0.02/0.08 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.02/0.08 % Command : tptp2X_and_run_prover9 %d %s 0.07/0.27 % Computer : n003.cluster.edu 0.07/0.27 % Model : x86_64 x86_64 0.07/0.27 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.07/0.27 % Memory : 8042.1875MB 0.07/0.27 % OS : Linux 3.10.0-693.el7.x86_64 0.07/0.27 % CPULimit : 1440 0.07/0.27 % WCLimit : 180 0.07/0.27 % DateTime : Mon Jul 3 07:51:01 EDT 2023 0.07/0.27 % CPUTime : 1.68/2.01 ============================== Prover9 =============================== 1.68/2.01 Prover9 (32) version 2009-11A, November 2009. 1.68/2.01 Process 25283 was started by sandbox2 on n003.cluster.edu, 1.68/2.01 Mon Jul 3 07:51:02 2023 1.68/2.01 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 1440 -f /tmp/Prover9_25129_n003.cluster.edu". 1.68/2.01 ============================== end of head =========================== 1.68/2.01 1.68/2.01 ============================== INPUT ================================= 1.68/2.01 1.68/2.01 % Reading from file /tmp/Prover9_25129_n003.cluster.edu 1.68/2.01 1.68/2.01 set(prolog_style_variables). 1.68/2.01 set(auto2). 1.68/2.01 % set(auto2) -> set(auto). 1.68/2.01 % set(auto) -> set(auto_inference). 1.68/2.01 % set(auto) -> set(auto_setup). 1.68/2.01 % set(auto_setup) -> set(predicate_elim). 1.68/2.01 % set(auto_setup) -> assign(eq_defs, unfold). 1.68/2.01 % set(auto) -> set(auto_limits). 1.68/2.01 % set(auto_limits) -> assign(max_weight, "100.000"). 1.68/2.01 % set(auto_limits) -> assign(sos_limit, 20000). 1.68/2.01 % set(auto) -> set(auto_denials). 1.68/2.01 % set(auto) -> set(auto_process). 1.68/2.01 % set(auto2) -> assign(new_constants, 1). 1.68/2.01 % set(auto2) -> assign(fold_denial_max, 3). 1.68/2.01 % set(auto2) -> assign(max_weight, "200.000"). 1.68/2.01 % set(auto2) -> assign(max_hours, 1). 1.68/2.01 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.68/2.01 % set(auto2) -> assign(max_seconds, 0). 1.68/2.01 % set(auto2) -> assign(max_minutes, 5). 1.68/2.01 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.68/2.01 % set(auto2) -> set(sort_initial_sos). 1.68/2.01 % set(auto2) -> assign(sos_limit, -1). 1.68/2.01 % set(auto2) -> assign(lrs_ticks, 3000). 1.68/2.01 % set(auto2) -> assign(max_megs, 400). 1.68/2.01 % set(auto2) -> assign(stats, some). 1.68/2.01 % set(auto2) -> clear(echo_input). 1.68/2.01 % set(auto2) -> set(quiet). 1.68/2.01 % set(auto2) -> clear(print_initial_clauses). 1.68/2.01 % set(auto2) -> clear(print_given). 1.68/2.01 assign(lrs_ticks,-1). 1.68/2.01 assign(sos_limit,10000). 1.68/2.01 assign(order,kbo). 1.68/2.01 set(lex_order_vars). 1.68/2.01 clear(print_given). 1.68/2.01 1.68/2.01 % formulas(sos). % not echoed (830 formulas) 1.68/2.01 1.68/2.01 ============================== end of input ========================== 1.68/2.01 1.68/2.01 % From the command line: assign(max_seconds, 1440). 1.68/2.01 1.68/2.01 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.68/2.01 1.68/2.01 % Formulas that are not ordinary clauses: 1.68/2.01 1 (all Ts all Ga all T_1 (hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,T_1),bot_bo1181479936a_bool))) -> (hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),Ts)) -> hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,T_1),Ts)))))) # label(fact_6_hoare__derivs_Oinsert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 2 (all A (bot_bot_fun_nat_bool = A <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),bot_bot_fun_nat_bool)))) # label(fact_457_subset__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 3 (all A_1 all B hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila447562797e_bool,collec727977250_state(hAPP_H1645666623e_bool(hAPP_f262880489e_bool(cOMBC_1424981238e_bool,fequal1531560888_state),A_1))),B)) # label(fact_573_insert__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 4 (all F_1 all A_1 hAPP_f124283079iple_a(finite1382394752iple_a(F_1),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool)) = A_1) # label(fact_421_fold1__singleton) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 5 (all F_1 all A (A = bot_bo1055319631e_bool <-> image_129517430iple_a(F_1,A) = bot_bo1181479936a_bool)) # label(fact_218_image__is__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 6 (all Y_21 all X_25 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_25),Y_21)))) # label(fact_488_inf__sup__ord_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 7 (all P all Q (hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) | -hBOOL(Q))) # label(help_fimplies_2_1_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 8 (all Pa all A all F (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,F)) -> (hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,F),A)) -> (hBOOL(hAPP_f1760790145l_bool(Pa,bot_bo1055319631e_bool)) -> ((all A_7 all F_2 (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,F_2)) -> (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_7),A)) -> (-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_7),F_2)) -> (hBOOL(hAPP_f1760790145l_bool(Pa,F_2)) -> hBOOL(hAPP_f1760790145l_bool(Pa,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_7),F_2)))))))) -> hBOOL(hAPP_f1760790145l_bool(Pa,F))))))) # label(fact_596_finite__subset__induct) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 9 (all F_1 all A (bot_bot_fun_nat_bool = A <-> image_1782127643iple_a(F_1,A) = bot_bo1181479936a_bool)) # label(fact_227_empty__is__image) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 10 (all X_1 all A all F_1 all F (hBOOL(finite1669978781iple_a(F_1,F)) -> (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> (bot_bo1181479936a_bool != A -> hAPP_f124283079iple_a(F,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A)) = hAPP_H678412245iple_a(hAPP_H1455657330iple_a(F_1,X_1),hAPP_f124283079iple_a(F,A))))))) # label(fact_308_folding__one_Oinsert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 11 (all A all B hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),B) = collec268032053iple_a(hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fconj),hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,member1713797107iple_a),A))),hAPP_f1591852335a_bool(hAPP_f34030599a_bool(cOMBB_1799513916iple_a,fNot),hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,member1713797107iple_a),B))))) # label(fact_362_set__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 12 (all F_1 all X_1 all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,hAPP_H1600811558iple_a(F_1,X_1)),image_129517430iple_a(F_1,A))))) # label(fact_205_imageI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 13 (all A_1 hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool) = collec268032053iple_a(hAPP_H1190454433a_bool(fequal879838495iple_a,A_1))) # label(fact_27_singleton__conv2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 14 (all B all X_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),B)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),B))))) # label(fact_531_set__rev__mp) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 15 (all U all F_1 ((all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(F_1,N_1)))) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_less_eq_nat),F_1)),U)))))) # label(fact_606_finite__less__ub) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 16 (all C_6 all B_9 all A_13 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_9),A_13)) -> (C_6 = B_9 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_6),A_13))))) # label(fact_519_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 17 (all P all Q all R hAPP_b2019457360e_bool(hAPP_a723219176e_bool(P,R),Q) = hAPP_a2036067514e_bool(hAPP_b540892988e_bool(hAPP_f1824947087e_bool(cOMBC_41962815e_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000t__a_000tc__HOL__Obool_000tc__fun_Itc__Com__Ostate_Mtc__) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 18 (all P all Q (hBOOL(Q) | hBOOL(P) | -hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)))) # label(help_fdisj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 19 (all B all A_1 hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)))) # label(fact_462_subset__insertI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 20 (all Loc_3 all Loc_2 (is_loc(Loc_3) & is_loc(Loc_2) -> (loc(Loc_3) = loc(Loc_2) <-> Loc_3 = Loc_2))) # label(fact_252_vname_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 21 (all P all Q all R hAPP_f800510211t_bool(hAPP_f1730770594t_bool(P,R),Q) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(hAPP_f1311642927t_bool(cOMBC_646315179t_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__018) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 22 (all A_9 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,bot_bot_nat),A_9))) # label(fact_551_bot__least) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 23 (all X_1 all Y_7 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_7),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_7)) <-> X_1 = Y_7)) # label(fact_525_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 24 (all Ba all F_1 all X_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> (Ba = hAPP_H678412245iple_a(F_1,X_1) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ba),image_306007685iple_a(F_1,A)))))) # label(fact_195_rev__image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 25 (all X_2 all Xa hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_2),Xa) = collec268032053iple_a(hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fdisj),hAPP_H1190454433a_bool(hAPP_f1371755681a_bool(cOMBC_839220826a_bool,fequal879838495iple_a),X_2))),hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,member1713797107iple_a),Xa)))) # label(fact_99_insert__compr__raw) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 26 (all B all Ca all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),A)) -> (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),B)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),B)))))) # label(fact_353_DiffI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 27 (all Ca all A ((A = bot_bo1055319631e_bool -> image_650584225_state(hAPP_H521649881_state(cOMBK_690957994_state,Ca),A) = bot_bo1055319631e_bool) & (A != bot_bo1055319631e_bool -> image_650584225_state(hAPP_H521649881_state(cOMBK_690957994_state,Ca),A) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Ca),bot_bo1055319631e_bool)))) # label(fact_166_image__constant__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 28 (all A_1 all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),A)) -> (exists B_15 (A = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B_15) & -hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),B_15)))))) # label(fact_141_mk__disjoint__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 29 (all B all X_1 all A (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),B)) <-> hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),B)))))) # label(fact_470_subset__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 30 (all N all H all F_1 all F (hBOOL(finite950012314iple_a(F_1,F)) -> ((all X_2 all Y_11 hAPP_H678412245iple_a(hAPP_H1455657330iple_a(F_1,hAPP_H678412245iple_a(H,X_2)),hAPP_H678412245iple_a(H,Y_11)) = hAPP_H678412245iple_a(H,hAPP_H678412245iple_a(hAPP_H1455657330iple_a(F_1,X_2),Y_11))) -> (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,N)) -> (N != bot_bo1181479936a_bool -> hAPP_H678412245iple_a(H,hAPP_f124283079iple_a(F,N)) = hAPP_f124283079iple_a(F,image_306007685iple_a(H,N))))))) # label(fact_402_folding__one__idem_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 31 (all A_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) -> A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A))) # label(fact_68_insert__absorb) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 32 (all N_3 all Pa all Ca all Q_1 (hBOOL(hAPP_H513860823e_bool(hoare_592710359_state(N_3),hoare_1575745797_state(Pa,Ca,Q_1))) <-> (all Z_7 all S_2 (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_7),S_2)) -> (all S_3 (hBOOL(evaln(Ca,S_2,N_3,S_3)) -> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_1,Z_7),S_3)))))))) # label(fact_304_triple__valid__def2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 33 (all Pa ((all X_2 -hBOOL(hAPP_nat_bool(Pa,X_2))) <-> bot_bot_fun_nat_bool = collect_nat(Pa))) # label(fact_46_Collect__empty__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 34 (all A A = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila447562797e_bool,A),bot_bo1055319631e_bool)) # label(fact_507_Un__empty__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 35 (all P all Q all R hAPP_v594194232_state(hAPP_s712361723_state(P,R),Q) = hAPP_s1892499976_state(hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__Com__Ovname_000tc__fun_Itc__Nat__) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 36 (all F_1 all X_1 all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_H716259088te_nat(F_1,X_1)),image_1410872416te_nat(F_1,A))))) # label(fact_203_imageI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 37 (all F_1 all A all X_1 (hBOOL(hAPP_H513860823e_bool(finite9525415_state(F_1,A),X_1)) -> bot_bo1055319631e_bool != A)) # label(fact_263_fold1Set__nonempty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 38 (all P all Q all R hAPP_f162060345e_bool(P,hAPP_a2036067514e_bool(Q,R)) = hAPP_a1224971408e_bool(hAPP_f1406200875e_bool(hAPP_f635443597e_bool(cOMBB_2139825703bool_a,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_022) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 39 (all P all Q all R hAPP_bool_bool(P,hAPP_state_bool(Q,R)) = hAPP_state_bool(hAPP_f1759915619e_bool(hAPP_f2073279419e_bool(cOMBB_160679318_state,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Ostate_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 40 (all A_1 all B hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B) = collec268032053iple_a(hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fdisj),hAPP_H1190454433a_bool(hAPP_f1371755681a_bool(cOMBC_839220826a_bool,fequal879838495iple_a),A_1))),hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,member1713797107iple_a),B)))) # label(fact_92_insert__compr) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 41 (all B all A_1 all A (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),A)) -> hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B)) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),B))) # label(fact_653_Int__insert__right__if0) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 42 (all B all Ca all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B)))))) # label(fact_622_IntI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 43 (all B_1_1 all B_2_1 all B_3_1 all B_4_1 is_bool(evaln(B_1_1,B_2_1,B_3_1,B_4_1))) # label(gsy_c_Natural_Oevaln) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 44 (all Ts all Ga (hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,Ts),Ga)) -> hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),Ts)))) # label(fact_570_asm) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 45 (all B all A_1 all A ((-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),A)) -> hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B)) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,A),B)) & (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),A)) -> hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B)) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,A),B))))) # label(fact_661_Int__insert__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 46 (all P all Q hAPP_n178040171iple_a(hAPP_H1450464520iple_a(cOMBK_1505147640_a_nat,P),Q) = P) # label(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_It__a_J_00) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 47 (all B all A all F_1 all F (hBOOL(finite795500164em_nat(F_1,F)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (bot_bot_fun_nat_bool != B -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) -> hAPP_f22106695ol_nat(F,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,hAPP_f22106695ol_nat(F,B)),hAPP_f22106695ol_nat(F,A))))))) # label(fact_442_folding__one__idem_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 48 (all Pa all F (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,F)) -> (F != bot_bo1181479936a_bool -> ((all X_2 hBOOL(hAPP_f540970102l_bool(Pa,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_2),bot_bo1181479936a_bool)))) -> ((all X_2 all F_2 (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,F_2)) -> (bot_bo1181479936a_bool != F_2 -> (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_2),F_2)) -> (hBOOL(hAPP_f540970102l_bool(Pa,F_2)) -> hBOOL(hAPP_f540970102l_bool(Pa,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_2),F_2)))))))) -> hBOOL(hAPP_f540970102l_bool(Pa,F))))))) # label(fact_343_finite__ne__induct) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 49 (all X_15 all A_6 all B_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_15),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_6),B_5))) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_15),A_6)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_15),B_5))))) # label(fact_626_le__infE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 50 (all X_1 (hBOOL(X_1) <-> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,bot_bot_bool),X_1)))) # label(fact_499_sup__bot__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 51 (all B_1 all A_2 all X_9 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_2),X_9)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_2),B_1)),X_9)))) # label(fact_633_le__infI1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 52 (all F_1 all A (image_194810223_a_nat(F_1,A) = bot_bot_fun_nat_bool <-> A = bot_bo1181479936a_bool)) # label(fact_229_empty__is__image) # label(axiom) # label(non_clause). [assumption]. 1.68/2.01 53 (all B all A all F_1 all F (hBOOL(finite950012314iple_a(F_1,F)) -> (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> (bot_bo1181479936a_bool != B -> (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,B),A)) -> hAPP_f124283079iple_a(F,A) = hAPP_H678412245iple_a(hAPP_H1455657330iple_a(F_1,hAPP_f124283079iple_a(F,B)),hAPP_f124283079iple_a(F,A))))))) # label(fact_444_folding__one__idem_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 54 (all X_1 all Y_7 all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_7),Z_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Z_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_1),Y_7)),Z_2)))) # label(fact_485_le__sup__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 55 (all X_1 (hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_1)) <-> hBOOL(bot_bot_bool))) # label(fact_125_bot__apply) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 56 (all P all Q all R hAPP_state_bool(hAPP_f1759915619e_bool(hAPP_f644196280e_bool(cOMBS_1378840469l_bool,P),Q),R) = hAPP_bool_bool(hAPP_s58564346l_bool(P,R),hAPP_state_bool(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 57 (all Ba all A all B (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),B)) -> hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ba),B))))) # label(fact_473_subset__insertI2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 58 (all X_1 all A (bot_bot_fun_nat_bool != A -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),A)))))) # label(fact_417_fold1__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 59 (all Ga all Ca all Q_1 all Pa ((all Z_7 all S_2 (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_7),S_2)) -> hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(hAPP_f1863945078e_bool(cOMBK_631994958_state,hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,fequal_state),S_2)),Ca,hAPP_f1863945078e_bool(cOMBK_631994958_state,hAPP_s1806633685e_bool(Q_1,Z_7)))),bot_bo1055319631e_bool))))) -> hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(Pa,Ca,Q_1)),bot_bo1055319631e_bool))))) # label(fact_11_escape) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 60 (all F_1 all A_1 all B image_1782127643iple_a(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hAPP_n178040171iple_a(F_1,A_1)),image_1782127643iple_a(F_1,B))) # label(fact_239_image__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 61 (all A_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),A)) -> hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),A) = A)) # label(fact_67_insert__absorb) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 62 (all A all Ba all F_1 all X_1 (Ba = hAPP_nat_nat(F_1,X_1) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_nat_nat(F_1,A)))))) # label(fact_179_image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 63 (all B hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,bot_bo1181479936a_bool),B) = bot_bo1181479936a_bool) # label(fact_685_Int__empty__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 64 (all Ba all A_1 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_1),Ba)),hAPP_f22106695ol_nat(finite_fold_nat_nat(semila972727038up_nat,Ba),A)))))) # label(fact_456_sup__le__fold__sup) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 65 (all Pa ((all X_2 -hBOOL(hAPP_H1421470952a_bool(Pa,X_2))) <-> bot_bo1181479936a_bool = collec268032053iple_a(Pa))) # label(fact_47_Collect__empty__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 66 (all A_1 all A (bot_bot_fun_nat_bool = A -> -hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)))) # label(fact_45_equals0D) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 67 (all A all X_1 all B (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) -> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B))) # label(fact_374_insert__Diff1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 68 (all A_1 all C all D_2 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C),D_2)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),C)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),D_2))))) # label(fact_474_insert__mono) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 69 (all A all B (hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B) = bot_bot_fun_nat_bool <-> A = bot_bot_fun_nat_bool & bot_bot_fun_nat_bool = B)) # label(fact_509_Un__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 70 (all B all A_1 all C (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),C)) -> hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,B),C)) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B)),C))) # label(fact_643_Int__insert__left__if1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 71 (all B bot_bot_fun_nat_bool = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,bot_bot_fun_nat_bool),B)) # label(fact_683_Int__empty__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 72 (all Ca all A all B (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),B))) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),A)))) # label(fact_673_IntD1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 73 (all A all X_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))) -> bot_bot_fun_nat_bool = A | A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))) # label(fact_582_subset__singletonD) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 74 (all X all Y (-hBOOL(hAPP_H513860823e_bool(hAPP_H1645666623e_bool(fequal1531560888_state,X),Y)) | Y = X)) # label(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_Itc__Com) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 75 (all A all Ba ((hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ba),bot_bo1181479936a_bool))) -> Ba = hAPP_f124283079iple_a(partia1866111638iple_a(Ba),A)) & (-hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ba),bot_bo1181479936a_bool))) -> hAPP_f124283079iple_a(the_Ho746640593iple_a,hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,member1713797107iple_a),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ba),bot_bo1181479936a_bool)))) = hAPP_f124283079iple_a(partia1866111638iple_a(Ba),A)))) # label(fact_602_flat__lub__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 76 (all Z_2 all F_1 all A ((exists X_2 (hAPP_H678412245iple_a(F_1,X_2) = Z_2 & hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_2),A)))) <-> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Z_2),image_306007685iple_a(F_1,A))))) # label(fact_211_image__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 77 (all X_2 (hBOOL(hAPP_H1421470952a_bool(bot_bo1181479936a_bool,X_2)) <-> hBOOL(bot_bot_bool))) # label(fact_127_bot__fun__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 78 (all X_1 X_1 = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila447562797e_bool,bot_bo1055319631e_bool),X_1)) # label(fact_501_sup__bot__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 