%   ORIGINAL: h4/bag/mlt__INSERT__CANCEL__I
% 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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> 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/relation/TC__lifts__monotonicities: !f R. (!x y. R x y ==> R (f x) (f y)) ==> (!x y. h4/relation/TC R x y ==> h4/relation/TC R (f x) (f y))
% Assm: h4/bag/BAG__UNION__INSERT_c0: !e b2 b1. h4/bag/BAG__UNION (h4/bag/BAG__INSERT e b1) b2 = h4/bag/BAG__INSERT e (h4/bag/BAG__UNION b1 b2)
% Assm: h4/bag/BAG__UNION__INSERT_c1: !e b2 b1. h4/bag/BAG__UNION b1 (h4/bag/BAG__INSERT e b2) = h4/bag/BAG__INSERT e (h4/bag/BAG__UNION b1 b2)
% Assm: h4/bag/C__BAG__INSERT__ONE__ONE: !y x b. h4/bag/BAG__INSERT x b = h4/bag/BAG__INSERT y b <=> x = y
% Assm: h4/bag/BAG__UNION__EMPTY_c0: !b. h4/bag/BAG__UNION b h4/bag/EMPTY__BAG = b
% Assm: h4/bag/FINITE__BAG__THM_c1: !e b. h4/bag/FINITE__BAG (h4/bag/BAG__INSERT e b) <=> h4/bag/FINITE__BAG b
% Assm: h4/bag/mlt1__def: !r b2 b1. h4/bag/mlt1 r b1 b2 <=> h4/bag/FINITE__BAG b1 /\ h4/bag/FINITE__BAG b2 /\ (?e rep res. b1 = h4/bag/BAG__UNION rep res /\ b2 = h4/bag/BAG__UNION res (h4/bag/BAG__INSERT e h4/bag/EMPTY__BAG) /\ (!e_27. h4/bag/BAG__IN e_27 rep ==> r e_27 e))
% Goal: !e b a R. h4/relation/TC (h4/bag/mlt1 R) a b ==> h4/relation/TC (h4/bag/mlt1 R) (h4/bag/BAG__INSERT e a) (h4/bag/BAG__INSERT e b)
%   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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> 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_relations_TCu_u_liftsu_u_monotonicities]: !f R. (!x y. happ (happ R x) y ==> happ (happ R (happ f x)) (happ f y)) ==> (!x y. h4/relation/TC R x y ==> h4/relation/TC R (happ f x) (happ f y))
% Assm [h4s_bags_BAGu_u_UNIONu_u_INSERTu_c0]: !e b2 b1. h4/bag/BAG__UNION (h4/bag/BAG__INSERT e b1) b2 = h4/bag/BAG__INSERT e (h4/bag/BAG__UNION b1 b2)
% Assm [h4s_bags_BAGu_u_UNIONu_u_INSERTu_c1]: !e b2 b1. h4/bag/BAG__UNION b1 (h4/bag/BAG__INSERT e b2) = h4/bag/BAG__INSERT e (h4/bag/BAG__UNION b1 b2)
% Assm [h4s_bags_Cu_u_BAGu_u_INSERTu_u_ONEu_u_ONE]: !y x b. h4/bag/BAG__INSERT x b = h4/bag/BAG__INSERT y b <=> x = y
% Assm [h4s_bags_BAGu_u_UNIONu_u_EMPTYu_c0]: !b. h4/bag/BAG__UNION b h4/bag/EMPTY__BAG = b
% Assm [h4s_bags_FINITEu_u_BAGu_u_THMu_c1]: !e b. h4/bag/FINITE__BAG (h4/bag/BAG__INSERT e b) <=> h4/bag/FINITE__BAG b
% Assm [h4s_bags_mlt1u_u_def]: !r b2 b1. happ (happ (h4/bag/mlt1 r) b1) b2 <=> h4/bag/FINITE__BAG b1 /\ h4/bag/FINITE__BAG b2 /\ (?e rep res. b1 = h4/bag/BAG__UNION rep res /\ b2 = h4/bag/BAG__UNION res (h4/bag/BAG__INSERT e h4/bag/EMPTY__BAG) /\ (!e_27. h4/bag/BAG__IN e_27 rep ==> happ (happ r e_27) e))
% Goal: !e b a R. h4/relation/TC (h4/bag/mlt1 R) a b ==> h4/relation/TC (h4/bag/mlt1 R) (h4/bag/BAG__INSERT e a) (h4/bag/BAG__INSERT e b)
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_Q233104,TV_Q233100]: ![V_f, V_g]: (![V_x]: s(TV_Q233100,happ(s(t_fun(TV_Q233104,TV_Q233100),V_f),s(TV_Q233104,V_x))) = s(TV_Q233100,happ(s(t_fun(TV_Q233104,TV_Q233100),V_g),s(TV_Q233104,V_x))) => s(t_fun(TV_Q233104,TV_Q233100),V_f) = s(t_fun(TV_Q233104,TV_Q233100),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_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_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> 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_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_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,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_relations_TCu_u_liftsu_u_monotonicities, axiom, ![TV_u_27a]: ![V_f, V_R]: (![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x))))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_y))))))) => ![V_x, V_y]: (p(s(t_bool,h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x),s(TV_u_27a,V_y)))) => p(s(t_bool,h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_y))))))))).
fof(ah4s_bags_BAGu_u_UNIONu_u_INSERTu_c0, axiom, ![TV_u_27a]: ![V_e, V_b2, V_b1]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_union(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_union(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2)))))).
fof(ah4s_bags_BAGu_u_UNIONu_u_INSERTu_c1, axiom, ![TV_u_27a]: ![V_e, V_b2, V_b1]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_union(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2))))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_union(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2)))))).
fof(ah4s_bags_Cu_u_BAGu_u_INSERTu_u_ONEu_u_ONE, axiom, ![TV_u_27a]: ![V_y, V_x, V_b]: (s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_bags_BAGu_u_UNIONu_u_EMPTYu_c0, axiom, ![TV_u_27a]: ![V_b]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_union(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))) = s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)).
fof(ah4s_bags_FINITEu_u_BAGu_u_THMu_c1, axiom, ![TV_u_27a]: ![V_e, V_b]: s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))))) = s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))).
fof(ah4s_bags_mlt1u_u_def, axiom, ![TV_u_27a]: ![V_r, V_b2, V_b1]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool)),h4s_bags_mlt1(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2)))) <=> (p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1)))) & (p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2)))) & ?[V_e, V_rep, V_res]: (s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_union(s(t_fun(TV_u_27a,t_h4s_nums_num),V_rep),s(t_fun(TV_u_27a,t_h4s_nums_num),V_res))) & (s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_union(s(t_fun(TV_u_27a,t_h4s_nums_num),V_res),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))))) & ![V_eu_27]: (p(s(t_bool,h4s_bags_bagu_u_in(s(TV_u_27a,V_eu_27),s(t_fun(TV_u_27a,t_h4s_nums_num),V_rep)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r),s(TV_u_27a,V_eu_27))),s(TV_u_27a,V_e))))))))))).
fof(ch4s_bags_mltu_u_INSERTu_u_CANCELu_u_I, conjecture, ![TV_u_27a]: ![V_e, V_b, V_a, V_R]: (p(s(t_bool,h4s_relations_tc(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool)),h4s_bags_mlt1(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_a),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) => p(s(t_bool,h4s_relations_tc(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool)),h4s_bags_mlt1(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_a))),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))))))).
