%   ORIGINAL: h4/alist/ALOOKUP__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/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/alist/ALOOKUP__prefix_c0: !v ls2 ls k. h4/alist/ALOOKUP ls k = h4/option/SOME v ==> h4/alist/ALOOKUP (h4/list/APPEND ls ls2) k = h4/option/SOME v
% Assm: h4/alist/ALOOKUP__prefix_c1: !ls2 ls k. h4/alist/ALOOKUP ls k = h4/option/NONE ==> h4/alist/ALOOKUP (h4/list/APPEND ls ls2) k = h4/alist/ALOOKUP ls2 k
% Goal: !l2 l1 k. h4/alist/ALOOKUP (h4/list/APPEND l1 l2) k = h4/option/option__CASE (h4/alist/ALOOKUP l1 k) (h4/alist/ALOOKUP l2 k) (\v. h4/option/SOME v)
%   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_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = happ f x
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_alists_ALOOKUPu_u_prefixu_c0]: !v ls2 ls k. h4/alist/ALOOKUP ls k = h4/option/SOME v ==> h4/alist/ALOOKUP (h4/list/APPEND ls ls2) k = h4/option/SOME v
% Assm [h4s_alists_ALOOKUPu_u_prefixu_c1]: !ls2 ls k. h4/alist/ALOOKUP ls k = h4/option/NONE ==> h4/alist/ALOOKUP (h4/list/APPEND ls ls2) k = h4/alist/ALOOKUP ls2 k
% Goal: !_0. (!v. happ _0 v = h4/option/SOME v) ==> (!l2 l1 k. h4/alist/ALOOKUP (h4/list/APPEND l1 l2) k = h4/option/option__CASE (h4/alist/ALOOKUP l1 k) (h4/alist/ALOOKUP l2 k) _0)
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_Q270615,TV_Q270611]: ![V_f, V_g]: (![V_x]: s(TV_Q270611,happ(s(t_fun(TV_Q270615,TV_Q270611),V_f),s(TV_Q270615,V_x))) = s(TV_Q270611,happ(s(t_fun(TV_Q270615,TV_Q270611),V_g),s(TV_Q270615,V_x))) => s(t_fun(TV_Q270615,TV_Q270611),V_f) = s(t_fun(TV_Q270615,TV_Q270611),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_options_optionu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_options_optionu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_alists_ALOOKUPu_u_prefixu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_ls2, V_ls, V_k]: (s(t_h4s_options_option(TV_u_27b),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_ls),s(TV_u_27a,V_k))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_v))) => s(t_h4s_options_option(TV_u_27b),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_append(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_ls),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_ls2))),s(TV_u_27a,V_k))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_v))))).
fof(ah4s_alists_ALOOKUPu_u_prefixu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_ls2, V_ls, V_k]: (s(t_h4s_options_option(TV_u_27b),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_ls),s(TV_u_27a,V_k))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none) => s(t_h4s_options_option(TV_u_27b),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_append(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_ls),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_ls2))),s(TV_u_27a,V_k))) = s(t_h4s_options_option(TV_u_27b),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_ls2),s(TV_u_27a,V_k))))).
fof(ch4s_alists_ALOOKUPu_u_APPEND, conjecture, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_v]: s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27b)),V_uu_0),s(TV_u_27b,V_v))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_v))) => ![V_l2, V_l1, V_k]: s(t_h4s_options_option(TV_u_27b),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_append(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l1),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l2))),s(TV_u_27a,V_k))) = s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27b),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l1),s(TV_u_27a,V_k))),s(t_h4s_options_option(TV_u_27b),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l2),s(TV_u_27a,V_k))),s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27b)),V_uu_0))))).
