))))))).
fof(ch4s_lbtrees_finiteu_u_cases, conjecture, ![TV_u_27a]: ![V_a0]: (p(s(t_bool,h4s_lbtrees_finite(s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a0)))) <=> (s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a0) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf) | ?[V_a, V_t1, V_t2]: (s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a0) = 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))) & (p(s(t_bool,h4s_lbtrees_finite(s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1)))) & p(s(t_bool,h4s_lbtrees_finite(s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))))))))).
