%   ORIGINAL: h4/sptree/domain__inter
% 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/EXISTS__SIMP: !t. (?x. t) <=> 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_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/EXISTS__REFL: !a. ?x. x = a
% 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/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/pred__set/IN__INTER: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm: h4/sptree/lookup__inter: !m2 m1 k. h4/sptree/lookup k (h4/sptree/inter m1 m2) = h4/pair/pair__CASE (h4/pair/_2C (h4/sptree/lookup k m1) (h4/sptree/lookup k m2)) (\v3 v4. h4/option/option__CASE v3 h4/option/NONE (\v. h4/option/option__CASE v4 h4/option/NONE (\w. h4/option/SOME v)))
% Assm: h4/sptree/domain__lookup: !t k. h4/bool/IN k (h4/sptree/domain t) <=> (?v. h4/sptree/lookup k t = h4/option/SOME v)
% Goal: !t2 t1. h4/sptree/domain (h4/sptree/inter t1 t2) = h4/pred__set/INTER (h4/sptree/domain t1) (h4/sptree/domain t2)
%   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_EXISTSu_u_SIMP]: !t. (?x. t) <=> 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_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_EXISTSu_u_REFL]: !a. ?x. x = a
% 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_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_predu_u_sets_INu_u_INTER]: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm [h4s_sptrees_lookupu_u_inter]: !_3. (!v w. happ (happ _3 v) w = h4/option/SOME v) ==> (!_2. (!v4 v. happ (happ _2 v4) v = h4/option/option__CASE v4 h4/option/NONE (happ _3 v)) ==> (!_1. (!v3 v4. happ (happ _1 v3) v4 = h4/option/option__CASE v3 h4/option/NONE (happ _2 v4)) ==> (!_0. (!v3. happ _0 v3 = happ _1 v3) ==> (!m2 m1 k. h4/sptree/lookup k (h4/sptree/inter m1 m2) = h4/pair/pair__CASE (h4/pair/_2C (h4/sptree/lookup k m1) (h4/sptree/lookup k m2)) _0))))
% Assm [h4s_sptrees_domainu_u_lookup]: !t k. h4/bool/IN k (h4/sptree/domain t) <=> (?v. h4/sptree/lookup k t = h4/option/SOME v)
% Goal: !t2 t1. h4/sptree/domain (h4/sptree/inter t1 t2) = h4/pred__set/INTER (h4/sptree/domain t1) (h4/sptree/domain t2)
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_Q82079,TV_Q82075]: ![V_f, V_g]: (![V_x]: s(TV_Q82075,happ(s(t_fun(TV_Q82079,TV_Q82075),V_f),s(TV_Q82079,V_x))) = s(TV_Q82075,happ(s(t_fun(TV_Q82079,TV_Q82075),V_g),s(TV_Q82079,V_x))) => s(t_fun(TV_Q82079,TV_Q82075),V_f) = s(t_fun(TV_Q82079,TV_Q82075),V_g))).
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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EXISTSu_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_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_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_EXISTSu_u_REFL, axiom, ![TV_u_27a]: ![V_a]: ?[V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_a)).
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_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_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_predu_u_sets_INu_u_INTER, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_sptrees_lookupu_u_inter, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_3]: (![V_v, V_w]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))),V_uu_3),s(TV_u_27a,V_v))),s(TV_u_27b,V_w))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_v))) => ![V_uu_2]: (![V_v4, V_v]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_options_option(TV_u_27b),t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))),V_uu_2),s(t_h4s_options_option(TV_u_27b),V_v4))),s(TV_u_27a,V_v))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27b),V_v4),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))),V_uu_3),s(TV_u_27a,V_v))))) => ![V_uu_1]: (![V_v3, V_v4]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_uu_1),s(t_h4s_options_option(TV_u_27a),V_v3))),s(t_h4s_options_option(TV_u_27b),V_v4))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_v3),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_options_option(TV_u_27b),t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))),V_uu_2),s(t_h4s_options_option(TV_u_27b),V_v4))))) => ![V_uu_0]: (![V_v3]: s(t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_uu_0),s(t_h4s_options_option(TV_u_27a),V_v3))) = s(t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_uu_1),s(t_h4s_options_option(TV_u_27a),V_v3))) => ![V_m2, V_m1, V_k]: 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),h4s_sptrees_inter(s(t_h4s_sptrees_spt(TV_u_27a),V_m1),s(t_h4s_sptrees_spt(TV_u_27b),V_m2))))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_pairs_u_2c(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_m1))),s(t_h4s_options_option(TV_u_27b),h4s_sptrees_lookup(s(t_h4s_nums_num,V_k),s(t_h4s_sptrees_spt(TV_u_27b),V_m2))))),s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_uu_0)))))))).
fof(ah4s_sptrees_domainu_u_lookup, axiom, ![TV_u_27a]: ![V_t, V_k]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_k),s(t_fun(t_h4s_nums_num,t_bool),h4s_sptrees_domain(s(t_h4s_sptrees_spt(TV_u_27a),V_t)))))) <=> ?[V_v]: 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_domainu_u_inter, conjecture, ![TV_u_27a,TV_u_27b]: ![V_t2, V_t1]: s(t_fun(t_h4s_nums_num,t_bool),h4s_sptrees_domain(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_inter(s(t_h4s_sptrees_spt(TV_u_27a),V_t1),s(t_h4s_sptrees_spt(TV_u_27b),V_t2))))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_inter(s(t_fun(t_h4s_nums_num,t_bool),h4s_sptrees_domain(s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_fun(t_h4s_nums_num,t_bool),h4s_sptrees_domain(s(t_h4s_sptrees_spt(TV_u_27b),V_t2)))))).
