%   ORIGINAL: h4/sptree/lookup__fromAList__toAList
% 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/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/sptree/ALOOKUP__toAList: !x t. h4/alist/ALOOKUP (h4/sptree/toAList t) x = h4/sptree/lookup x t
% Assm: h4/sptree/lookup__fromAList: !x ls. h4/sptree/lookup x (h4/sptree/fromAList ls) = h4/alist/ALOOKUP ls x
% Goal: !x t. h4/sptree/lookup x (h4/sptree/fromAList (h4/sptree/toAList t)) = h4/sptree/lookup x 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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_sptrees_ALOOKUPu_u_toAList]: !x t. h4/alist/ALOOKUP (h4/sptree/toAList t) x = h4/sptree/lookup x t
% Assm [h4s_sptrees_lookupu_u_fromAList]: !x ls. h4/sptree/lookup x (h4/sptree/fromAList ls) = h4/alist/ALOOKUP ls x
% Goal: !x t. h4/sptree/lookup x (h4/sptree/fromAList (h4/sptree/toAList t)) = h4/sptree/lookup x 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_Q84451,TV_Q84447]: ![V_f, V_g]: (![V_x]: s(TV_Q84447,happ(s(t_fun(TV_Q84451,TV_Q84447),V_f),s(TV_Q84451,V_x))) = s(TV_Q84447,happ(s(t_fun(TV_Q84451,TV_Q84447),V_g),s(TV_Q84451,V_x))) => s(t_fun(TV_Q84451,TV_Q84447),V_f) = s(t_fun(TV_Q84451,TV_Q84447),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
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_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_sptrees_ALOOKUPu_u_toAList, axiom, ![TV_u_27a]: ![V_x, V_t]: s(t_h4s_options_option(TV_u_27a),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),h4s_sptrees_toalist(s(t_h4s_sptrees_spt(TV_u_27a),V_t))),s(t_h4s_nums_num,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_sptrees_lookup(s(t_h4s_nums_num,V_x),s(t_h4s_sptrees_spt(TV_u_27a),V_t)))).
fof(ah4s_sptrees_lookupu_u_fromAList, axiom, ![TV_u_27a]: ![V_x, V_ls]: s(t_h4s_options_option(TV_u_27a),h4s_sptrees_lookup(s(t_h4s_nums_num,V_x),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_fromalist(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),V_ls))))) = s(t_h4s_options_option(TV_u_27a),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),V_ls),s(t_h4s_nums_num,V_x)))).
fof(ch4s_sptrees_lookupu_u_fromAListu_u_toAList, conjecture, ![TV_u_27a]: ![V_x, V_t]: s(t_h4s_options_option(TV_u_27a),h4s_sptrees_lookup(s(t_h4s_nums_num,V_x),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_fromalist(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),h4s_sptrees_toalist(s(t_h4s_sptrees_spt(TV_u_27a),V_t))))))) = s(t_h4s_options_option(TV_u_27a),h4s_sptrees_lookup(s(t_h4s_nums_num,V_x),s(t_h4s_sptrees_spt(TV_u_27a),V_t)))).