79 (all P all Q all R hAPP_H727730819e_bool(hAPP_f531275309e_bool(hAPP_f1470644835e_bool(cOMBB_946224375_state,P),Q),R) = hAPP_f1583986009e_bool(P,hAPP_H1645666623e_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____ddpglwnxwg__Otriple_I_033) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 80 (all P all Q all R hAPP_b589554111l_bool(P,hAPP_H1421470952a_bool(Q,R)) = hAPP_H1017515220l_bool(hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_014) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 81 (all Q_1 all Pa (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,collec268032053iple_a(Q_1))) | hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,collec268032053iple_a(Pa))) -> hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,collec268032053iple_a(hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fconj),Pa)),Q_1)))))) # label(fact_309_finite__Collect__conjI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 82 (all C_2 all S_1 all T (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C_2),S_1),T)) -> (exists N_1 hBOOL(evaln(C_2,S_1,N_1,T))))) # label(fact_298_evalc__evaln) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 83 (all A_1 all Ba all A all X_1 (hBOOL(hAPP_nat_bool(finite929467206at_nat(times_times_nat,Ba,A),X_1)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),A)) -> hBOOL(hAPP_nat_bool(finite929467206at_nat(times_times_nat,A_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)))),X_1)))))) # label(fact_410_fold__graph__permute__diff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 84 (all X_1 all F_1 all F (hBOOL(finite774711482_state(F_1,F)) -> hAPP_f2143211163_state(F,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),bot_bo1055319631e_bool)) = X_1)) # label(fact_247_folding__one_Osingleton) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 85 (all F_1 image_129517430iple_a(F_1,bot_bo1055319631e_bool) = bot_bo1181479936a_bool) # label(fact_224_image__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 86 (all A_1 (A_1 = bot_bo1181479936a_bool | (exists A_19 exists A_7 (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A_19)) & A_1 = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_7),A_19))) <-> hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A_1)))) # label(fact_336_finite_Osimps) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 87 (all R_1 all S (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,R_1),S)) <-> hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,member1713797107iple_a),R_1)),hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,member1713797107iple_a),S))))) # label(fact_543_pred__subset__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 88 (all A all Ba all F_1 all X_1 (Ba = hAPP_H1600811558iple_a(F_1,X_1) -> (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ba),image_129517430iple_a(F_1,A)))))) # label(fact_183_image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 89 (all B_1_1 is_bool(finite419198954a_bool(B_1_1))) # label(gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____ddpglwnxwg__Otri) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 90 (all F_1 all A_1 hAPP_f2143211163_state(finite1346402327_state(F_1),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),bot_bo1055319631e_bool)) = A_1) # label(fact_420_fold1__singleton) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 91 (all X_6 all Y_4 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_6),Y_4)),X_6))) # label(fact_638_inf__le1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 92 (all P all Q all R hAPP_H513860823e_bool(hAPP_f1456107715e_bool(hAPP_f1548785833e_bool(cOMBB_1653402815_state,P),Q),R) = hAPP_f1760790145l_bool(P,hAPP_H1645666623e_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____ddpglwnxwg__Otriple_I_026) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 93 (all B_1_1 (is_loc(B_1_1) -> is_vname(loc(B_1_1)))) # label(gsy_c_Com_Ovname_OLoc) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 94 (all X_35 all A_1 all S_4 all N_3 hBOOL(evaln(ass(X_35,A_1),S_4,N_3,hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S_4),X_35),hAPP_state_nat(A_1,S_4))))) # label(fact_280_evaln_OAssign) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 95 (all Pa all F (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F)) -> (bot_bot_fun_nat_bool != F -> ((all X_2 hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),bot_bot_fun_nat_bool)))) -> ((all X_2 all F_2 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2)) -> (bot_bot_fun_nat_bool != F_2 -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),F_2)) -> (hBOOL(hAPP_f54304608l_bool(Pa,F_2)) -> hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),F_2)))))))) -> hBOOL(hAPP_f54304608l_bool(Pa,F))))))) # label(fact_341_finite__ne__induct) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 96 (all Z_2 all F_1 all A ((exists X_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) & hAPP_n178040171iple_a(F_1,X_2) = Z_2)) <-> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Z_2),image_1782127643iple_a(F_1,A))))) # label(fact_213_image__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 97 (all N (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N)) <-> (exists M all X_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),M)))))) # label(fact_605_finite__nat__set__iff__bounded__le) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 98 (all A_1 all Ba (hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),bot_bo1055319631e_bool) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Ba),bot_bo1055319631e_bool) -> Ba = A_1)) # label(fact_105_singleton__inject) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 99 (all Y_9 all X_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_9),X_12)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_12),Y_9) = Y_9)) # label(fact_630_inf__absorb2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 100 (all F_1 all X_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(F_1,X_1)),image_nat_nat(F_1,A))))) # label(fact_201_imageI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 101 (all B all X_1 all A (-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> (-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),B)) -> (hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),A) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),B) <-> B = A)))) # label(fact_75_insert__ident) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 102 (all A ((all X_2 -hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_2),A))) <-> bot_bo1055319631e_bool = A)) # label(fact_61_all__not__in__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 103 (all Ca all A ((A = bot_bo1055319631e_bool -> bot_bo1181479936a_bool = image_129517430iple_a(hAPP_H377435237iple_a(cOMBK_2022976521_state,Ca),A)) & (bot_bo1055319631e_bool != A -> hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ca),bot_bo1181479936a_bool) = image_129517430iple_a(hAPP_H377435237iple_a(cOMBK_2022976521_state,Ca),A)))) # label(fact_162_image__constant__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 104 (all F_1 image_194810223_a_nat(F_1,bot_bo1181479936a_bool) = bot_bot_fun_nat_bool) # label(fact_223_image__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 105 (all Y_7 all A all X_1 (hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_7),A),X_1)) <-> Y_7 = X_1 | hBOOL(hAPP_nat_bool(A,X_1)))) # label(fact_77_insert__code) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 106 (all A_1 all Ba all Ca all D_3 (hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Ca),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,D_3),bot_bo1055319631e_bool)) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Ba),bot_bo1055319631e_bool)) <-> D_3 = Ba & Ca = A_1 | A_1 = D_3 & Ba = Ca)) # label(fact_111_doubleton__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 107 (all Z_4 all X_19 all Y_15 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_19),Y_15)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_15),Z_4)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_19),Z_4))))) # label(fact_517_order__trans) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 108 (all P all Q all R hAPP_f746301080e_bool(hAPP_a1224971408e_bool(P,R),Q) = hAPP_a2036067514e_bool(hAPP_f100967412e_bool(hAPP_f1460451647e_bool(cOMBC_524597097e_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__Com__Ostate_J_000t) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 109 (all P all Q all R hAPP_bool_bool(P,hAPP_H1421470952a_bool(Q,R)) = hAPP_H1421470952a_bool(hAPP_f1591852335a_bool(hAPP_f34030599a_bool(cOMBB_1799513916iple_a,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Hoare____Mirabel) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 110 (all Ba all A_1 (A_1 = Ba <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool))))) # label(fact_113_singleton__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 111 (all Pa all A_1 ((-hBOOL(hAPP_H513860823e_bool(Pa,A_1)) -> collec727977250_state(hAPP_f921536533e_bool(hAPP_f760664097e_bool(cOMBS_1248383340l_bool,hAPP_f1558728829l_bool(hAPP_f1442918689l_bool(cOMBB_1382207997_state,fconj),hAPP_H1645666623e_bool(fequal1531560888_state,A_1))),Pa)) = bot_bo1055319631e_bool) & (hBOOL(hAPP_H513860823e_bool(Pa,A_1)) -> hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),bot_bo1055319631e_bool) = collec727977250_state(hAPP_f921536533e_bool(hAPP_f760664097e_bool(cOMBS_1248383340l_bool,hAPP_f1558728829l_bool(hAPP_f1442918689l_bool(cOMBB_1382207997_state,fconj),hAPP_H1645666623e_bool(fequal1531560888_state,A_1))),Pa))))) # label(fact_38_Collect__conv__if2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 112 (all R_1 all S all X_2 (hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),R_1)),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S)),X_2)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,R_1),S))))) # label(fact_664_inf__Int__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 113 (all X_1 all Y_7 (bot_bot_fun_nat_bool = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_1),Y_7) <-> bot_bot_fun_nat_bool = X_1 & Y_7 = bot_bot_fun_nat_bool)) # label(fact_492_sup__eq__bot__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 114 (all B_1_1 all B_2_1 is_bool(hAPP_f1760790145l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____ddpglwnxwg__Otriple_Itc__Com__O) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 115 (all Z_6 all Y_25 all X_32 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_25),X_32)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_6),X_32)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,Y_25),Z_6)),X_32))))) # label(fact_479_sup__least) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 116 (all A all B (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))))) # label(fact_373_finite__Diff2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 117 (all Y_7 all X_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_7),X_1)) -> (X_1 = Y_7 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_7))))) # label(fact_523_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 118 (all X_1 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> (A != bot_bot_fun_nat_bool -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_1),hAPP_f22106695ol_nat(big_la43341705in_nat,A)) = hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)))))) # label(fact_612_Sup__fin_Oinsert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 119 (all A all F_1 all F (hBOOL(finite988810631ne_nat(F_1,F)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hAPP_f22106695ol_nat(F,A) = hAPP_f22106695ol_nat(finite_fold1_nat(F_1),A)))) # label(fact_427_folding__one_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 120 (all X_1 bot_bo1181479936a_bool = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,bot_bo1181479936a_bool),X_1)) # label(fact_689_inf__bot__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 121 (all P all Q all R hAPP_H1641355846a_bool(hAPP_f485051996a_bool(hAPP_f2085120383a_bool(cOMBB_2080540641iple_a,P),Q),R) = hAPP_f1026156344a_bool(P,hAPP_H1190454433a_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____ddpglwnxwg__Otriple_I_032) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 122 (all A_1 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(semila972727038up_nat,A_1),A))) # label(fact_615_Sup__fin_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 123 (all A_1 all A (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A)) -> hBOOL(hAPP_f1760790145l_bool(finite784854244_state,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),A))))) # label(fact_315_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 124 (all F_1 all X_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,hAPP_H928324994_state(F_1,X_1)),image_1604514514_state(F_1,A))))) # label(fact_200_imageI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 125 (all B all A_1 all A ((hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) -> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B))) & (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) -> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B))))) # label(fact_660_Int__insert__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 126 (all X_5 all Y_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_5),Y_3)),X_5))) # label(fact_639_inf__sup__ord_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 127 (all Ba all A_1 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool))) -> A_1 = Ba)) # label(fact_107_singletonE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 128 (all A_1 all Pa hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_1),collect_fun_nat_bool(Pa)) = collect_fun_nat_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,fimplies),hAPP_f1246832597l_bool(hAPP_f1164891443l_bool(cOMBB_238756964t_bool,fNot),hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,fequal_fun_nat_bool),A_1)))),Pa))) # label(fact_90_insert__Collect) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 129 (all B_1_1 all B_2_1 (is_bool(B_2_1) -> is_bool(hAPP_bool_bool(B_1_1,B_2_1)))) # label(gsy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 130 (all A_1 hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_1),bot_bo1701429464l_bool) = collect_fun_nat_bool(hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,fequal_fun_nat_bool),A_1))) # label(fact_33_singleton__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 131 (all Y_24 all X_30 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_24),X_30)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_30),Y_24) = X_30)) # label(fact_481_sup__absorb1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 132 (all P all Q all R hAPP_a1200519163e_bool(hAPP_f963367678e_bool(hAPP_f375255701e_bool(cOMBB_145932198bool_a,P),Q),R) = hAPP_f644196280e_bool(P,hAPP_a849909144l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H_028) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 133 (all Ca all X_1 all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) = image_1410872416te_nat(hAPP_n436597305te_nat(cOMBK_1950023923_state,Ca),A))) # label(fact_173_image__constant) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 134 (all X_1 all F_1 all F (hBOOL(finite988810631ne_nat(F_1,F)) -> X_1 = hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) # label(fact_248_folding__one_Osingleton) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 135 (all A_1 all Pa hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),collec727977250_state(Pa)) = collec727977250_state(hAPP_f921536533e_bool(hAPP_f760664097e_bool(cOMBS_1248383340l_bool,hAPP_f1558728829l_bool(hAPP_f1442918689l_bool(cOMBB_1382207997_state,fimplies),hAPP_f921536533e_bool(hAPP_f1037965299e_bool(cOMBB_416661851_state,fNot),hAPP_H1645666623e_bool(hAPP_f262880489e_bool(cOMBC_1424981238e_bool,fequal1531560888_state),A_1)))),Pa))) # label(fact_91_insert__Collect) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 136 (all A all F_1 all F (hBOOL(big_se275732192ig_nat(F_1,F)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hAPP_f22106695ol_nat(F,A) = hAPP_f22106695ol_nat(finite_fold1_nat(F_1),A)))) # label(fact_441_semilattice__big_OF__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 137 (all X_2 all Xa collec727977250_state(hAPP_f921536533e_bool(hAPP_f760664097e_bool(cOMBS_1248383340l_bool,hAPP_f1558728829l_bool(hAPP_f1442918689l_bool(cOMBB_1382207997_state,fdisj),hAPP_H1645666623e_bool(hAPP_f262880489e_bool(cOMBC_1424981238e_bool,fequal1531560888_state),X_2))),hAPP_f921536533e_bool(hAPP_f1262649863e_bool(cOMBC_764456866l_bool,member1758697444_state),Xa))) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_2),Xa)) # label(fact_102_insert__compr__raw) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 138 (all Z_2 all X_1 all A all V all F_1 (hBOOL(finite1200705745iple_a(F_1)) -> (hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,Z_2,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A)),V)) -> (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> -(all Y_11 (hAPP_H678412245iple_a(hAPP_H1455657330iple_a(F_1,X_1),Y_11) = V -> -hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,Z_2,A),Y_11)))))))) # label(fact_416_comp__fun__commute_Ofold__graph__insertE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 139 (all A_1 all B hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)))) # label(fact_97_insertI1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 140 (all A all B collec268032053iple_a(hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fdisj),hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,member1713797107iple_a),A))),hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,member1713797107iple_a),B))) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,A),B)) # label(fact_541_Un__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 141 (all A_1 all A bot_bot_fun_nat_bool != hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A)) # label(fact_116_insert__not__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 142 (all B all A_1 all A ((hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),A)) -> hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B)) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),B))) & (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),A)) -> hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B)) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),B)))) # label(fact_662_Int__insert__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 143 (all P all Q all R hAPP_H513860823e_bool(hAPP_H1645666623e_bool(hAPP_f262880489e_bool(cOMBC_1424981238e_bool,P),Q),R) = hAPP_H513860823e_bool(hAPP_H1645666623e_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_Itc__Com__) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 144 (all B_1_1 (is_glb(B_1_1) -> is_vname(glb(B_1_1)))) # label(gsy_c_Com_Ovname_OGlb) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 145 (all A bot_bot_fun_nat_bool = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),A)) # label(fact_364_Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 146 (all Pa all Q_1 (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,collec268032053iple_a(hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fdisj),Pa)),Q_1)))) <-> hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,collec268032053iple_a(Q_1))) & hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,collec268032053iple_a(Pa))))) # label(fact_317_finite__Collect__disjI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 147 (all A_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),bot_bot_nat)) <-> bot_bot_nat = A_1)) # label(fact_556_bot__unique) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 148 (all B all Ca all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))))) # label(fact_354_DiffI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 149 (all Ba all A_1 (Ba = A_1 <-> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,Ba),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),bot_bo1055319631e_bool))))) # label(fact_114_singleton__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 150 (all A_1 hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool) = collect_nat(hAPP_n1699378549t_bool(fequal_nat,A_1))) # label(fact_28_singleton__conv2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 151 (all Ca all A all B (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),B))) -> -(hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),B))))) # label(fact_351_DiffE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 152 (all A_1 all Ba all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Ba),A))) <-> A_1 = Ba | hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),A)))) # label(fact_81_insert__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 153 (all F_1 all A_1 all B hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hAPP_H928324994_state(F_1,A_1)),image_1604514514_state(F_1,B)) = image_1604514514_state(F_1,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B))) # label(fact_236_image__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 154 (all X_1 (hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila310582991f_bool,bot_bot_bool),X_1)) <-> hBOOL(bot_bot_bool))) # label(fact_686_inf__bot__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 155 (all A_1 all Ba all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Ba),A))) -> (Ba != A_1 -> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),A))))) # label(fact_20_insertE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 156 (all X_35 all A_1 all S_4 all N_3 all T_1 (hBOOL(evaln(ass(X_35,A_1),S_4,N_3,T_1)) -> T_1 = hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S_4),X_35),hAPP_state_nat(A_1,S_4)))) # label(fact_281_evaln__elim__cases_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 157 (all Ga all Pa all X_35 all A_1 hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(hAPP_f1838002347e_bool(hAPP_f1706273077e_bool(cOMBC_867582640e_bool,hAPP_f289738463e_bool(hAPP_f1151843515e_bool(cOMBB_1941618714_state,cOMBB_844853809_state),Pa)),hAPP_f871651461_state(hAPP_f851239890_state(cOMBS_777315357_state,hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,update),X_35)),A_1)),ass(X_35,A_1),Pa)),bot_bo1055319631e_bool)))) # label(fact_160_Ass) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 158 (all F_1 all A_1 all A all X_1 (hBOOL(hAPP_nat_bool(finite929467206at_nat(F_1,A_1,A),X_1)) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) -> hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A)),X_1))))) # label(fact_272_fold1Set_Ointros) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 159 (all P all Q all R hAPP_H513860823e_bool(hAPP_f921536533e_bool(hAPP_f1262649863e_bool(cOMBC_764456866l_bool,P),Q),R) = hAPP_f1760790145l_bool(hAPP_H248360617l_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_Itc__Com___027) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 160 (all X_1 all A all B (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),B)) -> (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),B))))) # label(fact_529_set__mp) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 161 (all X_1 all Y_7 all A hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Y_7),A)) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Y_7),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A))) # label(fact_82_insert__commute) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 162 (all C_4 all A_11 all B_7 (A_11 = B_7 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_4),B_7)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_4),A_11))))) # label(fact_521_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 163 (all X_1 all Y_7 all Pa ((hBOOL(Pa) -> hAPP_f124283079iple_a(the_Ho746640593iple_a,hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fconj),hAPP_f1591852335a_bool(hAPP_f34030599a_bool(cOMBB_1799513916iple_a,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_H1190454433a_bool(hAPP_f1371755681a_bool(cOMBC_839220826a_bool,fequal879838495iple_a),X_1)))),hAPP_f1591852335a_bool(hAPP_f34030599a_bool(cOMBB_1799513916iple_a,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_H1190454433a_bool(hAPP_f1371755681a_bool(cOMBC_839220826a_bool,fequal879838495iple_a),Y_7)))) = X_1) & (-hBOOL(Pa) -> Y_7 = hAPP_f124283079iple_a(the_Ho746640593iple_a,hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fconj),hAPP_f1591852335a_bool(hAPP_f34030599a_bool(cOMBB_1799513916iple_a,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_H1190454433a_bool(hAPP_f1371755681a_bool(cOMBC_839220826a_bool,fequal879838495iple_a),X_1)))),hAPP_f1591852335a_bool(hAPP_f34030599a_bool(cOMBB_1799513916iple_a,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_H1190454433a_bool(hAPP_f1371755681a_bool(cOMBC_839220826a_bool,fequal879838495iple_a),Y_7))))))) # label(fact_242_If__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 