%   ORIGINAL: h4/words/WORD__SET__INDUCT
% 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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/pred__set/FINITE__INDUCT: !P. P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ P s ==> (!e. ~h4/bool/IN e s ==> P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> P s)
% Assm: h4/words/WORD__FINITE: !s. h4/pred__set/FINITE s
% Goal: !P. P h4/pred__set/EMPTY /\ (!s. P s ==> (!e. ~h4/bool/IN e s ==> P (h4/pred__set/INSERT e s))) ==> (!s. P s)
%   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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_predu_u_sets_FINITEu_u_INDUCT]: !P. happ P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ happ P s ==> (!e. ~h4/bool/IN e s ==> happ P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> happ P s)
% Assm [h4s_wordss_WORDu_u_FINITE]: !s. h4/pred__set/FINITE s
% Goal: !P. happ P h4/pred__set/EMPTY /\ (!s. happ P s ==> (!e. ~h4/bool/IN e s ==> happ P (h4/pred__set/INSERT e s))) ==> (!s. happ P s)
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_Q130387,TV_Q130383]: ![V_f, V_g]: (![V_x]: s(TV_Q130383,happ(s(t_fun(TV_Q130387,TV_Q130383),V_f),s(TV_Q130387,V_x))) = s(TV_Q130383,happ(s(t_fun(TV_Q130387,TV_Q130383),V_g),s(TV_Q130387,V_x))) => s(t_fun(TV_Q130387,TV_Q130383),V_f) = s(t_fun(TV_Q130387,TV_Q130383),V_g))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_FINITEu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) & ![V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))) => ![V_e]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))))))) => ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_wordss_WORDu_u_FINITE, axiom, ![TV_u_27a]: ![V_s]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),V_s))))).
fof(ch4s_wordss_WORDu_u_SETu_u_INDUCT, conjecture, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),t_bool),V_P),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),h4s_predu_u_sets_empty)))) & ![V_s]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),t_bool),V_P),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),V_s)))) => ![V_e]: (~ (p(s(t_bool,h4s_bools_in(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_e),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),V_s))))) => p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),t_bool),V_P),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),h4s_predu_u_sets_insert(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_e),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),V_s))))))))) => ![V_s]: p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),t_bool),V_P),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),V_s)))))).
