%   ORIGINAL: h4/list/SNOC__Axiom
% 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/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. 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/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/list/list__Axiom__old: !x f. h4/bool/_3F_21 (\fn1. fn1 h4/list/NIL = x /\ (!h t. fn1 (h4/list/CONS h t) = f (fn1 t) h t))
% Assm: h4/list/REVERSE__DEF_c0: h4/list/REVERSE h4/list/NIL = h4/list/NIL
% Assm: h4/list/REVERSE__REVERSE: !l. h4/list/REVERSE (h4/list/REVERSE l) = l
% Assm: h4/list/REVERSE__SNOC: !x l. h4/list/REVERSE (h4/list/SNOC x l) = h4/list/CONS x (h4/list/REVERSE l)
% Goal: !f e. ?fn. fn h4/list/NIL = e /\ (!x l. fn (h4/list/SNOC x l) = f x l (fn l))
%   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_EXISTSu_u_UNIQUEu_u_DEF]: !x. h4/bool/_3F_21 x <=> $exists x /\ (!x y. happ x x /\ happ x y ==> x = y)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. 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_combins_ou_u_DEF]: !g f x. h4/combin/o f g x = happ f (happ g x)
% Assm [h4s_lists_listu_u_Axiomu_u_old]: !_0. (!x f fn1. happ (happ (happ _0 x) f) fn1 <=> happ fn1 h4/list/NIL = x /\ (!h t. happ fn1 (h4/list/CONS h t) = happ (happ (happ f (happ fn1 t)) h) t)) ==> (!x f. h4/bool/_3F_21 (happ (happ _0 x) f))
% Assm [h4s_lists_REVERSEu_u_DEFu_c0]: h4/list/REVERSE h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_REVERSEu_u_REVERSE]: !l. h4/list/REVERSE (h4/list/REVERSE l) = l
% Assm [h4s_lists_REVERSEu_u_SNOC]: !x l. h4/list/REVERSE (h4/list/SNOC x l) = h4/list/CONS x (h4/list/REVERSE l)
% Goal: !f e. ?fn. happ fn h4/list/NIL = e /\ (!x l. happ fn (h4/list/SNOC x l) = happ (happ (happ f x) l) (happ fn l))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q245609,TV_Q245605]: ![V_f, V_g]: (![V_x]: s(TV_Q245605,happ(s(t_fun(TV_Q245609,TV_Q245605),V_f),s(TV_Q245609,V_x))) = s(TV_Q245605,happ(s(t_fun(TV_Q245609,TV_Q245605),V_g),s(TV_Q245609,V_x))) => s(t_fun(TV_Q245609,TV_Q245605),V_f) = s(t_fun(TV_Q245609,TV_Q245605),V_g))).
fof(ah4s_bools_EXISTSu_u_UNIQUEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),V_x)))) <=> (p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x)))) & ![V_x0, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x0)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x0) = s(TV_u_27a,V_y))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
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_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_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_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_lists_listu_u_Axiomu_u_old, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_x, V_f, V_fn1]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b))),t_fun(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),t_bool)),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b))),t_fun(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),t_bool))),V_uu_0),s(TV_u_27b,V_x))),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b))),V_f))),s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn1)))) <=> (s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn1),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_x) & ![V_h, V_t]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn1),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(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b))),V_f),s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn1),s(t_h4s_lists_list(TV_u_27a),V_t))))),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) => ![V_x, V_f]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b))),t_fun(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),t_bool)),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b))),t_fun(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),t_bool))),V_uu_0),s(TV_u_27b,V_x))),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b))),V_f)))))))).
fof(ah4s_lists_REVERSEu_u_DEFu_c0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_lists_REVERSEu_u_REVERSE, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
fof(ah4s_lists_REVERSEu_u_SNOC, axiom, ![TV_u_27a]: ![V_x, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = 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),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ch4s_lists_SNOCu_u_Axiom, conjecture, ![TV_u_27b,TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_e) & ![V_x, V_l]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,TV_u_27b))),V_f),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))),s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),V_l))))))).