164 (all R_1 all S all X_2 (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_2),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,R_1),S))) <-> hBOOL(hAPP_H1421470952a_bool(hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,member1713797107iple_a),R_1)),hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,member1713797107iple_a),S)),X_2)))) # label(fact_663_inf__Int__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 165 (all A_1 all G all F_1 (finite_fold1_nat(F_1) = G -> A_1 = hAPP_f22106695ol_nat(G,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)))) # label(fact_422_fold1__singleton__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 166 (all A all Ba all F_1 all X_1 (hAPP_H716259088te_nat(F_1,X_1) = Ba -> (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_1410872416te_nat(F_1,A)))))) # label(fact_181_image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 167 (all P all Q all R hAPP_f1759915619e_bool(P,hAPP_s1806633685e_bool(Q,R)) = hAPP_s1806633685e_bool(hAPP_f817621513e_bool(hAPP_f512427579e_bool(cOMBB_572666224_state,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_012) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 168 (all A hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,bot_bo1181479936a_bool),A))) # label(fact_454_empty__subsetI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 169 (all X_1 (hBOOL(bot_bot_bool) <-> hBOOL(hAPP_H1421470952a_bool(bot_bo1181479936a_bool,X_1)))) # label(fact_124_bot__apply) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 170 (all X_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),bot_bot_fun_nat_bool)) <-> hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_2)))) # label(fact_158_bot__empty__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 171 (all A all B hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B) = collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B)))) # label(fact_670_Int__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 172 (all B all A all F_1 all F (hBOOL(finite506823037_state(F_1,F)) -> (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A)) -> (bot_bo1055319631e_bool != A -> (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,B)) -> (B != bot_bo1055319631e_bool -> hAPP_f2143211163_state(F,hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila447562797e_bool,A),B)) = hAPP_H563960305_state(hAPP_H521649881_state(F_1,hAPP_f2143211163_state(F,A)),hAPP_f2143211163_state(F,B)))))))) # label(fact_592_folding__one__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 173 (all X_1 all Y_7 all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,Y_7),Z_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Z_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_7)))) # label(fact_634_le__inf__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 174 (all B all A (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A)) -> hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila447562797e_bool,A),B) = hAPP_f921536533e_bool(finite512563852e_bool(insert1835143293_state,B),A))) # label(fact_580_union__fold__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 175 (all A_1 all Ba all Ca all D_3 (Ca = A_1 & Ba = D_3 | D_3 = A_1 & Ba = Ca <-> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,D_3),bot_bot_fun_nat_bool)))) # label(fact_110_doubleton__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 176 (all B all A all F_1 all F (hBOOL(finite506823037_state(F_1,F)) -> (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A)) -> (B != bot_bo1055319631e_bool -> (hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,B),A)) -> hAPP_f2143211163_state(F,A) = hAPP_H563960305_state(hAPP_H521649881_state(F_1,hAPP_f2143211163_state(F,B)),hAPP_f2143211163_state(F,A))))))) # label(fact_443_folding__one__idem_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 177 (all A_1 all Ga all Pa all Ca all Q_1 all X_35 all S_5 (hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(Pa,Ca,hAPP_f1838002347e_bool(hAPP_f1706273077e_bool(cOMBC_867582640e_bool,hAPP_f289738463e_bool(hAPP_f1151843515e_bool(cOMBB_1941618714_state,cOMBB_844853809_state),Q_1)),hAPP_n1547241352_state(hAPP_f1848060885_state(cOMBC_1777403949_state,hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,update),loc(X_35))),getlocs(S_5,X_35))))),bot_bo1055319631e_bool))) -> hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(hAPP_f817621513e_bool(hAPP_f512427579e_bool(cOMBB_572666224_state,hAPP_f644196280e_bool(cOMBS_1378840469l_bool,hAPP_f1259673775l_bool(hAPP_f1561913689l_bool(cOMBB_188601460_state,fconj),hAPP_s1806633685e_bool(fequal_state,S_5)))),hAPP_f1838002347e_bool(hAPP_f1706273077e_bool(cOMBC_867582640e_bool,hAPP_f289738463e_bool(hAPP_f1151843515e_bool(cOMBB_1941618714_state,cOMBB_844853809_state),Pa)),hAPP_f871651461_state(hAPP_f851239890_state(cOMBS_777315357_state,hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,update),loc(X_35))),A_1))),local(X_35,A_1,Ca),Q_1)),bot_bo1055319631e_bool))))) # label(fact_250_hoare__derivs_OLocal) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 178 (all Ca -hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),bot_bot_fun_nat_bool))) # label(fact_52_empty__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 179 (all A_1 all Pa ((exists X_2 (hBOOL(hAPP_H1421470952a_bool(Pa,X_2)) & (all Y_11 (hBOOL(hAPP_H1421470952a_bool(Pa,Y_11)) -> X_2 = Y_11)))) -> (hBOOL(hAPP_H1421470952a_bool(Pa,A_1)) -> A_1 = hAPP_f124283079iple_a(the_Ho746640593iple_a,Pa)))) # label(fact_267_the1__equality) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 180 (all Ts all Ga all Ts_1 (hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),Ts_1)) -> (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,Ts),Ts_1)) -> hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),Ts))))) # label(fact_569_weaken) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 181 (all N all H ((all X_2 all Y_11 hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,hAPP_nat_nat(H,X_2)),hAPP_nat_nat(H,Y_11)) = hAPP_nat_nat(H,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_2),Y_11))) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N)) -> (N != bot_bot_fun_nat_bool -> hAPP_nat_nat(H,hAPP_f22106695ol_nat(big_la43341705in_nat,N)) = hAPP_f22106695ol_nat(big_la43341705in_nat,image_nat_nat(H,N)))))) # label(fact_618_Sup__fin_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 182 (all Ca all A all B (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),B))) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),A)))) # label(fact_358_DiffD1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 183 (all Ba all A_1 (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ba),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool))) -> A_1 = Ba)) # label(fact_106_singletonE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 184 (all Ca all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B))) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)))) # label(fact_674_IntD1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 185 (all X_1 bot_bo1055319631e_bool = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,bot_bo1055319631e_bool),X_1)) # label(fact_688_inf__bot__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 186 (all P all Q all R hAPP_bool_bool(P,hAPP_f54304608l_bool(Q,R)) = hAPP_f54304608l_bool(hAPP_f1246832597l_bool(hAPP_f1164891443l_bool(cOMBB_238756964t_bool,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__fun_Itc__Nat__On) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 187 (all Ca all A all B (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),B))) <-> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),A)) & -hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),B)))) # label(fact_360_Diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 188 (all X_16 all Y_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_12),X_16)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_16),Y_12)))) # label(fact_526_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 189 (all A_1 all G all F_1 (finite1382394752iple_a(F_1) = G -> hAPP_f124283079iple_a(G,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool)) = A_1)) # label(fact_424_fold1__singleton__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 190 (all P P = hAPP_H678412245iple_a(cOMBI_1218222237iple_a,P)) # label(help_COMBI_1_1_COMBI_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_It__a_J_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 191 (all A ((exists X_2 hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))) <-> bot_bot_fun_nat_bool != A)) # label(fact_59_ex__in__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 192 (all Ba all A_1 (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,Ba),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),bot_bo1055319631e_bool))) -> Ba = A_1)) # label(fact_108_singletonE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 193 (all B all Ca all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,A),B))))) # label(fact_537_UnI1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 194 (all R_1 all S all X_2 (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_2),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,R_1),S))) <-> hBOOL(hAPP_H1421470952a_bool(hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,member1713797107iple_a),R_1)),hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,member1713797107iple_a),S)),X_2)))) # label(fact_545_sup__Un__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 195 (all X_1 all A all Pa (hBOOL(hAPP_H1421470952a_bool(Pa,X_1)) & hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) <-> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),collec268032053iple_a(Pa)))))) # label(fact_667_Int__Collect) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 196 (all A_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),bot_bot_fun_nat_bool)) -> bot_bot_fun_nat_bool = A_1)) # label(fact_558_le__bot) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 197 (all Ca -hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,Ca),bot_bo1055319631e_bool))) # label(fact_51_empty__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 198 (all A_1 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)))) # label(fact_320_finite__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 199 (all X_1 hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,bot_bot_fun_nat_bool),X_1) = X_1) # label(fact_500_sup__bot__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 200 (all A hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A))) # label(fact_452_empty__subsetI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 201 (all Y_27 all A_1 all Ca all S_4 all T_1 (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(local(Y_27,A_1,Ca)),S_4),T_1)) -> -(all S1_1 (T_1 = hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S1_1),loc(Y_27)),getlocs(S_4,Y_27)) -> -hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(Ca),hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S_4),loc(Y_27)),hAPP_state_nat(A_1,S_4))),S1_1)))))) # label(fact_290_evalc__elim__cases_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 202 (all B_1_1 all B_2_1 is_bool(finite774711482_state(B_1_1,B_2_1))) # label(gsy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_002) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 203 (all Q all P (-hBOOL(P) | -hBOOL(Q) | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)))) # label(help_fconj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 204 (all A_1 hAPP_f124283079iple_a(the_Ho746640593iple_a,hAPP_H1190454433a_bool(hAPP_f1371755681a_bool(cOMBC_839220826a_bool,fequal879838495iple_a),A_1)) = A_1) # label(fact_241_the__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 205 (all B_1_1 all B_2_1 is_bool(finite506823037_state(B_1_1,B_2_1))) # label(gsy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____ddpglwnxwg__O_003) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 206 (all Ca all S0_1 all Y_27 all A_1 all S1_2 (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(Ca),hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S0_1),loc(Y_27)),hAPP_state_nat(A_1,S0_1))),S1_2)) -> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(local(Y_27,A_1,Ca)),S0_1),hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S1_2),loc(Y_27)),getlocs(S0_1,Y_27)))))) # label(fact_269_evalc_OLocal) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 207 (all B all A (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> hAPP_f1591852335a_bool(finite1979045230a_bool(hAPP_f485051996a_bool(hAPP_f2085120383a_bool(cOMBB_2080540641iple_a,hAPP_f450029403a_bool(cOMBC_314077933a_bool,minus_469558085a_bool)),hAPP_f1665407592a_bool(hAPP_f549683569a_bool(cOMBC_708118077a_bool,insert873085594iple_a),bot_bo1181479936a_bool)),B),A) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,B),A))) # label(fact_439_minus__fold__remove) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 208 (all Q_1 all Pa (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Q_1))) | hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Pa))) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),Pa)),Q_1)))))) # label(fact_310_finite__Collect__conjI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 209 (all Ca all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)) & hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B))))) # label(fact_672_Int__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 210 (all F_1 all Z_2 all A1 all A2 (Z_2 = A2 & bot_bo1181479936a_bool = A1 | (exists X_2 exists A_19 exists Y_11 (hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_2),A_19) = A1 & A2 = hAPP_H678412245iple_a(hAPP_H1455657330iple_a(F_1,X_2),Y_11) & -hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_2),A_19)) & hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,Z_2,A_19),Y_11)))) <-> hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,Z_2,A1),A2)))) # label(fact_302_fold__graph_Osimps) # label(axiom) # label(non_clause). [assumption]. 1.68/2.02 211 (all Ts all Ga all T_1 (hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,T_1),bot_bo1055319631e_bool))) -> (hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),Ts)) -> hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,T_1),Ts)))))) # label(fact_7_hoare__derivs_Oinsert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 212 (all X_1 all A (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = A)) # label(fact_383_Diff__insert__absorb) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 213 (all X_17 all Y_13 (Y_13 = X_17 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_17),Y_13)))) # label(fact_524_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 214 (all A ((all Y_11 -hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Y_11),A))) -> A = bot_bot_fun_nat_bool)) # label(fact_144_equals0I) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 215 (all A_1 all C all D_2 (hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,C),D_2)) -> hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),C)),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),D_2))))) # label(fact_475_insert__mono) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 216 (all X_1 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (bot_bot_fun_nat_bool != A -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_1),hAPP_f22106695ol_nat(big_la43341705in_nat,A)) = hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A))))) # label(fact_611_Sup__fin_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 217 (all X_1 all A all F_1 all F (hBOOL(finite988810631ne_nat(F_1,F)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> (A != bot_bot_fun_nat_bool -> hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,X_1),hAPP_f22106695ol_nat(F,A))))))) # label(fact_306_folding__one_Oinsert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 218 (all A (A != bot_bo1055319631e_bool <-> (exists X_2 hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_2),A))))) # label(fact_58_ex__in__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 219 (all P all Q all R hAPP_H1645666623e_bool(hAPP_f762269719e_bool(hAPP_f964290431e_bool(cOMBC_488258100e_bool,P),Q),R) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_Itc__Com___030) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 220 (all P all Q all R hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 221 (all Ba all F_1 all X_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> (Ba = hAPP_H928324994_state(F_1,X_1) -> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,Ba),image_1604514514_state(F_1,A)))))) # label(fact_191_rev__image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 222 (all A_1 all A hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A) != bot_bot_fun_nat_bool) # label(fact_119_empty__not__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 223 (all X_35 all A_1 all S_4 hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(ass(X_35,A_1)),S_4),hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S_4),X_35),hAPP_state_nat(A_1,S_4))))) # label(fact_282_evalc_OAssign) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 224 (all X_1 all Y_7 (X_1 = bot_bo1181479936a_bool & bot_bo1181479936a_bool = Y_7 <-> bot_bo1181479936a_bool = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,X_1),Y_7))) # label(fact_494_sup__eq__bot__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 225 (all Y_7 all A all X_1 (hBOOL(hAPP_H513860823e_bool(hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Y_7),A),X_1)) <-> X_1 = Y_7 | hBOOL(hAPP_H513860823e_bool(A,X_1)))) # label(fact_78_insert__code) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 226 (all X_1 bot_bo1055319631e_bool = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,X_1),bot_bo1055319631e_bool)) # label(fact_692_inf__bot__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 227 (all P all Q (-hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) | hBOOL(Q))) # label(help_fconj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 228 (all P all Q all R hAPP_n1512601776t_bool(hAPP_f10625010t_bool(hAPP_f1115950719t_bool(cOMBB_1654519265ol_nat,P),Q),R) = hAPP_f1730770594t_bool(P,hAPP_n1699378549t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 229 (all B all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))))) # label(fact_355_finite__Diff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 230 (all B all A all F_1 all F (hBOOL(finite795500164em_nat(F_1,F)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (A != bot_bot_fun_nat_bool -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) -> (bot_bot_fun_nat_bool != B -> hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,hAPP_f22106695ol_nat(F,A)),hAPP_f22106695ol_nat(F,B)))))))) # label(fact_591_folding__one__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 231 (all A_1 hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,bot_bot_bool),A_1))) # label(fact_547_bot__least) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 232 (all B all A (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,A),B) = hAPP_f1591852335a_bool(finite1979045230a_bool(insert873085594iple_a,B),A))) # label(fact_581_union__fold__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 233 (all F_1 all X_1 all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,hAPP_H563960305_state(F_1,X_1)),image_650584225_state(F_1,A))))) # label(fact_198_imageI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 234 (all F_1 all A_1 all X_35 all X_1 (hBOOL(hAPP_H513860823e_bool(finite9525415_state(F_1,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),X_35)),X_1)) -> -(all A_7 all A_19 (hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_7),A_19) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),X_35) -> (hBOOL(hAPP_H513860823e_bool(finite1935632226_state(F_1,A_7,A_19),X_1)) -> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_7),A_19))))))) # label(fact_295_insert__fold1SetE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 235 (all A all Ca all B ((-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B))))) # label(fact_449_UnCI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 236 (all X_1 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hAPP_f22106695ol_nat(big_la43341705in_nat,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_1),hAPP_f22106695ol_nat(big_la43341705in_nat,A))))) # label(fact_609_Sup__fin_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 237 (all A all A_1 all B (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)))))) # label(fact_395_finite__Diff__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 238 (all P all Q P = hAPP_H678412245iple_a(hAPP_H1455657330iple_a(cOMBK_1824972302iple_a,P),Q)) # label(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_It__a_J_00_013) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 239 (all A_1 (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A_1),bot_bo1181479936a_bool)) <-> A_1 = bot_bo1181479936a_bool)) # label(fact_555_bot__unique) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 240 (all A all B collec268032053iple_a(hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fconj),hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,member1713797107iple_a),A))),hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,member1713797107iple_a),B))) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),B)) # label(fact_669_Int__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 241 (all A_1 all Z_2 all A all Y_7 all F_1 (hBOOL(finite1200705745iple_a(F_1)) -> (hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,Z_2,A),Y_7)) -> (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),A)) -> (exists Y_26 (hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,Z_2,hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool))),Y_26)) & hAPP_H678412245iple_a(hAPP_H1455657330iple_a(F_1,A_1),Y_26) = Y_7)))))) # label(fact_409_comp__fun__commute_Ofold__graph__insertE__aux) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 242 (all A_1 (A_1 = bot_bo1055319631e_bool <-> hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,A_1),bot_bo1055319631e_bool)))) # label(fact_554_bot__unique) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 243 (all X_1 bot_bo1181479936a_bool = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,X_1),bot_bo1181479936a_bool)) # label(fact_693_inf__bot__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 244 (all Ts all Ga all Ts_1 (hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),Ts_1)) -> (hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,Ts),Ts_1)) -> hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),Ts))))) # label(fact_568_weaken) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 245 (all P all Q P = hAPP_a2036067514e_bool(hAPP_f762886889e_bool(cOMBK_1458035955bool_a,P),Q)) # label(help_COMBK_1_1_COMBK_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000t__a_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 246 (all P all Q all R hAPP_s1874344717e_bool(hAPP_f873506917e_bool(hAPP_f1398071125e_bool(cOMBB_1757942702_state,P),Q),R) = hAPP_f167292325e_bool(P,hAPP_s2001034685l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H_024) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 247 (all P (-hBOOL(hAPP_bool_bool(fNot,P)) | -hBOOL(P))) # label(help_fNot_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 248 (all A hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,A),bot_bo1181479936a_bool) = A) # label(fact_508_Un__empty__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 249 (all B all A_1 all A (-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),A)) -> hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,A),B) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B)))) # label(fact_652_Int__insert__right__if0) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 250 (all Fun1_2 all Com_1 all Fun2_2 all Fun1_1 all Com all Fun2_1 (hoare_1575745797_state(Fun1_1,Com,Fun2_1) = hoare_1575745797_state(Fun1_2,Com_1,Fun2_2) <-> Fun2_2 = Fun2_1 & Com_1 = Com & Fun1_2 = Fun1_1)) # label(fact_2_triple_Oinject) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 251 (all X_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) <-> hBOOL(hAPP_H1421470952a_bool(A,X_1)))) # label(fact_230_mem__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 252 (all A_1 all B hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B)))) # label(fact_96_insertI1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 253 (all N all H all F_1 all F (hBOOL(finite795500164em_nat(F_1,F)) -> ((all X_2 all Y_11 hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,hAPP_nat_nat(H,X_2)),hAPP_nat_nat(H,Y_11)) = hAPP_nat_nat(H,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,X_2),Y_11))) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N)) -> (N != bot_bot_fun_nat_bool -> hAPP_f22106695ol_nat(F,image_nat_nat(H,N)) = hAPP_nat_nat(H,hAPP_f22106695ol_nat(F,N))))))) # label(fact_400_folding__one__idem_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 254 (all A_1 all A hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),bot_bo1055319631e_bool))) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),A)) # label(fact_387_insert__Diff__single) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 255 (all P all Q all R hAPP_state_bool(P,hAPP_state_state(Q,R)) = hAPP_state_bool(hAPP_f746301080e_bool(hAPP_f162060345e_bool(cOMBB_844853809_state,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Com__Ostate_000tc__HOL__Obool_000tc__Com__Ostate_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 256 (all A all A_1 all B hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila447562797e_bool,A),B)) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila447562797e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B))) # label(fact_563_Un__insert__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 257 (all A all Ba all F_1 all X_1 (hAPP_H738206399_a_nat(F_1,X_1) = Ba -> (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_194810223_a_nat(F_1,A)))))) # label(fact_180_image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 258 (all X_1 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(semila972727038up_nat,X_1),A)))) # label(fact_616_Sup__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 259 (all X_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> -(all B_15 (hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),B_15) = A -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),B_15)))))) # label(fact_136_Set_Oset__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 260 (all A_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_8),bot_bot_nat)) -> A_8 = bot_bot_nat)) # label(fact_561_le__bot) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 261 (all P all Q (is_bool(P) -> hAPP_f54304608l_bool(hAPP_b1630757474l_bool(cOMBK_1994329625t_bool,P),Q) = P)) # label(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 262 (all B all A_1 all C (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),C)) -> hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B)),C) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,B),C)))) # label(fact_644_Int__insert__left__if1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 263 (all Pa all A_1 ((-hBOOL(hAPP_H1421470952a_bool(Pa,A_1)) -> bot_bo1181479936a_bool = collec268032053iple_a(hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fconj),hAPP_H1190454433a_bool(fequal879838495iple_a,A_1))),Pa))) & (hBOOL(hAPP_H1421470952a_bool(Pa,A_1)) -> hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool) = collec268032053iple_a(hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fconj),hAPP_H1190454433a_bool(fequal879838495iple_a,A_1))),Pa))))) # label(fact_35_Collect__conv__if2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 264 (all P all Q all R hAPP_f540970102l_bool(hAPP_H1840393229l_bool(P,R),Q) = hAPP_H1421470952a_bool(hAPP_f1591852335a_bool(hAPP_f672239281a_bool(cOMBC_1552014468l_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_It__a_J_00_023) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 265 (all Y_7 all A all X_1 (hBOOL(hAPP_H1421470952a_bool(hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Y_7),A),X_1)) <-> hBOOL(hAPP_H1421470952a_bool(A,X_1)) | X_1 = Y_7)) # label(fact_76_insert__code) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 266 (all Ca all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)) & -hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)))) # label(fact_361_Diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 267 (all F_1 all A_1 all A all X_1 (hBOOL(hAPP_H513860823e_bool(finite1935632226_state(F_1,A_1,A),X_1)) -> (-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),A)) -> hBOOL(hAPP_H513860823e_bool(finite9525415_state(F_1,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),A)),X_1))))) # label(fact_271_fold1Set_Ointros) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 268 (all A_1 collec727977250_state(hAPP_H1645666623e_bool(fequal1531560888_state,A_1)) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),bot_bo1055319631e_bool)) # label(fact_30_singleton__conv2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 269 (all Ba all A_1 all B (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),B)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),B))))) # label(fact_71_insertI2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 270 (all X all Y (Y = X | -hBOOL(hAPP_H1421470952a_bool(hAPP_H1190454433a_bool(fequal879838495iple_a,X),Y)))) # label(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_It__a_J_) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 271 (all X_2 all Xa hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,X_2),Xa) = collect_fun_nat_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,fdisj),hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,fequal_fun_nat_bool),X_2))),hAPP_f1246832597l_bool(hAPP_f1443436725l_bool(cOMBC_1245412066l_bool,member_fun_nat_bool),Xa)))) # label(fact_101_insert__compr__raw) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 272 (all P all Q (is_bool(P) -> P = hAPP_nat_bool(hAPP_b1013836512t_bool(cOMBK_bool_nat,P),Q))) # label(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 273 (all C1 all S2 all C0 all S0 all S1 (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C0),S0),S1)) -> (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C1),S1),S2)) -> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(semi(C0,C1)),S0),S2))))) # label(fact_277_evalc_OSemi) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 274 (all A_1 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1)) <-> (exists A_19 exists A_7 (A_1 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_7),A_19) & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_19)))) | bot_bot_fun_nat_bool = A_1)) # label(fact_334_finite_Osimps) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 275 (all A ((all X_2 -hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_2),A))) <-> A = bot_bo1181479936a_bool)) # label(fact_60_all__not__in__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 276 (all B all A_1 all C (-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),C)) -> hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B)),C) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,B),C))) # label(fact_649_Int__insert__left__if0) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 277 (all F_1 all Z_2 all Y_7 all X_1 all A (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> (hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,Z_2,A),Y_7)) -> hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,Z_2,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A)),hAPP_H678412245iple_a(hAPP_H1455657330iple_a(F_1,X_1),Y_7)))))) # label(fact_289_fold__graph_OinsertI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 278 (all X_1 all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))))) # label(fact_530_set__mp) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 279 (all X_22 all Y_18 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_22),Y_18)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_18),X_22)))) # label(fact_514_linorder__le__cases) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 280 (all A_1 collec727977250_state(hAPP_H1645666623e_bool(hAPP_f262880489e_bool(cOMBC_1424981238e_bool,fequal1531560888_state),A_1)) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),bot_bo1055319631e_bool)) # label(fact_34_singleton__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 281 (all Ga hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),bot_bo1181479936a_bool))) # label(fact_0_empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 282 (all F_1 all A1 all A2 ((exists A_7 exists A_19 exists X_2 (A1 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_7),A_19) & X_2 = A2 & -hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_7),A_19)) & hBOOL(hAPP_nat_bool(finite929467206at_nat(F_1,A_7,A_19),X_2)))) <-> hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F_1,A1),A2)))) # label(fact_299_fold1Set_Osimps) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 283 (all B all Ca all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B))))) # label(fact_538_UnI1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 284 (all B all A all F_1 all F (hBOOL(finite950012314iple_a(F_1,F)) -> (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> (A != bot_bo1181479936a_bool -> (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,B)) -> (bot_bo1181479936a_bool != B -> hAPP_f124283079iple_a(F,hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,A),B)) = hAPP_H678412245iple_a(hAPP_H1455657330iple_a(F_1,hAPP_f124283079iple_a(F,A)),hAPP_f124283079iple_a(F,B)))))))) # label(fact_593_folding__one__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 285 (all A_1 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) -> hAPP_f22106695ol_nat(finite_fold_nat_nat(times_times_nat,A_1),A) = hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A))))) # label(fact_418_fold1__eq__fold) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 286 (all A all A_1 all B hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B))) # label(fact_562_Un__insert__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 287 (all A all A_1 all B hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool))),B) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B))) # label(fact_391_Diff__insert2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 288 (all X all Y (-hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(fequal1258664663a_bool,X),Y)) | Y = X)) # label(help_fequal_1_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____ddpglwnxwg__Otriple) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 289 (all Pa all Ga all P_2 all Ca all Q_1 (hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(P_2,Ca,Q_1)),bot_bo1055319631e_bool))) -> ((all Z_7 all S_2 (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_7),S_2)) -> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_2,Z_7),S_2)))) -> hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(Pa,Ca,Q_1)),bot_bo1055319631e_bool)))))) # label(fact_15_conseq1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 290 (all X_2 all Xa hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),Xa) = collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),X_2))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),Xa)))) # label(fact_100_insert__compr__raw) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 291 (all B hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,bot_bot_fun_nat_bool),B) = B) # label(fact_503_Un__empty__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 292 (all P all Q (-hBOOL(Q) | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)))) # label(help_fdisj_2_1_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 293 (all B all X_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))))) # label(fact_532_set__rev__mp) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 294 (all A_1 all A bot_bo1181479936a_bool != hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),A)) # label(fact_118_empty__not__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 295 (all A all A_1 all B (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B)))) <-> hBOOL(hAPP_f1760790145l_bool(finite784854244_state,hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),B))))) # label(fact_396_finite__Diff__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 296 (all Ba all F_1 all X_1 all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> (Ba = hAPP_H563960305_state(F_1,X_1) -> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,Ba),image_650584225_state(F_1,A)))))) # label(fact_189_rev__image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 297 (all A_1 all A hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),A) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila447562797e_bool,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),bot_bo1055319631e_bool)),A)) # label(fact_576_insert__is__Un) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 298 (all F_1 all A_1 all Ba (hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)),Ba)) <-> Ba = A_1)) # label(fact_244_fold1Set__sing) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 299 (all F_1 all A_1 all B image_129517430iple_a(F_1,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B)) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hAPP_H1600811558iple_a(F_1,A_1)),image_129517430iple_a(F_1,B))) # label(fact_238_image__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 300 (all F_1 all A_1 all Ba (hBOOL(hAPP_H513860823e_bool(finite9525415_state(F_1,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),bot_bo1055319631e_bool)),Ba)) <-> A_1 = Ba)) # label(fact_243_fold1Set__sing) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 301 (all A all Ba ((-hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Ba),bot_bo1055319631e_bool))) -> hAPP_f2143211163_state(partia415982977_state(Ba),A) = hAPP_f2143211163_state(the_Ho10452358_state,hAPP_f921536533e_bool(hAPP_f1262649863e_bool(cOMBC_764456866l_bool,member1758697444_state),hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Ba),bot_bo1055319631e_bool))))) & (hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Ba),bot_bo1055319631e_bool))) -> hAPP_f2143211163_state(partia415982977_state(Ba),A) = Ba))) # label(fact_604_flat__lub__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 302 (all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,ord_le1568362934t_bool),A)))))) # label(fact_455_finite__Collect__subsets) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 303 (all U_1 all C_2 all S_1 all T (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C_2),S_1),T)) -> (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C_2),S_1),U_1)) -> U_1 = T))) # label(fact_285_com__det) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 304 (all A all B (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))))) # label(fact_460_rev__finite__subset) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 305 (all P (hBOOL(P) | hBOOL(hAPP_bool_bool(fNot,P)))) # label(help_fNot_2_1_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 306 (all X_1 all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B)))) # label(fact_465_insert__subset) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 307 (all Ca all A all B (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,A),B))) <-> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),B)) | hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),A)))) # label(fact_539_Un__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 308 (all B all A ((all X_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B)))) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)))) # label(fact_599_subsetI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 309 (all Ba all F_1 all X_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> (Ba = hAPP_n178040171iple_a(F_1,X_1) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ba),image_1782127643iple_a(F_1,A)))))) # label(fact_197_rev__image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 310 (all Ga all Pa hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(Pa,skip,Pa)),bot_bo1055319631e_bool)))) # label(fact_131_hoare__derivs_OSkip) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 311 (all Loc_2 all Fun all Com local(Loc_2,Fun,Com) != skip) # label(fact_259_com_Osimps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 312 (all X_1 all A hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),A)) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),A)) # label(fact_87_insert__absorb2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 313 (all A_1 all Ba all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ba),A))) <-> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),A)) | Ba = A_1)) # label(fact_79_insert__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 314 (all B all X_1 all A (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),B)) -> (hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),B) <-> B = A)))) # label(fact_73_insert__ident) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 315 (all X_1 all Y_7 all A hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Y_7),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),A)) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Y_7),A))) # label(fact_84_insert__commute) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 316 (all Ga all Pa all Ca all Q_1 all C ((hBOOL(C) -> hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(Pa,Ca,Q_1)),bot_bo1055319631e_bool)))) -> hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(hAPP_b1095269219e_bool(hAPP_f701449317e_bool(cOMBC_471052088e_bool,hAPP_f873506917e_bool(hAPP_f1398071125e_bool(cOMBB_1757942702_state,cOMBC_231445413l_bool),hAPP_f1283379615l_bool(hAPP_f1345202233l_bool(cOMBB_1759179140_state,hAPP_f1561913689l_bool(cOMBB_188601460_state,fconj)),Pa))),C),Ca,Q_1)),bot_bo1055319631e_bool))))) # label(fact_9_constant) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 317 (all A bot_bo1181479936a_bool = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),bot_bo1181479936a_bool)) # label(fact_682_Int__empty__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 318 (all A_1 collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_1)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)) # label(fact_32_singleton__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 319 (all Q all P (hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) | hBOOL(P))) # label(help_fimplies_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 320 (all X all Y (Y = X | -hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y)))) # label(help_fequal_1_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 321 (all A ((all Y_11 -hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,Y_11),A))) -> bot_bo1055319631e_bool = A)) # label(fact_143_equals0I) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 322 (all F_1 all A (A = bot_bo1181479936a_bool <-> image_1604514514_state(F_1,A) = bot_bo1055319631e_bool)) # label(fact_220_image__is__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 323 (all X all Y (hBOOL(hAPP_H513860823e_bool(hAPP_H1645666623e_bool(fequal1531560888_state,X),Y)) | Y != X)) # label(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_Itc__Com) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 324 (all P all Q all R hAPP_bool_bool(P,hAPP_H513860823e_bool(Q,R)) = hAPP_H513860823e_bool(hAPP_f921536533e_bool(hAPP_f1037965299e_bool(cOMBB_416661851_state,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Hoare____Mirabel_009) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 325 (all A_1 all A all B hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),B)) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),A)),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B))) # label(fact_656_insert__inter__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 326 (all Y_22 all X_26 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_22),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_26),Y_22)))) # label(fact_487_sup__ge2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 327 (all A (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),bot_bo1181479936a_bool)) <-> A = bot_bo1181479936a_bool)) # label(fact_459_subset__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 328 (all Z_2 all F_1 all A ((exists X_2 (hAPP_H928324994_state(F_1,X_2) = Z_2 & hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_2),A)))) <-> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,Z_2),image_1604514514_state(F_1,A))))) # label(fact_208_image__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 329 (all F_1 all A (image_1604514514_state(F_1,A) = bot_bo1055319631e_bool <-> A = bot_bo1181479936a_bool)) # label(fact_228_empty__is__image) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 330 (all Pa all Ga all P_2 all Ca all Q_1 (hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(P_2,Ca,Q_1)),bot_bo1181479936a_bool))) -> ((all Z_7 all S_2 (hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_7),S_2)) -> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_2,Z_7),S_2)))) -> hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(Pa,Ca,Q_1)),bot_bo1181479936a_bool)))))) # label(fact_14_conseq1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 331 (all Z_2 all A all F_1 (hBOOL(finite1200705745iple_a(F_1)) -> (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,Z_2,A),hAPP_f124283079iple_a(finite978536264iple_a(F_1,Z_2),A)))))) # label(fact_435_comp__fun__commute_Ofold__graph__fold) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 332 (all P all Q all R hAPP_H226398757l_bool(hAPP_f1558728829l_bool(hAPP_f1442918689l_bool(cOMBB_1382207997_state,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_H513860823e_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_017) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 333 (all A all A_1 all B hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B)) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,A),B))) # label(fact_564_Un__insert__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 334 (all B_12 all A_16 all X_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_16),X_31)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_12),X_31)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_16),B_12)),X_31))))) # label(fact_480_le__supI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 335 (all X_1 bot_bot_fun_nat_bool = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,bot_bot_fun_nat_bool),X_1)) # label(fact_687_inf__bot__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 336 (all X_34 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_34),X_34))) # label(fact_445_order__refl) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 337 (all Ba all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),B))))) # label(fact_471_subset__insertI2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 338 (all B all A_1 all C (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),C)) -> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,B),C) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)),C))) # label(fact_648_Int__insert__left__if0) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 339 (all R_1 all S (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,R_1),S)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),R_1)),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S))))) # label(fact_544_pred__subset__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 340 (all A_1 all B all C hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila447562797e_bool,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B)),C) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila447562797e_bool,B),C))) # label(fact_566_Un__insert__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 341 (all A_1 all Ba (hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ba),bot_bo1181479936a_bool) -> Ba = A_1)) # label(fact_103_singleton__inject) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 342 (all A all Ba all F_1 all X_1 (Ba = hAPP_H563960305_state(F_1,X_1) -> (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,Ba),image_650584225_state(F_1,A)))))) # label(fact_176_image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 343 (all C1 all C2 all S_1 all T (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(semi(C1,C2)),S_1),T)) -> -(all S1_1 (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C1),S_1),S1_1)) -> -hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C2),S1_1),T)))))) # label(fact_292_evalc__elim__cases_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 344 (all A all B (bot_bo1181479936a_bool = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),B) <-> (all X_2 (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_2),A)) -> (all Xa (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Xa),B)) -> Xa != X_2)))))) # label(fact_679_disjoint__iff__not__equal) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 345 (all A_1 ((hBOOL(A_1) <-> hBOOL(bot_bot_bool)) <-> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A_1),bot_bot_bool)))) # label(fact_552_bot__unique) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 346 (all X all Y (hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(fequal1746921144e_bool,X),Y)) | Y != X)) # label(help_fequal_2_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____ddpglwnxwg__Otriple_008) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 347 (all X_1 (hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_1),bot_bot_bool)) <-> hBOOL(X_1))) # label(fact_495_sup__bot__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 348 (all A all Ba ((-hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool))) -> hAPP_f22106695ol_nat(partial_flat_lub_nat(Ba),A) = hAPP_f22106695ol_nat(the_nat,hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool))))) & (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool))) -> Ba = hAPP_f22106695ol_nat(partial_flat_lub_nat(Ba),A)))) # label(fact_603_flat__lub__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 349 (all B all A_1 all C ((hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),C)) -> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)),C) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,B),C))) & (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),C)) -> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)),C) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,B),C)))) # label(fact_657_Int__insert__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 350 (all D_3 all R_1 all Ga all Pa all Ca all Q_1 (hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(Pa,Ca,Q_1)),bot_bo1181479936a_bool))) -> (hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(Q_1,D_3,R_1)),bot_bo1181479936a_bool))) -> hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(Pa,semi(Ca,D_3),R_1)),bot_bo1181479936a_bool)))))) # label(fact_132_Comp) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 351 (all Ba all F_1 all X_1 all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> (Ba = hAPP_H716259088te_nat(F_1,X_1) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_1410872416te_nat(F_1,A)))))) # label(fact_194_rev__image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.03 352 (all Ca all X_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) = image_194810223_a_nat(hAPP_n1800114674_a_nat(cOMBK_944981412iple_a,Ca),A))) # label(fact_172_image__constant) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 353 (all Ca all S0_1 all Y_27 all A_1 all N_3 all S1_2 (hBOOL(evaln(Ca,hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S0_1),loc(Y_27)),hAPP_state_nat(A_1,S0_1)),N_3,S1_2)) -> hBOOL(evaln(local(Y_27,A_1,Ca),S0_1,N_3,hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S1_2),loc(Y_27)),getlocs(S0_1,Y_27)))))) # label(fact_270_evaln_OLocal) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 354 (all B all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (B != bot_bot_fun_nat_bool -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) -> hAPP_f22106695ol_nat(big_la43341705in_nat,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,hAPP_f22106695ol_nat(big_la43341705in_nat,B)),hAPP_f22106695ol_nat(big_la43341705in_nat,A)))))) # label(fact_613_Sup__fin_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 355 (all P all Q all R hAPP_s2001034685l_bool(hAPP_f1283379615l_bool(hAPP_f1345202233l_bool(cOMBB_1759179140_state,P),Q),R) = hAPP_f1259673775l_bool(P,hAPP_s1806633685e_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_021) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 356 (all F_1 bot_bo1181479936a_bool = image_1782127643iple_a(F_1,bot_bot_fun_nat_bool)) # label(fact_225_image__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 357 (all A bot_bot_fun_nat_bool = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),bot_bot_fun_nat_bool)) # label(fact_680_Int__empty__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 358 (all X_1 all A all F_1 all F (hBOOL(finite950012314iple_a(F_1,F)) -> (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> (A != bot_bo1181479936a_bool -> hAPP_f124283079iple_a(F,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A)) = hAPP_H678412245iple_a(hAPP_H1455657330iple_a(F_1,X_1),hAPP_f124283079iple_a(F,A)))))) # label(fact_340_folding__one__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 359 (all A_1 all A bot_bo1181479936a_bool != hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),A)) # label(fact_115_insert__not__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 360 (all X_7 all Y_5 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_7),Y_5)),Y_5))) # label(fact_637_inf__sup__ord_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 361 (all F_1 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (A != bot_bot_fun_nat_bool -> (exists X1 hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F_1,A),X1)))))) # label(fact_328_finite__nonempty__imp__fold1Set) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 362 (all A all F_1 all F (hBOOL(finite988810631ne_nat(F_1,F)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (bot_bot_fun_nat_bool != A -> ((all X_2 all Y_11 hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,X_2),Y_11)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_11),bot_bot_fun_nat_bool))))) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_f22106695ol_nat(F,A)),A))))))) # label(fact_325_folding__one_Oclosed) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 363 (all B all A_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),A)) -> hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),B)) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B)))) # label(fact_647_Int__insert__right__if1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 364 (all B_1_1 all B_2_1 is_bool(finite1669978781iple_a(B_1_1,B_2_1))) # label(gsy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 365 (all A_1 all A all F_1 all F (hBOOL(finite795500164em_nat(F_1,F)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F_1,A_1),A)))) # label(fact_432_folding__one__idem_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 366 (all A_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) -> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool))) = A)) # label(fact_380_insert__Diff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 367 (all Y_27 Y_27 = image_nat_nat(cOMBI_nat,Y_27)) # label(fact_185_image__ident) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 368 (all Ba all A_1 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hAPP_f22106695ol_nat(finite_fold_nat_nat(semila972727038up_nat,Ba),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_1),hAPP_f22106695ol_nat(finite_fold_nat_nat(semila972727038up_nat,Ba),A)))) # label(fact_578_fold__sup__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 369 (all Ca all X_1 all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> image_650584225_state(hAPP_H521649881_state(cOMBK_690957994_state,Ca),A) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Ca),bot_bo1055319631e_bool))) # label(fact_174_image__constant) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 370 (all X_1 X_1 = hAPP_f124283079iple_a(the_el1436340927iple_a,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),bot_bo1181479936a_bool))) # label(fact_121_the__elem__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 371 (all P all Q all R hAPP_b589554111l_bool(P,hAPP_f54304608l_bool(Q,R)) = hAPP_f1748468828l_bool(hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_011) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 372 (all Ba all F_1 all X_1 all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> (Ba = hAPP_H1600811558iple_a(F_1,X_1) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ba),image_129517430iple_a(F_1,A)))))) # label(fact_196_rev__image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 373 (all B_1_1 all B_2_1 is_bool(hAPP_state_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Com__Ostate_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 374 (all B all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hAPP_f800510211t_bool(finite1578363458t_bool(insert_nat,B),A) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B))) # label(fact_579_union__fold__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 375 (all A all A_1 all B hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B)) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),B)),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),bot_bo1055319631e_bool))) # label(fact_393_Diff__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 376 (all A_18 all B_14 all X_33 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_18),B_14)),X_33)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_18),X_33)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_14),X_33))))) # label(fact_477_le__supE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 377 (all Com1 all Com2 all Vname all Fun_1 semi(Com1,Com2) != ass(Vname,Fun_1)) # label(fact_215_com_Osimps_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 378 (all Q_1 all Ga all Ca all Pa ((all Z_7 all S_2 (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_7),S_2)) -> (exists P_1 exists Q_2 ((all S_3 ((all Z_8 (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_1,Z_8),S_2)) -> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_2,Z_8),S_3)))) -> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_1,Z_7),S_3)))) & hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(P_1,Ca,Q_2)),bot_bo1055319631e_bool))))))) -> hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(Pa,Ca,Q_1)),bot_bo1055319631e_bool))))) # label(fact_146_conseq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 379 (all Vname all Fun_1 all Com1 all Com2 ass(Vname,Fun_1) != semi(Com1,Com2)) # label(fact_214_com_Osimps_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 380 (all A_1 all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),A)) -> hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),bot_bo1055319631e_bool))) = A)) # label(fact_381_insert__Diff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 381 (all P all Q all R hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q) = hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 382 (all Ca all A ((bot_bo1181479936a_bool != A -> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) = image_194810223_a_nat(hAPP_n1800114674_a_nat(cOMBK_944981412iple_a,Ca),A)) & (A = bot_bo1181479936a_bool -> image_194810223_a_nat(hAPP_n1800114674_a_nat(cOMBK_944981412iple_a,Ca),A) = bot_bot_fun_nat_bool))) # label(fact_163_image__constant__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 383 (all A all Ba all F_1 all X_1 (hAPP_H928324994_state(F_1,X_1) = Ba -> (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,Ba),image_1604514514_state(F_1,A)))))) # label(fact_178_image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 384 (all P all Q hAPP_H716259088te_nat(hAPP_n436597305te_nat(cOMBK_1950023923_state,P),Q) = P) # label(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__Hoare____Mirabelle____ddpglwnxwg__O_005) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 385 (all X all Y (Y != X | hBOOL(hAPP_H1421470952a_bool(hAPP_H1190454433a_bool(fequal879838495iple_a,X),Y)))) # label(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_It__a_J_) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 386 (all X_1 X_1 = hAPP_f2143211163_state(the_el23965208_state,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),bot_bo1055319631e_bool))) # label(fact_123_the__elem__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 387 (all B_1_1 is_bool(finite1317819144e_bool(B_1_1))) # label(gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____ddpglwnxwg__Otri_001) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 388 (all Ba all F_1 all X_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> (Ba = hAPP_n1126952044_state(F_1,X_1) -> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,Ba),image_1821565372_state(F_1,A)))))) # label(fact_190_rev__image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 389 (all A_1 -hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),bot_bo1055319631e_bool))) # label(fact_25_emptyE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 390 (all Pa all F (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,F)) -> (hBOOL(hAPP_f540970102l_bool(Pa,bot_bo1181479936a_bool)) -> ((all X_2 all F_2 (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,F_2)) -> (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_2),F_2)) -> (hBOOL(hAPP_f540970102l_bool(Pa,F_2)) -> hBOOL(hAPP_f540970102l_bool(Pa,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_2),F_2))))))) -> hBOOL(hAPP_f540970102l_bool(Pa,F)))))) # label(fact_333_finite__induct) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 391 (all P all Q hAPP_H928324994_state(hAPP_H558669354_state(cOMBK_547979437iple_a,P),Q) = P) # label(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_Itc__Com__) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 392 (all F_1 all A (bot_bo1181479936a_bool = A <-> image_194810223_a_nat(F_1,A) = bot_bot_fun_nat_bool)) # label(fact_221_image__is__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 393 (all X_1 all A hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A)) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A)) # label(fact_85_insert__absorb2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 394 (all C_3 all A_10 all B_6 (A_10 = B_6 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_6),C_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_10),C_3))))) # label(fact_522_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 395 (all B_1_1 all B_2_1 is_bool(hAPP_H513860823e_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_Itc__Com__Ostate_J_0) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 396 (all F_1 all X_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,hAPP_H678412245iple_a(F_1,X_1)),image_306007685iple_a(F_1,A))))) # label(fact_204_imageI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 397 (all X all Y (X != Y | hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(fequal_state,X),Y)))) # label(help_fequal_2_1_fequal_000tc__Com__Ostate_T) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 398 (all A all B (A = bot_bo1055319631e_bool & B = bot_bo1055319631e_bool <-> bot_bo1055319631e_bool = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila447562797e_bool,A),B))) # label(fact_510_Un__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 399 (all A_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) -> (exists B_15 (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),B_15)) & A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B_15))))) # label(fact_140_mk__disjoint__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 400 (all Vname_1 all Fun skip != ass(Vname_1,Fun)) # label(fact_217_com_Osimps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 401 (all B all Ca all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),A)) -> (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),B)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),B)))))) # label(fact_621_IntI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 402 (all X_35 hAPP_f124283079iple_a(the_el1436340927iple_a,X_35) = hAPP_f124283079iple_a(the_Ho746640593iple_a,hAPP_f772297704a_bool(hAPP_f390613447a_bool(cOMBB_1501709507iple_a,hAPP_f430043647l_bool(fequal1258664663a_bool,X_35)),hAPP_f1665407592a_bool(hAPP_f549683569a_bool(cOMBC_708118077a_bool,insert873085594iple_a),bot_bo1181479936a_bool)))) # label(fact_149_the__elem__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 403 (all Y_16 all X_20 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_16),X_20)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_20),Y_16)) -> X_20 = Y_16))) # label(fact_516_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 404 (all Ga all G_1 all Ts (hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(G_1),Ts)) -> (hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),G_1)) -> hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),Ts))))) # label(fact_5_cut) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 405 (all A_1 all B all C hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,B),C)) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B)),C)) # label(fact_567_Un__insert__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 406 (all Pa all A_1 ((-hBOOL(hAPP_nat_bool(Pa,A_1)) -> bot_bot_fun_nat_bool = collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_1))),Pa))) & (hBOOL(hAPP_nat_bool(Pa,A_1)) -> collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_1))),Pa)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)))) # label(fact_40_Collect__conv__if) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 407 (all X_3 all Y_1 all Z hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_3),Y_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_3),Z))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,Y_1),Z))))) # label(fact_641_distrib__inf__le) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 408 (all A_1 all A all F_1 all F (hBOOL(finite506823037_state(F_1,F)) -> (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A)) -> hAPP_f2143211163_state(finite202520804_state(F_1,A_1),A) = hAPP_f2143211163_state(F,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),A))))) # label(fact_433_folding__one__idem_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 409 (all Ca all X_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Ca),bot_bo1055319631e_bool) = image_1604514514_state(hAPP_H558669354_state(cOMBK_547979437iple_a,Ca),A))) # label(fact_175_image__constant) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 410 (all C_2 all S_1 all N_2 all T (hBOOL(evaln(C_2,S_1,N_2,T)) -> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C_2),S_1),T)))) # label(fact_286_evaln__evalc) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 411 (all F_1 all A (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> (bot_bo1181479936a_bool != A -> (exists X1 hBOOL(hAPP_H1421470952a_bool(finite1537818352iple_a(F_1,A),X1)))))) # label(fact_329_finite__nonempty__imp__fold1Set) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 412 (all A_1 all A (bot_bo1055319631e_bool = A -> -hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),A)))) # label(fact_44_equals0D) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 413 (all Q_1 all Ga all Pa all Ca all Q_3 (hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(Pa,Ca,Q_3)),bot_bo1055319631e_bool))) -> ((all Z_7 all S_2 (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_3,Z_7),S_2)) -> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_1,Z_7),S_2)))) -> hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(Pa,Ca,Q_1)),bot_bo1055319631e_bool)))))) # label(fact_13_conseq2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 414 (all B_1_1 all B_2_1 is_bool(hAPP_nat_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 415 (all Pa Pa = collec268032053iple_a(Pa)) # label(fact_232_Collect__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 416 (all S_1 hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(skip),S_1),S_1))) # label(fact_278_evalc_OSkip) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 417 (all Loc_1 all Glb_1 loc(Loc_1) != glb(Glb_1)) # label(fact_323_vname_Osimps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 418 (all Pa ((exists X_2 (hBOOL(hAPP_H1421470952a_bool(Pa,X_2)) & (all Y_11 (hBOOL(hAPP_H1421470952a_bool(Pa,Y_11)) -> X_2 = Y_11)))) -> hBOOL(hAPP_H1421470952a_bool(Pa,hAPP_f124283079iple_a(the_Ho746640593iple_a,Pa))))) # label(fact_268_theI_H) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 419 (all Vname_1 all Fun ass(Vname_1,Fun) != skip) # label(fact_216_com_Osimps_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 420 (all A_1 (hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A_1),bot_bot_bool)) -> (hBOOL(bot_bot_bool) <-> hBOOL(A_1)))) # label(fact_557_le__bot) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 421 (all X_11 all Y_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_11),Y_8)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_11),Y_8) = X_11)) # label(fact_631_inf__absorb1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 422 (all X_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> -(all B_15 (A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B_15) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_15)))))) # label(fact_137_Set_Oset__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 423 (all F_1 all A_1 all X_35 all X_1 (hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),X_35)),X_1)) -> -(all A_7 all A_19 (hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_7),A_19) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),X_35) -> (hBOOL(hAPP_nat_bool(finite929467206at_nat(F_1,A_7,A_19),X_1)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_7),A_19))))))) # label(fact_294_insert__fold1SetE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 424 (all Pa (collect_fun_nat_bool(Pa) = bot_bo1701429464l_bool <-> (all X_2 -hBOOL(hAPP_f54304608l_bool(Pa,X_2))))) # label(fact_48_Collect__empty__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 425 (all X_2 (hBOOL(bot_bot_bool) <-> hBOOL(hAPP_H513860823e_bool(bot_bo1055319631e_bool,X_2)))) # label(fact_129_bot__fun__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 426 (all Ca all X_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) = image_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,Ca),A))) # label(fact_171_image__constant) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 427 (all A_1 all A (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) <-> hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),A))))) # label(fact_322_finite__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 428 (all A_1 all A bot_bo1055319631e_bool != hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),A)) # label(fact_117_insert__not__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 429 (all Loc_3 all Fun_1 all Com_1 all Com1 all Com2 semi(Com1,Com2) != local(Loc_3,Fun_1,Com_1)) # label(fact_254_com_Osimps_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 430 (all F_1 all X_1 -hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F_1,bot_bot_fun_nat_bool),X_1))) # label(fact_261_empty__fold1SetE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 431 (all X_1 all A all B (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),B)) -> (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),B))))) # label(fact_533_in__mono) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 432 (all A all X_1 all B ((-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),B)) -> hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A)),B) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),B))) & (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),B)) -> hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),B) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A)),B)))) # label(fact_379_insert__Diff__if) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 433 (all F_1 all A_1 A_1 = hAPP_f22106695ol_nat(finite_fold1_nat(F_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool))) # label(fact_419_fold1__singleton) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 434 (all P all Q all R hAPP_f746301080e_bool(hAPP_s286259371e_bool(P,R),Q) = hAPP_s1806633685e_bool(hAPP_f1838002347e_bool(hAPP_f1706273077e_bool(cOMBC_867582640e_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__fun_Itc__Com__Ostate_Mtc__Com__Os) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 435 (all F all Ga (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,F),Ga))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,Ga)) & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F)))) # label(fact_512_finite__Un) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 436 (all Z_2 all F_1 all A ((exists X_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) & Z_2 = hAPP_nat_nat(F_1,X_2))) <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Z_2),image_nat_nat(F_1,A))))) # label(fact_209_image__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 437 (all B hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila447562797e_bool,bot_bo1055319631e_bool),B) = B) # label(fact_504_Un__empty__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 438 (all Y_7 -(all Fun1 all Com_2 all Fun2 