%   ORIGINAL: h4/sorting/PERM__FUN__CONS
% 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !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/sorting/PERM__CONS__IFF: !x l2 l1. h4/sorting/PERM (h4/list/CONS x l1) (h4/list/CONS x l2) <=> h4/sorting/PERM l1 l2
% Assm: h4/sorting/PERM__EQUIVALENCE__ALT__DEF: !y x. h4/sorting/PERM x y <=> h4/sorting/PERM x = h4/sorting/PERM y
% Goal: !x l1_27 l1. h4/sorting/PERM l1 = h4/sorting/PERM l1_27 ==> h4/sorting/PERM (h4/list/CONS x l1) = h4/sorting/PERM (h4/list/CONS x l1_27)
%   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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !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_sortings_PERMu_u_CONSu_u_IFF]: !x l2 l1. happ (h4/sorting/PERM (h4/list/CONS x l1)) (h4/list/CONS x l2) <=> happ (h4/sorting/PERM l1) l2
% Assm [h4s_sortings_PERMu_u_EQUIVALENCEu_u_ALTu_u_DEF]: !y x. happ (h4/sorting/PERM x) y <=> h4/sorting/PERM x = h4/sorting/PERM y
% Goal: !x l1_27 l1. h4/sorting/PERM l1 = h4/sorting/PERM l1_27 ==> h4/sorting/PERM (h4/list/CONS x l1) = h4/sorting/PERM (h4/list/CONS x l1_27)
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_Q276684,TV_Q276680]: ![V_f, V_g]: (![V_x]: s(TV_Q276680,happ(s(t_fun(TV_Q276684,TV_Q276680),V_f),s(TV_Q276684,V_x))) = s(TV_Q276680,happ(s(t_fun(TV_Q276684,TV_Q276680),V_g),s(TV_Q276684,V_x))) => s(t_fun(TV_Q276684,TV_Q276680),V_f) = s(t_fun(TV_Q276684,TV_Q276680),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> 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_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_sortings_PERMu_u_CONSu_u_IFF, axiom, ![TV_u_27a]: ![V_x, V_l2, V_l1]: s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l1))))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l2))))) = s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27a),V_l2)))).
fof(ah4s_sortings_PERMu_u_EQUIVALENCEu_u_ALTu_u_DEF, axiom, ![TV_u_27a]: ![V_y, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_x))),s(t_h4s_lists_list(TV_u_27a),V_y)))) <=> s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_x))) = s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_y))))).
fof(ch4s_sortings_PERMu_u_FUNu_u_CONS, conjecture, ![TV_u_27a]: ![V_x, V_l1u_27, V_l1]: (s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_l1u_27))) => s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l1))))) = s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l1u_27))))))).
