%   ORIGINAL: h4/bag/FINITE__BAG__DIFF
% 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/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bag/BAG__DELETE0: !e b0 b. h4/bag/BAG__DELETE b0 e b <=> b0 = h4/bag/BAG__INSERT e b
% Assm: h4/bag/BAG__IN__BAG__DELETE: !e b. h4/bag/BAG__IN e b ==> (?b_27. h4/bag/BAG__DELETE b e b_27)
% Assm: h4/bag/BAG__DIFF__EMPTY_c2: !b. h4/bag/BAG__DIFF h4/bag/EMPTY__BAG b = h4/bag/EMPTY__BAG
% Assm: h4/bag/BAG__DIFF__INSERT__same: !x b2 b1. h4/bag/BAG__DIFF (h4/bag/BAG__INSERT x b1) (h4/bag/BAG__INSERT x b2) = h4/bag/BAG__DIFF b1 b2
% Assm: h4/bag/BAG__DIFF__INSERT: !x b2 b1. ~h4/bag/BAG__IN x b1 ==> h4/bag/BAG__DIFF (h4/bag/BAG__INSERT x b2) b1 = h4/bag/BAG__INSERT x (h4/bag/BAG__DIFF b2 b1)
% Assm: h4/bag/STRONG__FINITE__BAG__INDUCT: !P. P h4/bag/EMPTY__BAG /\ (!b. h4/bag/FINITE__BAG b /\ P b ==> (!e. P (h4/bag/BAG__INSERT e b))) ==> (!b. h4/bag/FINITE__BAG b ==> P b)
% Assm: h4/bag/FINITE__BAG__THM_c0: h4/bag/FINITE__BAG h4/bag/EMPTY__BAG
% Assm: h4/bag/FINITE__BAG__THM_c1: !e b. h4/bag/FINITE__BAG (h4/bag/BAG__INSERT e b) <=> h4/bag/FINITE__BAG b
% Goal: !b1. h4/bag/FINITE__BAG b1 ==> (!b2. h4/bag/FINITE__BAG (h4/bag/BAG__DIFF b1 b2))
%   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_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~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_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bags_BAGu_u_DELETE0]: !e b0 b. h4/bag/BAG__DELETE b0 e b <=> b0 = h4/bag/BAG__INSERT e b
% Assm [h4s_bags_BAGu_u_INu_u_BAGu_u_DELETE]: !e b. h4/bag/BAG__IN e b ==> (?b_27. h4/bag/BAG__DELETE b e b_27)
% Assm [h4s_bags_BAGu_u_DIFFu_u_EMPTYu_c2]: !b. h4/bag/BAG__DIFF h4/bag/EMPTY__BAG b = h4/bag/EMPTY__BAG
% Assm [h4s_bags_BAGu_u_DIFFu_u_INSERTu_u_same]: !x b2 b1. h4/bag/BAG__DIFF (h4/bag/BAG__INSERT x b1) (h4/bag/BAG__INSERT x b2) = h4/bag/BAG__DIFF b1 b2
% Assm [h4s_bags_BAGu_u_DIFFu_u_INSERT]: !x b2 b1. ~h4/bag/BAG__IN x b1 ==> h4/bag/BAG__DIFF (h4/bag/BAG__INSERT x b2) b1 = h4/bag/BAG__INSERT x (h4/bag/BAG__DIFF b2 b1)
% Assm [h4s_bags_STRONGu_u_FINITEu_u_BAGu_u_INDUCT]: !P. happ P h4/bag/EMPTY__BAG /\ (!b. h4/bag/FINITE__BAG b /\ happ P b ==> (!e. happ P (h4/bag/BAG__INSERT e b))) ==> (!b. h4/bag/FINITE__BAG b ==> happ P b)
% Assm [h4s_bags_FINITEu_u_BAGu_u_THMu_c0]: h4/bag/FINITE__BAG h4/bag/EMPTY__BAG
% 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
% Goal: !b1. h4/bag/FINITE__BAG b1 ==> (!b2. h4/bag/FINITE__BAG (h4/bag/BAG__DIFF b1 b2))
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_Q230609,TV_Q230605]: ![V_f, V_g]: (![V_x]: s(TV_Q230605,happ(s(t_fun(TV_Q230609,TV_Q230605),V_f),s(TV_Q230609,V_x))) = s(TV_Q230605,happ(s(t_fun(TV_Q230609,TV_Q230605),V_g),s(TV_Q230609,V_x))) => s(t_fun(TV_Q230609,TV_Q230605),V_f) = s(t_fun(TV_Q230609,TV_Q230605),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_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (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_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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> 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_bags_BAGu_u_DELETE0, axiom, ![TV_u_27a]: ![V_e, V_b0, V_b]: (p(s(t_bool,h4s_bags_bagu_u_delete(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b0),s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) <=> s(t_fun(TV_u_27a,t_h4s_nums_num),V_b0) = 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))))).
fof(ah4s_bags_BAGu_u_INu_u_BAGu_u_DELETE, axiom, ![TV_u_27a]: ![V_e, V_b]: (p(s(t_bool,h4s_bags_bagu_u_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) => ?[V_bu_27]: p(s(t_bool,h4s_bags_bagu_u_delete(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b),s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_bu_27)))))).
fof(ah4s_bags_BAGu_u_DIFFu_u_EMPTYu_c2, axiom, ![TV_u_27c]: ![V_b]: s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_bags_bagu_u_diff(s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_bags_emptyu_u_bag),s(t_fun(TV_u_27c,t_h4s_nums_num),V_b))) = s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_bags_emptyu_u_bag)).
fof(ah4s_bags_BAGu_u_DIFFu_u_INSERTu_u_same, axiom, ![TV_u_27a]: ![V_x, V_b2, V_b1]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_diff(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_b1))),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_b2))))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_diff(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_DIFFu_u_INSERT, axiom, ![TV_u_27a]: ![V_x, V_b2, V_b1]: (~ (p(s(t_bool,h4s_bags_bagu_u_in(s(TV_u_27a,V_x),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_diff(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_b2))),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_x),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_diff(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1))))))).
fof(ah4s_bags_STRONGu_u_FINITEu_u_BAGu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)))) & ![V_b]: ((p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))))) => ![V_e]: p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_P),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)))))))) => ![V_b]: (p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))))))).
fof(ah4s_bags_FINITEu_u_BAGu_u_THMu_c0, axiom, ![TV_u_27a]: p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))))).
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(ch4s_bags_FINITEu_u_BAGu_u_DIFF, conjecture, ![TV_u_27a]: ![V_b1]: (p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1)))) => ![V_b2]: p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_diff(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2)))))))).
