%   ORIGINAL: h4/Encode/wf__encode__sum
% 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/FALSITY: !t. F ==> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> 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/sum/INR__INL__11_c0: !y x. h4/sum/INL x = h4/sum/INL y <=> x = y
% Assm: h4/sum/INR__INL__11_c1: !y x. h4/sum/INR x = h4/sum/INR y <=> x = y
% Assm: h4/sum/sum__CASES: !ss. (?x. ss = h4/sum/INL x) \/ (?y. ss = h4/sum/INR y)
% Assm: h4/sum/sum__case__def_c0: !x f1 f. h4/sum/sum__CASE (h4/sum/INL x) f f1 = f x
% Assm: h4/sum/sum__case__def_c1: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = f1 y
% Assm: h4/rich__list/IS__PREFIX_c2: !x2 x1 l2 l1. h4/list/isPREFIX (h4/list/CONS x2 l2) (h4/list/CONS x1 l1) <=> x1 = x2 /\ h4/list/isPREFIX l2 l1
% Assm: h4/Encode/wf__encoder__def: !p e. h4/Encode/wf__encoder p e <=> (!x y. p x /\ p y /\ h4/list/isPREFIX (e y) (e x) ==> x = y)
% Assm: h4/Encode/encode__sum__def_c0: !yb xb x. h4/Encode/encode__sum xb yb (h4/sum/INL x) = h4/list/CONS T (xb x)
% Assm: h4/Encode/encode__sum__def_c1: !yb y xb. h4/Encode/encode__sum xb yb (h4/sum/INR y) = h4/list/CONS F (yb y)
% Assm: h4/Encode/lift__sum__def: !x p2 p1. h4/Encode/lift__sum p1 p2 x <=> h4/sum/sum__CASE x (\x1. p1 x1) (\x2. p2 x2)
% Goal: !p2 p1 e2 e1. h4/Encode/wf__encoder p1 e1 /\ h4/Encode/wf__encoder p2 e2 ==> h4/Encode/wf__encoder (h4/Encode/lift__sum p1 p2) (h4/Encode/encode__sum e1 e2)
%   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_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> 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_sums_INRu_u_INLu_u_11u_c0]: !y x. h4/sum/INL x = h4/sum/INL y <=> x = y
% Assm [h4s_sums_INRu_u_INLu_u_11u_c1]: !y x. h4/sum/INR x = h4/sum/INR y <=> x = y
% Assm [h4s_sums_sumu_u_CASES]: !ss. (?x. ss = h4/sum/INL x) \/ (?y. ss = h4/sum/INR y)
% Assm [h4s_sums_sumu_u_caseu_u_defu_c0]: !x f1 f. h4/sum/sum__CASE (h4/sum/INL x) f f1 = happ f x
% Assm [h4s_sums_sumu_u_caseu_u_defu_c1]: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = happ f1 y
% Assm [h4s_richu_u_lists_ISu_u_PREFIXu_c2]: !x2 x1 l2 l1. h4/list/isPREFIX (h4/list/CONS x2 l2) (h4/list/CONS x1 l1) <=> x1 = x2 /\ h4/list/isPREFIX l2 l1
% Assm [h4s_Encodes_wfu_u_encoderu_u_def]: !p e. h4/Encode/wf__encoder p e <=> (!x y. happ p x /\ happ p y /\ h4/list/isPREFIX (happ e y) (happ e x) ==> x = y)
% Assm [h4s_Encodes_encodeu_u_sumu_u_defu_c0]: !yb xb x. happ (h4/Encode/encode__sum xb yb) (h4/sum/INL x) = h4/list/CONS T (happ xb x)
% Assm [h4s_Encodes_encodeu_u_sumu_u_defu_c1]: !yb y xb. happ (h4/Encode/encode__sum xb yb) (h4/sum/INR y) = h4/list/CONS F (happ yb y)
% Assm [h4s_Encodes_liftu_u_sumu_u_def]: !_1. (!p2 x2. happ (happ _1 p2) x2 <=> happ p2 x2) ==> (!_0. (!p1 x1. happ (happ _0 p1) x1 <=> happ p1 x1) ==> (!x p2 p1. happ (h4/Encode/lift__sum p1 p2) x <=> h4/sum/sum__CASE x (happ _0 p1) (happ _1 p2)))
% Goal: !p2 p1 e2 e1. h4/Encode/wf__encoder p1 e1 /\ h4/Encode/wf__encoder p2 e2 ==> h4/Encode/wf__encoder (h4/Encode/lift__sum p1 p2) (h4/Encode/encode__sum e1 e2)
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_Q203898,TV_Q203894]: ![V_f, V_g]: (![V_x]: s(TV_Q203894,happ(s(t_fun(TV_Q203898,TV_Q203894),V_f),s(TV_Q203898,V_x))) = s(TV_Q203894,happ(s(t_fun(TV_Q203898,TV_Q203894),V_g),s(TV_Q203898,V_x))) => s(t_fun(TV_Q203898,TV_Q203894),V_f) = s(t_fun(TV_Q203898,TV_Q203894),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => 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_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> 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_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
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_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_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_sums_INRu_u_INLu_u_11u_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_sums_INRu_u_INLu_u_11u_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_x))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))) <=> s(TV_u_27b,V_x) = s(TV_u_27b,V_y))).