hoare_1760757500iple_a(Fun1,Com_2,Fun2) != Y_7)) # label(fact_135_triple_Oexhaust) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 439 (all Pa all Q_1 hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,collect_nat(Pa)),collect_nat(Q_1)) = collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),Pa)),Q_1))) # label(fact_666_Collect__conj__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 440 (all Z_5 all Y_17 all X_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_17),X_21)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_5),Y_17)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_5),X_21))))) # label(fact_515_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 441 (all Ca all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B))) -> -(hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)) -> -hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B))))) # label(fact_624_IntE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 442 (all X_1 all A all F_1 all F (hBOOL(finite1669978781iple_a(F_1,F)) -> (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> (bot_bo1181479936a_bool != hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),bot_bo1181479936a_bool)) -> hAPP_f124283079iple_a(F,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A)) = hAPP_H678412245iple_a(hAPP_H1455657330iple_a(F_1,X_1),hAPP_f124283079iple_a(F,hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),bot_bo1181479936a_bool))))) & (bot_bo1181479936a_bool = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),bot_bo1181479936a_bool)) -> X_1 = hAPP_f124283079iple_a(F,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A)))))) # label(fact_350_folding__one_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 443 (all S_1 all N_2 hBOOL(evaln(skip,S_1,N_2,S_1))) # label(fact_275_evaln_OSkip) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 444 (all X all Y (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) | X = Y)) # label(help_fequal_1_1_fequal_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 445 (all A_1 all B all C hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)),C) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,B),C))) # label(fact_565_Un__insert__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 446 (all Ca all A all B (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),B))) -> -(hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),A)) -> -hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),B))))) # label(fact_623_IntE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 447 (all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hAPP_f22106695ol_nat(big_la43341705in_nat,A) = hAPP_f22106695ol_nat(finite_fold1_nat(semila972727038up_nat),A))) # label(fact_610_Sup__fin_OF__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 448 (all F_1 all A (bot_bot_fun_nat_bool = A <-> image_1782127643iple_a(F_1,A) = bot_bo1181479936a_bool)) # label(fact_219_image__is__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 449 (all Ca all X_1 all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ca),bot_bo1181479936a_bool) = image_129517430iple_a(hAPP_H377435237iple_a(cOMBK_2022976521_state,Ca),A))) # label(fact_170_image__constant) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 450 (all B all A_1 all C ((-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),C)) -> hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B)),C) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,B),C)) & (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),C)) -> hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,B),C)) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B)),C)))) # label(fact_658_Int__insert__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 451 (all A all A_1 all B hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),B)),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool)) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B))) # label(fact_394_Diff__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 452 (all Pa all A all F (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),A)) -> (hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool)) -> ((all A_7 all F_2 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_7),A)) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_7),F_2)) -> (hBOOL(hAPP_f54304608l_bool(Pa,F_2)) -> hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_7),F_2)))))))) -> hBOOL(hAPP_f54304608l_bool(Pa,F))))))) # label(fact_595_finite__subset__induct) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 453 (all X_1 all Y_7 all A hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_7),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_7),A))) # label(fact_83_insert__commute) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 454 (all X_1 all A all F_1 all F (hBOOL(finite774711482_state(F_1,F)) -> (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A)) -> (bot_bo1055319631e_bool != hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),bot_bo1055319631e_bool)) -> hAPP_H563960305_state(hAPP_H521649881_state(F_1,X_1),hAPP_f2143211163_state(F,hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),bot_bo1055319631e_bool)))) = hAPP_f2143211163_state(F,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),A))) & (bot_bo1055319631e_bool = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),bot_bo1055319631e_bool)) -> X_1 = hAPP_f2143211163_state(F,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),A)))))) # label(fact_349_folding__one_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 455 (all X_1 all A hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) # label(fact_86_insert__absorb2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 456 (all Pa (bot_bo1701429464l_bool = collect_fun_nat_bool(Pa) <-> (all X_2 -hBOOL(hAPP_f54304608l_bool(Pa,X_2))))) # label(fact_55_empty__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 457 (all Vname all Fun_1 all Vname_1 all Fun (is_vname(Vname_1) & is_vname(Vname) -> (Fun = Fun_1 & Vname_1 = Vname <-> ass(Vname_1,Fun) = ass(Vname,Fun_1)))) # label(fact_188_com_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 458 (all P all Q all R hAPP_n1699378549t_bool(P,hAPP_nat_nat(Q,R)) = hAPP_n1699378549t_bool(hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 459 (all Pa all Q_1 hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,collec268032053iple_a(Pa)),collec268032053iple_a(Q_1)) = collec268032053iple_a(hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fconj),Pa)),Q_1))) # label(fact_665_Collect__conj__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 460 (all A all B (bot_bot_fun_nat_bool = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B) -> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) = A)) # label(fact_694_Diff__triv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 461 (all A_1 all A hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),A) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool)))) # label(fact_388_insert__Diff__single) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 462 (all B all X_1 all A (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) -> (hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A) <-> A = B)))) # label(fact_74_insert__ident) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 463 (all X_1 all A all F_1 all F (hBOOL(finite988810631ne_nat(F_1,F)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hAPP_f22106695ol_nat(finite_fold_nat_nat(F_1,X_1),A) = hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)))))) # label(fact_429_folding__one_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 464 (all A_1 all B collec727977250_state(hAPP_f921536533e_bool(hAPP_f760664097e_bool(cOMBS_1248383340l_bool,hAPP_f1558728829l_bool(hAPP_f1442918689l_bool(cOMBB_1382207997_state,fdisj),hAPP_H1645666623e_bool(hAPP_f262880489e_bool(cOMBC_1424981238e_bool,fequal1531560888_state),A_1))),hAPP_f921536533e_bool(hAPP_f1262649863e_bool(cOMBC_764456866l_bool,member1758697444_state),B))) = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B)) # label(fact_95_insert__compr) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 465 (all Pa all A_1 ((-hBOOL(hAPP_H1421470952a_bool(Pa,A_1)) -> collec268032053iple_a(hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fconj),hAPP_H1190454433a_bool(hAPP_f1371755681a_bool(cOMBC_839220826a_bool,fequal879838495iple_a),A_1))),Pa)) = bot_bo1181479936a_bool) & (hBOOL(hAPP_H1421470952a_bool(Pa,A_1)) -> collec268032053iple_a(hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fconj),hAPP_H1190454433a_bool(hAPP_f1371755681a_bool(cOMBC_839220826a_bool,fequal879838495iple_a),A_1))),Pa)) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool)))) # label(fact_39_Collect__conv__if) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 466 (all Pa all Q_1 collec268032053iple_a(hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fdisj),Pa)),Q_1)) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,collec268032053iple_a(Pa)),collec268032053iple_a(Q_1))) # label(fact_527_Collect__disj__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 467 (all A all F_1 all F (hBOOL(finite1669978781iple_a(F_1,F)) -> (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> (A != bot_bo1181479936a_bool -> ((all X_2 all Y_11 hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,hAPP_H678412245iple_a(hAPP_H1455657330iple_a(F_1,X_2),Y_11)),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_2),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Y_11),bot_bo1181479936a_bool))))) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,hAPP_f124283079iple_a(F,A)),A))))))) # label(fact_327_folding__one_Oclosed) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 468 (all A_1 all B hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,collec268032053iple_a(hAPP_H1190454433a_bool(hAPP_f1371755681a_bool(cOMBC_839220826a_bool,fequal879838495iple_a),A_1))),B) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B)) # label(fact_574_insert__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 469 (all A_1 all A (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),A))) <-> hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A)))) # label(fact_321_finite__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 470 (all Pa all Q_1 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),Pa)),Q_1)))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Pa))) & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Q_1))))) # label(fact_318_finite__Collect__disjI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 471 (all X_1 all A all F_1 all F (hBOOL(finite988810631ne_nat(F_1,F)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) != bot_bot_fun_nat_bool -> hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,X_1),hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))))) & (bot_bot_fun_nat_bool = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) -> X_1 = hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)))))) # label(fact_348_folding__one_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 472 (all Ca all A ((A != bot_bo1181479936a_bool -> hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Ca),bot_bo1055319631e_bool) = image_1604514514_state(hAPP_H558669354_state(cOMBK_547979437iple_a,Ca),A)) & (A = bot_bo1181479936a_bool -> image_1604514514_state(hAPP_H558669354_state(cOMBK_547979437iple_a,Ca),A) = bot_bo1055319631e_bool))) # label(fact_164_image__constant__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 473 (all A_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),A)) -> hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool))) = A)) # label(fact_382_insert__Diff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 474 (all Pa all A (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> (hBOOL(hAPP_f540970102l_bool(Pa,A)) -> ((all A_7 all A_19 (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A_19)) -> (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_7),A_19)) -> (hBOOL(hAPP_f540970102l_bool(Pa,A_19)) -> hBOOL(hAPP_f540970102l_bool(Pa,hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A_19),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_7),bot_bo1181479936a_bool)))))))) -> hBOOL(hAPP_f540970102l_bool(Pa,bot_bo1181479936a_bool)))))) # label(fact_405_finite__empty__induct) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 475 (all A_1 all C all D_2 (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,C),D_2)) -> hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),C)),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),D_2))))) # label(fact_476_insert__mono) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 476 (all A_1 -hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),bot_bot_fun_nat_bool))) # label(fact_26_emptyE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 477 (all P all Q all R hAPP_f167292325e_bool(P,hAPP_a849909144l_bool(Q,R)) = hAPP_a723219176e_bool(hAPP_f340725611e_bool(hAPP_f1006724181e_bool(cOMBB_1348041619bool_a,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 478 (all B all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (A != bot_bot_fun_nat_bool -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) -> (bot_bot_fun_nat_bool != B -> (hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B) = bot_bot_fun_nat_bool -> hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,hAPP_f22106695ol_nat(big_la43341705in_nat,A)),hAPP_f22106695ol_nat(big_la43341705in_nat,B)))))))) # label(fact_620_Sup__fin_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 479 (all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (A != bot_bot_fun_nat_bool -> ((all X_2 all Y_11 hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_2),Y_11)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_11),bot_bot_fun_nat_bool))))) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_f22106695ol_nat(big_la43341705in_nat,A)),A)))))) # label(fact_619_Sup__fin_Oclosed) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 480 (all Z_2 all F_1 all A ((exists X_2 (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_2),A)) & Z_2 = hAPP_H738206399_a_nat(F_1,X_2))) <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Z_2),image_194810223_a_nat(F_1,A))))) # label(fact_210_image__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 481 (all X_1 all Y_7 ((hBOOL(X_1) <-> hBOOL(bot_bot_bool)) & (hBOOL(bot_bot_bool) <-> hBOOL(Y_7)) <-> (hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_1),Y_7)) <-> hBOOL(bot_bot_bool)))) # label(fact_491_sup__eq__bot__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 482 (all B B = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,bot_bo1181479936a_bool),B)) # label(fact_505_Un__empty__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 483 (all B_1_1 all B_2_1 is_bool(hAPP_f54304608l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 484 (all A hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,bot_bo1181479936a_bool),A) = bot_bo1181479936a_bool) # label(fact_372_empty__Diff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 485 (all Ca all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) | hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B))))) # label(fact_540_Un__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 486 (all A_1 all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),A)) -> hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),A) = A)) # label(fact_69_insert__absorb) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 487 (all X_1 X_1 = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila447562797e_bool,X_1),bot_bo1055319631e_bool)) # label(fact_497_sup__bot__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 488 (all Ga all G_1 all Ts (hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(G_1),Ts)) -> (hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),G_1)) -> hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),Ts))))) # label(fact_4_cut) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 489 (all Com1_2 all Com2_2 skip != semi(Com1_2,Com2_2)) # label(fact_148_com_Osimps_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 490 (all Ca all A all B (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),B)) -> (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),B))))) # label(fact_446_subsetD) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 491 (all Ca all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)))) # label(fact_359_DiffD1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 492 (all F_1 all Z_2 hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,Z_2,bot_bo1181479936a_bool),Z_2))) # label(fact_288_fold__graph_OemptyI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 493 (all Ts all Ga (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,Ts),Ga)) -> hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),Ts)))) # label(fact_571_asm) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 494 (all N all H all F_1 all F (hBOOL(finite506823037_state(F_1,F)) -> ((all X_2 all Y_11 hAPP_H563960305_state(hAPP_H521649881_state(F_1,hAPP_H563960305_state(H,X_2)),hAPP_H563960305_state(H,Y_11)) = hAPP_H563960305_state(H,hAPP_H563960305_state(hAPP_H521649881_state(F_1,X_2),Y_11))) -> (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,N)) -> (N != bot_bo1055319631e_bool -> hAPP_f2143211163_state(F,image_650584225_state(H,N)) = hAPP_H563960305_state(H,hAPP_f2143211163_state(F,N))))))) # label(fact_401_folding__one__idem_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 495 (all A all B (hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,A),B) = bot_bo1181479936a_bool <-> bot_bo1181479936a_bool = A & B = bot_bo1181479936a_bool)) # label(fact_511_Un__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 496 (all X all Y (X = Y | -hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(fequal1746921144e_bool,X),Y)))) # label(help_fequal_1_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____ddpglwnxwg__Otriple_007) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 497 (all Ba all F_1 all X_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> (hAPP_nat_nat(F_1,X_1) = Ba -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_nat_nat(F_1,A)))))) # label(fact_192_rev__image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 498 (all Y_27 image_650584225_state(cOMBI_1367665338_state,Y_27) = Y_27) # label(fact_186_image__ident) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 499 (all A_1 all A all F_1 all F (hBOOL(finite950012314iple_a(F_1,F)) -> (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> hAPP_f124283079iple_a(finite978536264iple_a(F_1,A_1),A) = hAPP_f124283079iple_a(F,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),A))))) # label(fact_434_folding__one__idem_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 500 (all X_1 X_1 = hAPP_f124283079iple_a(the_Ho746640593iple_a,hAPP_H1190454433a_bool(fequal879838495iple_a,X_1))) # label(fact_240_the__sym__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 501 (all A ((all X_2 -hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))) <-> A = bot_bot_fun_nat_bool)) # label(fact_62_all__not__in__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 502 (all A all X_1 all B (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),B)) -> hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),A)),B) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),B))) # label(fact_375_insert__Diff1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.04 503 (all X_4 all Y_2 all Z_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,Y_2),Z_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_4),Y_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_4),Z_1))))) # label(fact_640_distrib__sup__le) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 504 (all B_1_1 all B_2_1 is_bool(hAPP_H1421470952a_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_It__a_J_000tc__HOL__) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 505 (all A_1 all A hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),A) != bot_bo1055319631e_bool) # label(fact_120_empty__not__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 506 (all X_1 X_1 = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_1),bot_bot_fun_nat_bool)) # label(fact_496_sup__bot__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 507 (all A_1 hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,bot_bo1181479936a_bool),A_1))) # label(fact_550_bot__least) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 508 (all B all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (bot_bot_fun_nat_bool != A -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) -> (bot_bot_fun_nat_bool != B -> hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,hAPP_f22106695ol_nat(big_la43341705in_nat,A)),hAPP_f22106695ol_nat(big_la43341705in_nat,B))))))) # label(fact_614_Sup__fin_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 509 (all A all B collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B))) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B)) # label(fact_542_Un__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 510 (all P all Q hAPP_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,P),Q) = P) # label(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 511 (all Pa all F (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,F)) -> (bot_bo1055319631e_bool != F -> ((all X_2 hBOOL(hAPP_f1760790145l_bool(Pa,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_2),bot_bo1055319631e_bool)))) -> ((all X_2 all F_2 (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,F_2)) -> (bot_bo1055319631e_bool != F_2 -> (-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_2),F_2)) -> (hBOOL(hAPP_f1760790145l_bool(Pa,F_2)) -> hBOOL(hAPP_f1760790145l_bool(Pa,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_2),F_2)))))))) -> hBOOL(hAPP_f1760790145l_bool(Pa,F))))))) # label(fact_342_finite__ne__induct) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 512 (all Com1_2 all Com2_2 skip != semi(Com1_2,Com2_2)) # label(fact_147_com_Osimps_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 513 (all Q_1 all Pa all Ga all P_2 all Ca all Q_3 (hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(P_2,Ca,Q_3)),bot_bo1055319631e_bool))) -> ((all Z_7 all S_2 (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_7),S_2)) -> (all S_3 ((all Z_8 (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_2,Z_8),S_2)) -> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_3,Z_8),S_3)))) -> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_1,Z_7),S_3)))))) -> hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(Pa,Ca,Q_1)),bot_bo1055319631e_bool)))))) # label(fact_17_conseq12) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 514 (all P all Q all R hAPP_f54304608l_bool(hAPP_f103356543l_bool(P,R),Q) = hAPP_f54304608l_bool(hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 515 (all B all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hAPP_f800510211t_bool(finite1578363458t_bool(hAPP_f10625010t_bool(hAPP_f1115950719t_bool(cOMBB_1654519265ol_nat,hAPP_f1311642927t_bool(cOMBC_646315179t_bool,minus_2067140911t_bool)),hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,insert_nat),bot_bot_fun_nat_bool)),B),A) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,B),A))) # label(fact_437_minus__fold__remove) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 516 (all B_13 all D_1 all A_17 all C_7 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_17),C_7)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_13),D_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_17),B_13)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,C_7),D_1)))))) # label(fact_478_sup__mono) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 517 (all Ba all F_1 all X_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> (hAPP_H738206399_a_nat(F_1,X_1) = Ba -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_194810223_a_nat(F_1,A)))))) # label(fact_193_rev__image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 518 (all X_1 all A all F_1 all F (hBOOL(finite950012314iple_a(F_1,F)) -> (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hAPP_f124283079iple_a(F,A) = hAPP_H678412245iple_a(hAPP_H1455657330iple_a(F_1,X_1),hAPP_f124283079iple_a(F,A)))))) # label(fact_398_folding__one__idem_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 519 (all D_3 all R_1 all Ga all Pa all Ca all Q_1 (hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(Pa,Ca,Q_1)),bot_bo1055319631e_bool))) -> (hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(Q_1,D_3,R_1)),bot_bo1055319631e_bool))) -> hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,hoare_1575745797_state(Pa,semi(Ca,D_3),R_1)),bot_bo1055319631e_bool)))))) # label(fact_133_Comp) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 