%   ORIGINAL: h4/llist/LFINITE__STRONG__INDUCTION
% 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/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/llist/LFINITE__THM_c0: h4/llist/LFINITE h4/llist/LNIL <=> T
% Assm: h4/llist/LFINITE__THM_c1: !t h. h4/llist/LFINITE (h4/llist/LCONS h t) <=> h4/llist/LFINITE t
% Assm: h4/llist/LFINITE__INDUCTION: !P. P h4/llist/LNIL /\ (!h t. P t ==> P (h4/llist/LCONS h t)) ==> (!a0. h4/llist/LFINITE a0 ==> P a0)
% Goal: !P. P h4/llist/LNIL /\ (!h t. h4/llist/LFINITE t /\ P t ==> P (h4/llist/LCONS h t)) ==> (!a0. h4/llist/LFINITE a0 ==> P a0)
%   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_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_llists_LFINITEu_u_THMu_c0]: h4/llist/LFINITE h4/llist/LNIL <=> T
% Assm [h4s_llists_LFINITEu_u_THMu_c1]: !t h. h4/llist/LFINITE (h4/llist/LCONS h t) <=> h4/llist/LFINITE t
% Assm [h4s_llists_LFINITEu_u_INDUCTION]: !P. happ P h4/llist/LNIL /\ (!h t. happ P t ==> happ P (h4/llist/LCONS h t)) ==> (!a0. h4/llist/LFINITE a0 ==> happ P a0)
% Goal: !P. happ P h4/llist/LNIL /\ (!h t. h4/llist/LFINITE t /\ happ P t ==> happ P (h4/llist/LCONS h t)) ==> (!a0. h4/llist/LFINITE a0 ==> happ P a0)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q170374,TV_Q170370]: ![V_f, V_g]: (![V_x]: s(TV_Q170370,happ(s(t_fun(TV_Q170374,TV_Q170370),V_f),s(TV_Q170374,V_x))) = s(TV_Q170370,happ(s(t_fun(TV_Q170374,TV_Q170370),V_g),s(TV_Q170374,V_x))) => s(t_fun(TV_Q170374,TV_Q170370),V_f) = s(t_fun(TV_Q170374,TV_Q170370),V_g))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_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_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) <=> p(s(t_bool,V_t)))).
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_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_llists_LFINITEu_u_THMu_c0, axiom, ![TV_u_27a]: s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_bool,t0)).
fof(ah4s_llists_LFINITEu_u_THMu_c1, axiom, ![TV_u_27b]: ![V_t, V_h]: s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lcons(s(TV_u_27b,V_h),s(t_h4s_llists_llist(TV_u_27b),V_t))))) = s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27b),V_t)))).
fof(ah4s_llists_LFINITEu_u_INDUCTION, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil)))) & ![V_h, V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),V_t)))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h),s(t_h4s_llists_llist(TV_u_27a),V_t)))))))) => ![V_a0]: (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_a0)))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),V_a0))))))).
fof(ch4s_llists_LFINITEu_u_STRONGu_u_INDUCTION, conjecture, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil)))) & ![V_h, V_t]: ((p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_t)))) & p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),V_t))))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h),s(t_h4s_llists_llist(TV_u_27a),V_t)))))))) => ![V_a0]: (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_a0)))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),V_a0))))))).