fof(ah4s_sums_sumu_u_CASES, axiom, ![TV_u_27a,TV_u_27b]: ![V_ss]: (?[V_x]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_ss) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))) | ?[V_y]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_ss) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_y, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f1),s(TV_u_27b,V_y)))).
fof(ah4s_richu_u_lists_ISu_u_PREFIXu_c2, axiom, ![TV_u_27a]: ![V_x2, V_x1, V_l2, V_l1]: (p(s(t_bool,h4s_lists_isprefix(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x2),s(t_h4s_lists_list(TV_u_27a),V_l2))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x1),s(t_h4s_lists_list(TV_u_27a),V_l1)))))) <=> (s(TV_u_27a,V_x1) = s(TV_u_27a,V_x2) & p(s(t_bool,h4s_lists_isprefix(s(t_h4s_lists_list(TV_u_27a),V_l2),s(t_h4s_lists_list(TV_u_27a),V_l1))))))).
fof(ah4s_Encodes_wfu_u_encoderu_u_def, axiom, ![TV_u_27a]: ![V_p, V_e]: (p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)))) <=> ![V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_x)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_y)))) & p(s(t_bool,h4s_lists_isprefix(s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(TV_u_27a,V_y))),s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(TV_u_27a,V_x)))))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))).
fof(ah4s_Encodes_encodeu_u_sumu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_yb, V_xb, V_x]: s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_sum(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_xb),s(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),V_yb))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))))) = s(t_h4s_lists_list(t_bool),h4s_lists_cons(s(t_bool,t),s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_xb),s(TV_u_27a,V_x)))))).
fof(ah4s_Encodes_encodeu_u_sumu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_yb, V_y, V_xb]: s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_sum(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_xb),s(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),V_yb))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))))) = s(t_h4s_lists_list(t_bool),h4s_lists_cons(s(t_bool,f),s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),V_yb),s(TV_u_27b,V_y)))))).
fof(ah4s_Encodes_liftu_u_sumu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_p2, V_x2]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_p2))),s(TV_u_27b,V_x2))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_p2),s(TV_u_27b,V_x2))) => ![V_uu_0]: (![V_p1, V_x1]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_p1))),s(TV_u_27a,V_x1))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p1),s(TV_u_27a,V_x1))) => ![V_x, V_p2, V_p1]: s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),h4s_encodes_liftu_u_sum(s(t_fun(TV_u_27a,t_bool),V_p1),s(t_fun(TV_u_27b,t_bool),V_p2))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_x))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_x),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_p1))),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_p2)))))))).
fof(ch4s_Encodes_wfu_u_encodeu_u_sum, conjecture, ![TV_u_27a,TV_u_27b]: ![V_p2, V_p1, V_e2, V_e1]: ((p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27a,t_bool),V_p1),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e1)))) & p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27b,t_bool),V_p2),s(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),V_e2))))) => p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),h4s_encodes_liftu_u_sum(s(t_fun(TV_u_27a,t_bool),V_p1),s(t_fun(TV_u_27b,t_bool),V_p2))),s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_sum(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e1),s(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),V_e2)))))))).
