%   ORIGINAL: h4/sptree/ALOOKUP__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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/DISJ__COMM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/alist/ALOOKUP__FAILS: !x l. h4/alist/ALOOKUP l x = h4/option/NONE <=> (!k v. h4/bool/IN (h4/pair/_2C k v) (h4/list/LIST__TO__SET l) ==> ~(k = x))
% Assm: h4/alist/ALOOKUP__MEM: !v k al. h4/alist/ALOOKUP al k = h4/option/SOME v ==> h4/bool/IN (h4/pair/_2C k v) (h4/list/LIST__TO__SET al)
% Assm: h4/sptree/MEM__toAList: !v t k. h4/bool/IN (h4/pair/_2C k v) (h4/list/LIST__TO__SET (h4/sptree/toAList t)) <=> h4/sptree/lookup k t = h4/option/SOME v
% Goal: !x t. h4/alist/ALOOKUP (h4/sptree/toAList t) x = 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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_DISJu_u_COMM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_alists_ALOOKUPu_u_FAILS]: !x l. h4/alist/ALOOKUP l x = h4/option/NONE <=> (!k v. h4/bool/IN (h4/pair/_2C k v) (h4/list/LIST__TO__SET l) ==> ~(k = x))
% Assm [h4s_alists_ALOOKUPu_u_MEM]: !v k al. h4/alist/ALOOKUP al k = h4/option/SOME v ==> h4/bool/IN (h4/pair/_2C k v) (h4/list/LIST__TO__SET al)
% Assm [h4s_sptrees_MEMu_u_toAList]: !v t k. h4/bool/IN (h4/pair/_2C k v) (h4/list/LIST__TO__SET (h4/sptree/toAList t)) <=> h4/sptree/lookup k t = h4/option/SOME v
% Goal: !x t. h4/alist/ALOOKUP (h4/sptree/toAList t) x = 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_Q82319,TV_Q82315]: ![V_f, V_g]: (![V_x]: s(TV_Q82315,happ(s(t_fun(TV_Q82319,TV_Q82315),V_f),s(TV_Q82319,V_x))) = s(TV_Q82315,happ(s(t_fun(TV_Q82319,TV_Q82315),V_g),s(TV_Q82319,V_x))) => s(t_fun(TV_Q82319,TV_Q82315),V_f) = s(t_fun(TV_Q82319,TV_Q82315),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t0)))).
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_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_DISJu_u_COMM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_bools_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
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_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_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_alists_ALOOKUPu_u_FAILS, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_l]: (s(t_h4s_options_option(TV_u_27a),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_l),s(TV_u_27b,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> ![V_k, V_v]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27b,V_k),s(TV_u_27a,V_v))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_l)))))) => ~ (s(TV_u_27b,V_k) = s(TV_u_27b,V_x))))).
fof(ah4s_alists_ALOOKUPu_u_MEM, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_k, V_al]: (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_al),s(TV_u_27a,V_k))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_v))) => p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_k),s(TV_u_27b,V_v))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_al)))))))).
fof(ah4s_sptrees_MEMu_u_toAList, axiom, ![TV_u_27a]: ![V_v, V_t, V_k]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_nums_num,V_k),s(TV_u_27a,V_v))),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),t_bool),h4s_lists_listu_u_tou_u_set(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_k),s(t_h4s_sptrees_spt(TV_u_27a),V_t))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_v))))).
fof(ch4s_sptrees_ALOOKUPu_u_toAList, conjecture, ![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)))).
