%   ORIGINAL: h4/Encode/lift__blist__suc
% 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/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/prim__rec/INV__SUC__EQ: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm: h4/list/LENGTH0_c1: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm: h4/list/EVERY__DEF_c1: !t h P. h4/list/EVERY P (h4/list/CONS h t) <=> P h /\ h4/list/EVERY P t
% Assm: h4/Encode/lift__blist__def: !x p m. h4/Encode/lift__blist m p x <=> h4/list/EVERY p x /\ h4/list/LENGTH x = m
% Goal: !t p n h. h4/Encode/lift__blist (h4/num/SUC n) p (h4/list/CONS h t) <=> p h /\ h4/Encode/lift__blist n p t
%   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_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_primu_u_recs_INVu_u_SUCu_u_EQ]: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm [h4s_lists_LENGTH0u_c1]: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm [h4s_lists_EVERYu_u_DEFu_c1]: !t h P. h4/list/EVERY P (h4/list/CONS h t) <=> happ P h /\ h4/list/EVERY P t
% Assm [h4s_Encodes_liftu_u_blistu_u_def]: !x p m. h4/Encode/lift__blist m p x <=> h4/list/EVERY p x /\ h4/list/LENGTH x = m
% Goal: !t p n h. h4/Encode/lift__blist (h4/num/SUC n) p (h4/list/CONS h t) <=> happ p h /\ h4/Encode/lift__blist n p t
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_Q204073,TV_Q204069]: ![V_f, V_g]: (![V_x]: s(TV_Q204069,happ(s(t_fun(TV_Q204073,TV_Q204069),V_f),s(TV_Q204073,V_x))) = s(TV_Q204069,happ(s(t_fun(TV_Q204073,TV_Q204069),V_g),s(TV_Q204073,V_x))) => s(t_fun(TV_Q204073,TV_Q204069),V_f) = s(t_fun(TV_Q204073,TV_Q204069),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
fof(ah4s_bools_CONJu_u_ASSOC, 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_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,t0)))).
fof(ah4s_primu_u_recs_INVu_u_SUCu_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_lists_LENGTH0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_EVERYu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: (p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h)))) & p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_Encodes_liftu_u_blistu_u_def, axiom, ![TV_u_27a]: ![V_x, V_p, V_m]: (p(s(t_bool,h4s_encodes_liftu_u_blist(s(t_h4s_nums_num,V_m),s(t_fun(TV_u_27a,t_bool),V_p),s(t_h4s_lists_list(TV_u_27a),V_x)))) <=> (p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_p),s(t_h4s_lists_list(TV_u_27a),V_x)))) & s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_x))) = s(t_h4s_nums_num,V_m)))).
fof(ch4s_Encodes_liftu_u_blistu_u_suc, conjecture, ![TV_u_27a]: ![V_t, V_p, V_n, V_h]: (p(s(t_bool,h4s_encodes_liftu_u_blist(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_fun(TV_u_27a,t_bool),V_p),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_h)))) & p(s(t_bool,h4s_encodes_liftu_u_blist(s(t_h4s_nums_num,V_n),s(t_fun(TV_u_27a,t_bool),V_p),s(t_h4s_lists_list(TV_u_27a),V_t))))))).
