%   ORIGINAL: h4/llist/LFILTER__APPEND
% 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/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> 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/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% 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/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/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/llist/LCONS__11: !t2 t1 h2 h1. h4/llist/LCONS h1 t1 = h4/llist/LCONS h2 t2 <=> h1 = h2 /\ t1 = t2
% Assm: h4/llist/LAPPEND0_c0: !x. h4/llist/LAPPEND h4/llist/LNIL x = x
% Assm: h4/llist/LAPPEND0_c1: !x t h. h4/llist/LAPPEND (h4/llist/LCONS h t) x = h4/llist/LCONS h (h4/llist/LAPPEND t x)
% Assm: h4/llist/LFINITE__STRONG__INDUCTION: !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)
% Assm: h4/llist/LFILTER__THM_c0: !P. h4/llist/LFILTER P h4/llist/LNIL = h4/llist/LNIL
% Assm: h4/llist/LFILTER__THM_c1: !t h P. h4/llist/LFILTER P (h4/llist/LCONS h t) = h4/bool/COND (P h) (h4/llist/LCONS h (h4/llist/LFILTER P t)) (h4/llist/LFILTER P t)
% Goal: !ll2 ll1 P. h4/llist/LFINITE ll1 ==> h4/llist/LFILTER P (h4/llist/LAPPEND ll1 ll2) = h4/llist/LAPPEND (h4/llist/LFILTER P ll1) (h4/llist/LFILTER P ll2)
%   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_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> 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_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% 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_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_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_llists_LCONSu_u_11]: !t2 t1 h2 h1. h4/llist/LCONS h1 t1 = h4/llist/LCONS h2 t2 <=> h1 = h2 /\ t1 = t2
% Assm [h4s_llists_LAPPEND0u_c0]: !x. h4/llist/LAPPEND h4/llist/LNIL x = x
% Assm [h4s_llists_LAPPEND0u_c1]: !x t h. h4/llist/LAPPEND (h4/llist/LCONS h t) x = h4/llist/LCONS h (h4/llist/LAPPEND t x)
% Assm [h4s_llists_LFINITEu_u_STRONGu_u_INDUCTION]: !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)
% Assm [h4s_llists_LFILTERu_u_THMu_c0]: !P. h4/llist/LFILTER P h4/llist/LNIL = h4/llist/LNIL
% Assm [h4s_llists_LFILTERu_u_THMu_c1]: !t h P. h4/llist/LFILTER P (h4/llist/LCONS h t) = h4/bool/COND (happ P h) (h4/llist/LCONS h (h4/llist/LFILTER P t)) (h4/llist/LFILTER P t)
% Goal: !ll2 ll1 P. h4/llist/LFINITE ll1 ==> h4/llist/LFILTER P (h4/llist/LAPPEND ll1 ll2) = h4/llist/LAPPEND (h4/llist/LFILTER P ll1) (h4/llist/LFILTER P ll2)
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_Q171528,TV_Q171524]: ![V_f, V_g]: (![V_x]: s(TV_Q171524,happ(s(t_fun(TV_Q171528,TV_Q171524),V_f),s(TV_Q171528,V_x))) = s(TV_Q171524,happ(s(t_fun(TV_Q171528,TV_Q171524),V_g),s(TV_Q171528,V_x))) => s(t_fun(TV_Q171528,TV_Q171524),V_f) = s(t_fun(TV_Q171528,TV_Q171524),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,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
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_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_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_CONDu_u_CLAUSESu_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_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
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_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_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,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_llists_LCONSu_u_11, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_h2, V_h1]: (s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h1),s(t_h4s_llists_llist(TV_u_27a),V_t1))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h2),s(t_h4s_llists_llist(TV_u_27a),V_t2))) <=> (s(TV_u_27a,V_h1) = s(TV_u_27a,V_h2) & s(t_h4s_llists_llist(TV_u_27a),V_t1) = s(t_h4s_llists_llist(TV_u_27a),V_t2)))).
fof(ah4s_llists_LAPPEND0u_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lappend(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil),s(t_h4s_llists_llist(TV_u_27a),V_x))) = s(t_h4s_llists_llist(TV_u_27a),V_x)).
fof(ah4s_llists_LAPPEND0u_c1, axiom, ![TV_u_27a]: ![V_x, V_t, V_h]: s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lappend(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))),s(t_h4s_llists_llist(TV_u_27a),V_x))) = 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),h4s_llists_lappend(s(t_h4s_llists_llist(TV_u_27a),V_t),s(t_h4s_llists_llist(TV_u_27a),V_x)))))).
fof(ah4s_llists_LFINITEu_u_STRONGu_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,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))))))).
fof(ah4s_llists_LFILTERu_u_THMu_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lfilter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil)).
fof(ah4s_llists_LFILTERu_u_THMu_c1, axiom, ![TV_u_27b]: ![V_t, V_h, V_P]: s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lfilter(s(t_fun(TV_u_27b,t_bool),V_P),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_h4s_llists_llist(TV_u_27b),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_h))),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),h4s_llists_lfilter(s(t_fun(TV_u_27b,t_bool),V_P),s(t_h4s_llists_llist(TV_u_27b),V_t))))),s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lfilter(s(t_fun(TV_u_27b,t_bool),V_P),s(t_h4s_llists_llist(TV_u_27b),V_t)))))).
fof(ch4s_llists_LFILTERu_u_APPEND, conjecture, ![TV_u_27a]: ![V_ll2, V_ll1, V_P]: (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_ll1)))) => s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lfilter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lappend(s(t_h4s_llists_llist(TV_u_27a),V_ll1),s(t_h4s_llists_llist(TV_u_27a),V_ll2))))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lappend(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lfilter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),V_ll1))),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lfilter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),V_ll2))))))).