520 (all Pa (collec727977250_state(Pa) = bot_bo1055319631e_bool <-> (all X_2 -hBOOL(hAPP_H513860823e_bool(Pa,X_2))))) # label(fact_56_empty__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 521 (all B_1_1 all B_2_1 is_bool(finite795500164em_nat(B_1_1,B_2_1))) # label(gsy_c_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 522 (all Pa all A_1 (hBOOL(hAPP_H1421470952a_bool(Pa,A_1)) -> ((all X_2 (hBOOL(hAPP_H1421470952a_bool(Pa,X_2)) -> A_1 = X_2)) -> A_1 = hAPP_f124283079iple_a(the_Ho746640593iple_a,Pa)))) # label(fact_246_the__equality) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 523 (all X_1 (hBOOL(bot_bot_bool) <-> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila310582991f_bool,X_1),bot_bot_bool)))) # label(fact_690_inf__bot__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 524 (all Ca all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) -> -hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)))) # label(fact_357_DiffD2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 525 (all Z_2 all F_1 all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,Z_2),image_650584225_state(F_1,A))) <-> (exists X_2 (hAPP_H563960305_state(F_1,X_2) = Z_2 & hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_2),A)))))) # label(fact_207_image__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 526 (all Ga all Pa all Ca all Q_1 all C ((hBOOL(C) -> hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(Pa,Ca,Q_1)),bot_bo1181479936a_bool)))) -> hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(hAPP_b540892988e_bool(hAPP_f1824947087e_bool(cOMBC_41962815e_bool,hAPP_f340725611e_bool(hAPP_f1006724181e_bool(cOMBB_1348041619bool_a,cOMBC_231445413l_bool),hAPP_f1509969235l_bool(hAPP_f1178339559l_bool(cOMBB_1355796797bool_a,hAPP_f1561913689l_bool(cOMBB_188601460_state,fconj)),Pa))),C),Ca,Q_1)),bot_bo1181479936a_bool))))) # label(fact_8_constant) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 527 (all X_1 (hBOOL(bot_bot_bool) <-> hBOOL(hAPP_H513860823e_bool(bot_bo1055319631e_bool,X_1)))) # label(fact_126_bot__apply) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 528 (all C_5 all A_12 all B_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_12),B_8)) -> (C_5 = B_8 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_12),C_5))))) # label(fact_520_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 529 (all F_1 all Z_2 all A hAPP_f124283079iple_a(the_Ho746640593iple_a,finite2010942150iple_a(F_1,Z_2,A)) = hAPP_f124283079iple_a(finite978536264iple_a(F_1,Z_2),A)) # label(fact_426_fold__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 530 (all X_18 all Y_14 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_18),Y_14)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_14),X_18)) -> X_18 = Y_14))) # label(fact_518_order__antisym) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 531 (all A_1 hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,bot_bo1055319631e_bool),A_1))) # label(fact_549_bot__least) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 532 (all A all X_1 all B (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),B)) -> hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),B) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A)),B))) # label(fact_376_insert__Diff1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 533 (all A_1 all B hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_1))),B) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)) # label(fact_572_insert__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 534 (all Ba all A_1 all B (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),B)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ba),B))))) # label(fact_70_insertI2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 535 (all Ca all A all B (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,A),B))) -> (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),B))))) # label(fact_450_UnE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 536 (all Ca all A all B (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),B))) -> -hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),B)))) # label(fact_356_DiffD2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 537 (all Y_7 -(all Fun1 all Com_2 all Fun2 Y_7 != hoare_1575745797_state(Fun1,Com_2,Fun2))) # label(fact_134_triple_Oexhaust) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 538 (all A_1 all A all B hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B))) # label(fact_654_insert__inter__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 539 (all P all Q all R hAPP_b589554111l_bool(P,hAPP_state_bool(Q,R)) = hAPP_s58564346l_bool(hAPP_f1259673775l_bool(hAPP_f1561913689l_bool(cOMBB_188601460_state,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_004) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 540 (all A ((exists X_2 exists B_15 (hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),B_15) = A & -hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B_15)))) <-> A != bot_bot_fun_nat_bool)) # label(fact_154_nonempty__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 541 (all Ca all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) -> -(hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B))))) # label(fact_352_DiffE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 542 (all P (is_bool(P) -> fFalse = P | P = fTrue)) # label(help_fFalse_1_1_T) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 543 (all A_1 collec268032053iple_a(hAPP_H1190454433a_bool(hAPP_f1371755681a_bool(cOMBC_839220826a_bool,fequal879838495iple_a),A_1)) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool)) # label(fact_31_singleton__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 544 (all A all X_1 all B ((-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) -> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B)) & (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) -> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B)))) # label(fact_377_insert__Diff__if) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 545 (all X_24 all Y_20 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_24),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_24),Y_20)))) # label(fact_489_sup__ge1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 546 (all A_1 all A (bot_bo1181479936a_bool = A -> -hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),A)))) # label(fact_43_equals0D) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 547 (all A all B bot_bo1055319631e_bool = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,A),hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,B),A))) # label(fact_698_Diff__disjoint) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 548 (all P all Q all R hAPP_H1421470952a_bool(hAPP_H1190454433a_bool(P,R),Q) = hAPP_H1421470952a_bool(hAPP_H1190454433a_bool(hAPP_f1371755681a_bool(cOMBC_839220826a_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_It__a_J_00) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 549 (all S_1 all T (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(skip),S_1),T)) -> S_1 = T)) # label(fact_279_evalc__elim__cases_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 550 (all A_1 all A hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)),A)) # label(fact_575_insert__is__Un) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 551 (all Ga all F (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,Ga)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,F),Ga)))))) # label(fact_513_finite__UnI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 552 (all A all Ca all B ((-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),B)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),A))) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,A),B))))) # label(fact_448_UnCI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 553 (all A_1 all A (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),A))))) # label(fact_316_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 554 (all B all A_1 all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),A)) -> hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,A),B)) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B)))) # label(fact_646_Int__insert__right__if1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 555 (all P all Q all R hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(hAPP_f450029403a_bool(cOMBC_314077933a_bool,P),Q),R) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____ddpglwnxwg__Otriple_I) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 556 (all B all A_1 all C (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),C)) -> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,B),C)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)),C))) # label(fact_642_Int__insert__left__if1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 557 (all Q_1 all Ga all Pa all Ca all Q_3 (hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(Pa,Ca,Q_3)),bot_bo1181479936a_bool))) -> ((all Z_7 all S_2 (hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_3,Z_7),S_2)) -> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_7),S_2)))) -> hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(Pa,Ca,Q_1)),bot_bo1181479936a_bool)))))) # label(fact_12_conseq2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 558 (all Ba all A all B (hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,A),B)) -> hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Ba),B))))) # label(fact_472_subset__insertI2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 559 (all A bot_bot_fun_nat_bool = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,bot_bot_fun_nat_bool),A)) # label(fact_370_empty__Diff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 560 (all X_2 (hBOOL(hAPP_H1421470952a_bool(bot_bo1181479936a_bool,X_2)) <-> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_2),bot_bo1181479936a_bool)))) # label(fact_156_bot__empty__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 561 (all X_1 X_1 = hAPP_f22106695ol_nat(the_elem_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))) # label(fact_122_the__elem__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 562 (all B all A_1 hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,B),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B)))) # label(fact_464_subset__insertI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 563 (all A all B (bot_bo1181479936a_bool = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),B) -> hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),B) = A)) # label(fact_696_Diff__triv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 564 (all A (A = bot_bo1055319631e_bool <-> hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,A),bot_bo1055319631e_bool)))) # label(fact_458_subset__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 565 (all Ba all A_1 all B ((-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),B)) -> A_1 = Ba) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),B))))) # label(fact_22_insertCI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 566 (all P all Q (is_bool(P) -> hAPP_H513860823e_bool(hAPP_b1245957081e_bool(cOMBK_1079618832_state,P),Q) = P)) # label(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____ddpglwnxwg___006) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 567 (all A all B bot_bo1181479936a_bool = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,B),A))) # label(fact_699_Diff__disjoint) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 568 (all C1 all C2 all S_1 all N_2 all T (hBOOL(evaln(semi(C1,C2),S_1,N_2,T)) -> -(all S1_1 (hBOOL(evaln(C1,S_1,N_2,S1_1)) -> -hBOOL(evaln(C2,S1_1,N_2,T)))))) # label(fact_293_evaln__elim__cases_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 569 (all A hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,bot_bo1055319631e_bool),A))) # label(fact_453_empty__subsetI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 570 (all P all Q all R hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(P,R),Q) = hAPP_H1190454433a_bool(hAPP_f1665407592a_bool(hAPP_f549683569a_bool(cOMBC_708118077a_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_It__a_J_00_029) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 571 (all F_1 all X_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hAPP_n178040171iple_a(F_1,X_1)),image_1782127643iple_a(F_1,A)) = image_1782127643iple_a(F_1,A))) # label(fact_235_insert__image) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 572 (all A_1 all Ba all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),A))) <-> Ba = A_1 | hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)))) # label(fact_80_insert__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 573 (all B_1_1 is_bool(finite1200705745iple_a(B_1_1))) # label(gsy_c_Finite__Set_Ocomp__fun__commute_000tc__Hoare____Mirabelle____ddpglwnxwg__O) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 574 (all B all A_1 all A (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) -> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B))) # label(fact_651_Int__insert__right__if0) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 575 (all X all Y (X != Y | hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y)))) # label(help_fequal_2_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 576 (all A_1 all Ga all Pa all Ca all Q_1 all X_35 all S_5 (hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(Pa,Ca,hAPP_f100967412e_bool(hAPP_f1460451647e_bool(cOMBC_524597097e_bool,hAPP_f1406200875e_bool(hAPP_f635443597e_bool(cOMBB_2139825703bool_a,cOMBB_844853809_state),Q_1)),hAPP_n1547241352_state(hAPP_f1848060885_state(cOMBC_1777403949_state,hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,update),loc(X_35))),getlocs(S_5,X_35))))),bot_bo1181479936a_bool))) -> hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(hAPP_f769584981e_bool(hAPP_f1159960589e_bool(cOMBB_237455441bool_a,hAPP_f644196280e_bool(cOMBS_1378840469l_bool,hAPP_f1259673775l_bool(hAPP_f1561913689l_bool(cOMBB_188601460_state,fconj),hAPP_s1806633685e_bool(fequal_state,S_5)))),hAPP_f100967412e_bool(hAPP_f1460451647e_bool(cOMBC_524597097e_bool,hAPP_f1406200875e_bool(hAPP_f635443597e_bool(cOMBB_2139825703bool_a,cOMBB_844853809_state),Pa)),hAPP_f871651461_state(hAPP_f851239890_state(cOMBS_777315357_state,hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,update),loc(X_35))),A_1))),local(X_35,A_1,Ca),Q_1)),bot_bo1181479936a_bool))))) # label(fact_251_hoare__derivs_OLocal) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 577 (all A all B (bot_bo1055319631e_bool = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,A),B) <-> (all X_2 (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_2),A)) -> (all Xa (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,Xa),B)) -> Xa != X_2)))))) # label(fact_678_disjoint__iff__not__equal) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 578 (all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))))) # label(fact_461_finite__subset) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 579 (all A_3 all B_2 all X_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_2),X_10)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_3),B_2)),X_10)))) # label(fact_632_le__infI2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 580 (all Ga hBOOL(hAPP_f1760790145l_bool(hoare_659004819_state(Ga),bot_bo1055319631e_bool))) # label(fact_1_empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 581 (all F_1 all X_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,hAPP_n178040171iple_a(F_1,X_1)),image_1782127643iple_a(F_1,A))))) # label(fact_206_imageI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 582 (all Pa collect_nat(Pa) = Pa) # label(fact_233_Collect__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 583 (all B all A_1 hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,B),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B)))) # label(fact_463_subset__insertI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 584 (all B hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,bot_bo1055319631e_bool),B) = bot_bo1055319631e_bool) # label(fact_684_Int__empty__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 585 (all A (A != bot_bo1181479936a_bool <-> (exists X_2 exists B_15 (A = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_2),B_15) & -hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_2),B_15)))))) # label(fact_153_nonempty__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 586 (all A ((all Y_11 -hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Y_11),A))) -> A = bot_bo1181479936a_bool)) # label(fact_142_equals0I) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 587 (all X_2 (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_2),bot_bo1055319631e_bool)) <-> hBOOL(hAPP_H513860823e_bool(bot_bo1055319631e_bool,X_2)))) # label(fact_157_bot__empty__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 588 (all A_1 -hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),bot_bo1181479936a_bool))) # label(fact_24_emptyE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 589 (all X_1 all F_1 all F (hBOOL(finite1669978781iple_a(F_1,F)) -> X_1 = hAPP_f124283079iple_a(F,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),bot_bo1181479936a_bool)))) # label(fact_249_folding__one_Osingleton) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 590 (all P all Q all R hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) = hAPP_n1006566506l_bool(hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 591 (all F_1 all A (bot_bo1181479936a_bool = image_129517430iple_a(F_1,A) <-> A = bot_bo1055319631e_bool)) # label(fact_226_empty__is__image) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 592 (all A all X_1 all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B))) <-> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),B))) & (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))))) # label(fact_588_subset__insert__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 593 (all X_1 X_1 = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,bot_bo1181479936a_bool),X_1)) # label(fact_502_sup__bot__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 594 (all P all Q all R hAPP_state_state(hAPP_f871651461_state(hAPP_f851239890_state(cOMBS_777315357_state,P),Q),R) = hAPP_nat_state(hAPP_s1892499976_state(P,R),hAPP_state_nat(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Com__Ostate_000tc__Nat__Onat_000tc__Com__Ostate_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 595 (all P all Q all R hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(P,R),Q) = hAPP_f54304608l_bool(hAPP_f1246832597l_bool(hAPP_f1443436725l_bool(cOMBC_1245412066l_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__019) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 596 (all N_3 all Pa all Ca all Q_1 (hBOOL(hAPP_H1421470952a_bool(hoare_560051114alid_a(N_3),hoare_1760757500iple_a(Pa,Ca,Q_1))) <-> (all Z_7 all S_2 (hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_7),S_2)) -> (all S_3 (hBOOL(evaln(Ca,S_2,N_3,S_3)) -> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_7),S_3)))))))) # label(fact_305_triple__valid__def2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 597 (all Ba all A_1 all B ((-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),B)) -> Ba = A_1) -> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Ba),B))))) # label(fact_23_insertCI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 598 (all A all X_1 all B (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),B))) <-> (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),bot_bo1181479936a_bool))),B))) & (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),B))))) # label(fact_590_subset__insert__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 599 (all C2 all S2 all N2 all T2 all C1 all S1 all N1 all T1 (hBOOL(evaln(C1,S1,N1,T1)) -> (hBOOL(evaln(C2,S2,N2,T2)) -> (exists N_1 (hBOOL(evaln(C2,S2,N_1,T2)) & hBOOL(evaln(C1,S1,N_1,T1))))))) # label(fact_303_evaln__max2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 600 (all Pa all A all F (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,F)) -> (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,F),A)) -> (hBOOL(hAPP_f540970102l_bool(Pa,bot_bo1181479936a_bool)) -> ((all A_7 all F_2 (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,F_2)) -> (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_7),A)) -> (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_7),F_2)) -> (hBOOL(hAPP_f540970102l_bool(Pa,F_2)) -> hBOOL(hAPP_f540970102l_bool(Pa,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_7),F_2)))))))) -> hBOOL(hAPP_f540970102l_bool(Pa,F))))))) # label(fact_597_finite__subset__induct) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 601 (all A all X_1 (hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),bot_bo1055319631e_bool))) -> A = bot_bo1055319631e_bool | hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),bot_bo1055319631e_bool) = A)) # label(fact_583_subset__singletonD) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 602 (all A_1 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A))))) # label(fact_314_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 603 (all P all Q all R hAPP_f921536533e_bool(hAPP_f1583986009e_bool(P,R),Q) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(hAPP_f2049746453e_bool(cOMBC_872426556e_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____ddpglwnxwg__Otriple_I_031) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 604 (all P all Q (-hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) | hBOOL(Q) | -hBOOL(P))) # label(help_fimplies_3_1_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 605 (all P all Q all R hAPP_f1759915619e_bool(hAPP_a1200519163e_bool(P,R),Q) = hAPP_a2036067514e_bool(hAPP_f762886889e_bool(hAPP_f1261923407e_bool(cOMBC_892787026e_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 606 (all A_1 all Ba all Ca all D_3 (hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ba),bot_bo1181479936a_bool)) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ca),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,D_3),bot_bo1181479936a_bool)) <-> A_1 = Ca & D_3 = Ba | Ca = Ba & A_1 = D_3)) # label(fact_109_doubleton__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 607 (all X_1 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> (bot_bot_fun_nat_bool != hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_1),hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) = hAPP_f22106695ol_nat(big_la43341705in_nat,A)) & (bot_bot_fun_nat_bool = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) -> X_1 = hAPP_f22106695ol_nat(big_la43341705in_nat,A))))) # label(fact_607_Sup__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 608 (all Ba all A_1 all B ((-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),B)) -> A_1 = Ba) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ba),B))))) # label(fact_21_insertCI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 609 (all A_1 all Pa collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fimplies),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,fNot),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_1)))),Pa)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),collect_nat(Pa))) # label(fact_89_insert__Collect) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 610 (all F_1 all A hAPP_f124283079iple_a(finite1382394752iple_a(F_1),A) = hAPP_f124283079iple_a(the_Ho746640593iple_a,finite1537818352iple_a(F_1,A))) # label(fact_436_fold1__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 611 (all P hAPP_nat_nat(cOMBI_nat,P) = P) # label(help_COMBI_1_1_COMBI_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 612 (all Ca all X_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ca),bot_bo1181479936a_bool) = image_1782127643iple_a(hAPP_H1450464520iple_a(cOMBK_1505147640_a_nat,Ca),A))) # label(fact_168_image__constant) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 613 (all A all X_1 all B (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),bot_bo1181479936a_bool))),B)) -> (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),B)))))) # label(fact_587_diff__single__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 614 (all A all X_1 (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),bot_bo1181479936a_bool))) -> bot_bo1181479936a_bool = A | A = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),bot_bo1181479936a_bool))) # label(fact_584_subset__singletonD) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 615 (all A all Ca all B (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),B)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,A),B))))) # label(fact_535_UnI2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 616 (all X_1 all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))))) # label(fact_534_in__mono) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 617 (all X_1 all A all Pa (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),collect_nat(Pa)))) <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) & hBOOL(hAPP_nat_bool(Pa,X_1)))) # label(fact_668_Int__Collect) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 618 (all F_1 all X_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hAPP_H678412245iple_a(F_1,X_1)),image_306007685iple_a(F_1,A)) = image_306007685iple_a(F_1,A))) # label(fact_234_insert__image) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 619 (all P all Q all R hAPP_a2036067514e_bool(hAPP_f769584981e_bool(hAPP_f1159960589e_bool(cOMBB_237455441bool_a,P),Q),R) = hAPP_f1759915619e_bool(P,hAPP_a2036067514e_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 620 (all A_1 all A hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)))) # label(fact_386_insert__Diff__single) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 621 (all B all A_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) -> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B)))) # label(fact_645_Int__insert__right__if1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 622 (all P all Q all R hAPP_state_state(hAPP_n1547241352_state(hAPP_f1848060885_state(cOMBC_1777403949_state,P),Q),R) = hAPP_nat_state(hAPP_s1892499976_state(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__Nat__Onat_000tc__Com__Ostate_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 623 (all B_3 all X_13 all A_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_13),A_4)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_13),B_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_13),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_4),B_3)))))) # label(fact_629_le__infI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 624 (all X_1 all A all F_1 all F (hBOOL(finite1669978781iple_a(F_1,F)) -> (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hAPP_f124283079iple_a(finite978536264iple_a(F_1,X_1),A) = hAPP_f124283079iple_a(F,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A)))))) # label(fact_431_folding__one_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 625 (all A_1 (bot_bot_fun_nat_bool = A_1 <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),bot_bot_fun_nat_bool)))) # label(fact_553_bot__unique) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 626 (all X_23 all Y_19 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_23),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_23),Y_19)))) # label(fact_490_inf__sup__ord_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 627 (all Z_2 all A all Y_7 all F_1 (hBOOL(finite1200705745iple_a(F_1)) -> (hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,Z_2,A),Y_7)) -> Y_7 = hAPP_f124283079iple_a(finite978536264iple_a(F_1,Z_2),A)))) # label(fact_425_comp__fun__commute_Ofold__equality) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 628 (all X_1 all A all F_1 all F (hBOOL(finite795500164em_nat(F_1,F)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (A != bot_bot_fun_nat_bool -> hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,X_1),hAPP_f22106695ol_nat(F,A)))))) # label(fact_338_folding__one__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 629 (all B all X_1 all A (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B)))))) # label(fact_468_subset__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 630 (all X_1 X_1 = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,X_1),bot_bo1181479936a_bool)) # label(fact_498_sup__bot__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 631 (all P all Q all R hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q) = hAPP_nat_bool(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 632 (all Fun1_2 all Com_1 all Fun2_2 all Fun1_1 all Com all Fun2_1 (hoare_1760757500iple_a(Fun1_2,Com_1,Fun2_2) = hoare_1760757500iple_a(Fun1_1,Com,Fun2_1) <-> Fun2_2 = Fun2_1 & Com_1 = Com & Fun1_2 = Fun1_1)) # label(fact_3_triple_Oinject) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 633 (all Q all P (-hBOOL(P) | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)))) # label(help_fdisj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 634 (all A all F_1 all F (hBOOL(finite1669978781iple_a(F_1,F)) -> (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> hAPP_f124283079iple_a(finite1382394752iple_a(F_1),A) = hAPP_f124283079iple_a(F,A)))) # label(fact_428_folding__one_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 635 (all F_1 all A1 all A2 (hBOOL(hAPP_H1421470952a_bool(finite1537818352iple_a(F_1,A1),A2)) <-> (exists A_7 exists A_19 exists X_2 (X_2 = A2 & hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,A_7,A_19),X_2)) & -hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_7),A_19)) & A1 = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_7),A_19))))) # label(fact_301_fold1Set_Osimps) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 636 (all A all X_1 all B ((hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),B)) -> hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),B) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),A)),B)) & (-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),B)) -> hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),B)) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),A)),B)))) # label(fact_378_insert__Diff__if) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 637 (all A_1 all A hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),A) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila1498788770a_bool,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool)),A)) # label(fact_577_insert__is__Un) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 638 (all X_1 all Y_7 (Y_7 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_1),Y_7) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_7)))) # label(fact_486_le__iff__sup) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 639 (all Y_27 all A_1 all Ca all S_4 all N_3 all T_1 (hBOOL(evaln(local(Y_27,A_1,Ca),S_4,N_3,T_1)) -> -(all S1_1 (hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S1_1),loc(Y_27)),getlocs(S_4,Y_27)) = T_1 -> -hBOOL(evaln(Ca,hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S_4),loc(Y_27)),hAPP_state_nat(A_1,S_4)),N_3,S1_1)))))) # label(fact_291_evaln__elim__cases_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 640 (all Vname all Fun_1 all Loc_2 all Fun all Com ass(Vname,Fun_1) != local(Loc_2,Fun,Com)) # label(fact_257_com_Osimps_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 641 (all X_1 all A (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> A = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A)),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),bot_bo1181479936a_bool)))) # label(fact_385_Diff__insert__absorb) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 642 (all A all B hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,B),A)) = bot_bot_fun_nat_bool) # label(fact_697_Diff__disjoint) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 643 (all X_1 all A all F_1 all F (hBOOL(finite506823037_state(F_1,F)) -> (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A)) -> (A != bot_bo1055319631e_bool -> hAPP_f2143211163_state(F,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),A)) = hAPP_H563960305_state(hAPP_H521649881_state(F_1,X_1),hAPP_f2143211163_state(F,A)))))) # label(fact_339_folding__one__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 644 (all A all Ca all B (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B))))) # label(fact_536_UnI2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 645 (all P all Q all R hAPP_n1699378549t_bool(hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,P),Q),R) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__010) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 646 (all A_1 all Ba all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),A))) -> (Ba != A_1 -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A))))) # label(fact_19_insertE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 647 (all X_35 hAPP_f22106695ol_nat(the_elem_nat,X_35) = hAPP_f22106695ol_nat(the_nat,hAPP_f158894502t_bool(hAPP_f1777703707t_bool(cOMBB_955900739ol_nat,hAPP_f103356543l_bool(fequal_fun_nat_bool,X_35)),hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,insert_nat),bot_bot_fun_nat_bool)))) # label(fact_150_the__elem__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 648 (all P P = hAPP_H563960305_state(cOMBI_1367665338_state,P)) # label(help_COMBI_1_1_COMBI_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_Itc__Com__) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 649 (all B all A ((all X_2 (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_2),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_2),B)))) -> hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),B)))) # label(fact_598_subsetI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 650 (all X_8 all Y_6 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_8),Y_6)),Y_6))) # label(fact_636_inf__le2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 651 (all P all Q all R hAPP_bool_bool(hAPP_s58564346l_bool(P,R),Q) = hAPP_state_bool(hAPP_b2019457360e_bool(hAPP_f167292325e_bool(cOMBC_231445413l_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 652 (all F_1 all A all X_1 (hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F_1,A),X_1)) -> A != bot_bot_fun_nat_bool)) # label(fact_264_fold1Set__nonempty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 653 (all F_1 all A (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A)) -> (bot_bo1055319631e_bool != A -> (exists X1 hBOOL(hAPP_H513860823e_bool(finite9525415_state(F_1,A),X1)))))) # label(fact_330_finite__nonempty__imp__fold1Set) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 654 (all Pa all A_1 ((hBOOL(hAPP_nat_bool(Pa,A_1)) -> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool) = collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(fequal_nat,A_1))),Pa))) & (-hBOOL(hAPP_nat_bool(Pa,A_1)) -> bot_bot_fun_nat_bool = collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(fequal_nat,A_1))),Pa))))) # label(fact_36_Collect__conv__if2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 655 (all P all Q all R hAPP_b2019457360e_bool(hAPP_s1874344717e_bool(P,R),Q) = hAPP_s1806633685e_bool(hAPP_b1095269219e_bool(hAPP_f701449317e_bool(cOMBC_471052088e_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__HOL__Obool_000tc__fun_Itc__Com__O) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 656 (all F_1 all Z_2 all A (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> (exists X1 hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,Z_2,A),X1))))) # label(fact_337_finite__imp__fold__graph) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 657 (all B_4 all D all A_5 all C_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_5),C_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_4),D)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_5),B_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,C_1),D)))))) # label(fact_627_inf__mono) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 658 (all Glb_3 all Glb_2 (is_glb(Glb_3) & is_glb(Glb_2) -> (glb(Glb_3) = glb(Glb_2) <-> Glb_2 = Glb_3))) # label(fact_319_vname_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 659 (all Y_7 all Z_2 all A all X_1 all F_1 (hBOOL(finite1200705745iple_a(F_1)) -> (hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,Z_2,A),X_1)) -> (hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,Z_2,A),Y_7)) -> Y_7 = X_1)))) # label(fact_414_comp__fun__commute_Ofold__graph__determ) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 660 (all F_1 all X_1 -hBOOL(hAPP_H513860823e_bool(finite9525415_state(F_1,bot_bo1055319631e_bool),X_1))) # label(fact_260_empty__fold1SetE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 661 (all A_1 all B hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_1),B) = collect_fun_nat_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,fdisj),hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,fequal_fun_nat_bool),A_1))),hAPP_f1246832597l_bool(hAPP_f1443436725l_bool(cOMBC_1245412066l_bool,member_fun_nat_bool),B)))) # label(fact_94_insert__compr) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 662 (all K hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),K))))) # label(fact_601_finite__Collect__le__nat) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 663 (all X_1 all A all F_1 all F (hBOOL(finite1669978781iple_a(F_1,F)) -> (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,A)) -> (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> (hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),bot_bo1181479936a_bool)) != bot_bo1181479936a_bool -> hAPP_H678412245iple_a(hAPP_H1455657330iple_a(F_1,X_1),hAPP_f124283079iple_a(F,hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),bot_bo1181479936a_bool)))) = hAPP_f124283079iple_a(F,A)) & (bot_bo1181479936a_bool = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),bot_bo1181479936a_bool)) -> X_1 = hAPP_f124283079iple_a(F,A)))))) # label(fact_347_folding__one_Oremove) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 664 (all Pa all A_1 ((hBOOL(hAPP_f54304608l_bool(Pa,A_1)) -> collect_fun_nat_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,fconj),hAPP_f103356543l_bool(fequal_fun_nat_bool,A_1))),Pa)) = hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_1),bot_bo1701429464l_bool)) & (-hBOOL(hAPP_f54304608l_bool(Pa,A_1)) -> bot_bo1701429464l_bool = collect_fun_nat_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,fconj),hAPP_f103356543l_bool(fequal_fun_nat_bool,A_1))),Pa))))) # label(fact_37_Collect__conv__if2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 665 (all Com1 all Com2 all Loc_3 all Fun_1 all Com_1 local(Loc_3,Fun_1,Com_1) != semi(Com1,Com2)) # label(fact_255_com_Osimps_I35_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 666 (all A all A_1 all B (hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),B))) <-> hBOOL(hAPP_f540970102l_bool(finite1655202547iple_a,hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B)))))) # label(fact_397_finite__Diff__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 667 (all F_1 all X_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_H738206399_a_nat(F_1,X_1)),image_194810223_a_nat(F_1,A))))) # label(fact_202_imageI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 668 (all A_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),A)) -> (exists B_15 (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),B_15)) & hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B_15) = A)))) # label(fact_139_mk__disjoint__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 669 (all X_35 all A_1 all S_4 all T_1 (hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(ass(X_35,A_1)),S_4),T_1)) -> hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S_4),X_35),hAPP_state_nat(A_1,S_4)) = T_1)) # label(fact_283_evalc__elim__cases_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 670 (all B_1_1 all B_2_1 is_bool(finite988810631ne_nat(B_1_1,B_2_1))) # label(gsy_c_Finite__Set_Ofolding__one_000tc__Nat__Onat) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 671 (all F_1 image_1604514514_state(F_1,bot_bo1181479936a_bool) = bot_bo1055319631e_bool) # label(fact_222_image__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 672 (all X_1 bot_bot_fun_nat_bool = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,X_1),bot_bot_fun_nat_bool)) # label(fact_691_inf__bot__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 673 (all X_2 (hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_2)) <-> hBOOL(bot_bot_bool))) # label(fact_128_bot__fun__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 674 (all P all Q P = hAPP_H1600811558iple_a(hAPP_H377435237iple_a(cOMBK_2022976521_state,P),Q)) # label(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_It__a_J_00_015) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 675 (all Pa (bot_bot_fun_nat_bool = collect_nat(Pa) <-> (all X_2 -hBOOL(hAPP_nat_bool(Pa,X_2))))) # label(fact_53_empty__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 676 (all Ca -hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),bot_bo1181479936a_bool))) # label(fact_50_empty__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 677 (all F_1 all A_1 all A all X_1 (hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,A_1,A),X_1)) -> (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),A)) -> hBOOL(hAPP_H1421470952a_bool(finite1537818352iple_a(F_1,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),A)),X_1))))) # label(fact_273_fold1Set_Ointros) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 678 (all F_1 all A_1 all Ba (hBOOL(hAPP_H1421470952a_bool(finite1537818352iple_a(F_1,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool)),Ba)) <-> Ba = A_1)) # label(fact_245_fold1Set__sing) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 679 (all Ga all Pa all X_35 all A_1 hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(hAPP_f100967412e_bool(hAPP_f1460451647e_bool(cOMBC_524597097e_bool,hAPP_f1406200875e_bool(hAPP_f635443597e_bool(cOMBB_2139825703bool_a,cOMBB_844853809_state),Pa)),hAPP_f871651461_state(hAPP_f851239890_state(cOMBS_777315357_state,hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,update),X_35)),A_1)),ass(X_35,A_1),Pa)),bot_bo1181479936a_bool)))) # label(fact_159_Ass) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 680 (all Ca hoare_Mirabelle_MGT(Ca) = hoare_1575745797_state(fequal_state,Ca,evalc(Ca))) # label(fact_297_MGT__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 681 (all A A = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),bot_bo1055319631e_bool)) # label(fact_368_Diff__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 682 (all Ca all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B))) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)))) # label(fact_676_IntD2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 683 (all Ca all A all B (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),B)) & hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),A)) <-> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),B))))) # label(fact_671_Int__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 684 (all A_1 (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A_1),bot_bo1181479936a_bool)) -> bot_bo1181479936a_bool = A_1)) # label(fact_560_le__bot) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 685 (all Loc_3 all Fun_1 all Com_1 all Loc_2 all Fun all Com (is_loc(Loc_2) & is_loc(Loc_3) -> (Loc_2 = Loc_3 & Com_1 = Com & Fun = Fun_1 <-> local(Loc_3,Fun_1,Com_1) = local(Loc_2,Fun,Com)))) # label(fact_253_com_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 686 (all F_1 all A_1 all B image_194810223_a_nat(F_1,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,hAPP_H738206399_a_nat(F_1,A_1)),image_194810223_a_nat(F_1,B))) # label(fact_237_image__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 687 (all X_1 all A all F_1 all F (hBOOL(finite795500164em_nat(F_1,F)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,X_1),hAPP_f22106695ol_nat(F,A)) = hAPP_f22106695ol_nat(F,A))))) # label(fact_399_folding__one__idem_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 688 (all X_1 all A all B (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),B)) & hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,A),B)) <-> hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),A)),B)))) # label(fact_466_insert__subset) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 689 (all A all Ba all F_1 all X_1 (hAPP_H678412245iple_a(F_1,X_1) = Ba -> (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ba),image_306007685iple_a(F_1,A)))))) # label(fact_182_image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 690 (all Q_1 all Ga all Ca all Pa ((all Z_7 all S_2 (hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_7),S_2)) -> (exists P_1 exists Q_2 ((all S_3 ((all Z_8 (hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_1,Z_8),S_2)) -> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_2,Z_8),S_3)))) -> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_7),S_3)))) & hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(P_1,Ca,Q_2)),bot_bo1181479936a_bool))))))) -> hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(Pa,Ca,Q_1)),bot_bo1181479936a_bool))))) # label(fact_145_conseq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 691 (all B all X_1 all A (-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> (hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,A),B)) <-> hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),B)))))) # label(fact_469_subset__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 692 (all A_1 all A all B hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,A),B)) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),A)),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B))) # label(fact_655_insert__inter__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 693 (all B_10 all X_27 all A_14 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_27),A_14)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_27),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_14),B_10))))) # label(fact_484_le__supI1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 694 (all P all Q all R hAPP_s286259371e_bool(hAPP_f289738463e_bool(hAPP_f1151843515e_bool(cOMBB_1941618714_state,P),Q),R) = hAPP_f162060345e_bool(P,hAPP_s1806633685e_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_025) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 695 (all C1 all S2 all C0 all S0 all N_2 all S1 (hBOOL(evaln(C0,S0,N_2,S1)) -> (hBOOL(evaln(C1,S1,N_2,S2)) -> hBOOL(evaln(semi(C0,C1),S0,N_2,S2))))) # label(fact_274_evaln_OSemi) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 696 (all Pa all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (hBOOL(hAPP_f54304608l_bool(Pa,A)) -> ((all A_7 all A_19 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_19)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_7),A_19)) -> (hBOOL(hAPP_f54304608l_bool(Pa,A_19)) -> hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_19),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_7),bot_bot_fun_nat_bool)))))))) -> hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool)))))) # label(fact_403_finite__empty__induct) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 697 (all M_1 all C_2 all S_1 all N_2 all T (hBOOL(evaln(C_2,S_1,N_2,T)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),M_1)) -> hBOOL(evaln(C_2,S_1,M_1,T))))) # label(fact_600_evaln__nonstrict) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 698 (all A all Ba all F_1 all X_1 (hAPP_n178040171iple_a(F_1,X_1) = Ba -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ba),image_1782127643iple_a(F_1,A)))))) # label(fact_184_image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 699 (all S_1 all N_2 all T (hBOOL(evaln(skip,S_1,N_2,T)) -> S_1 = T)) # label(fact_276_evaln__elim__cases_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 700 (all Ca all A ((bot_bo1181479936a_bool = A -> bot_bo1181479936a_bool = image_306007685iple_a(hAPP_H1455657330iple_a(cOMBK_1824972302iple_a,Ca),A)) & (A != bot_bo1181479936a_bool -> hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ca),bot_bo1181479936a_bool) = image_306007685iple_a(hAPP_H1455657330iple_a(cOMBK_1824972302iple_a,Ca),A)))) # label(fact_167_image__constant__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 701 (all Loc_2 all Fun all Com local(Loc_2,Fun,Com) != skip) # label(fact_258_com_Osimps_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 702 (all Ca all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B))))) # label(fact_447_subsetD) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 703 (all Pa ((all X_2 -hBOOL(hAPP_H513860823e_bool(Pa,X_2))) <-> bot_bo1055319631e_bool = collec727977250_state(Pa))) # label(fact_49_Collect__empty__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 704 (all P all Q P = hAPP_H738206399_a_nat(hAPP_n1800114674_a_nat(cOMBK_944981412iple_a,P),Q)) # label(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__Hoare____Mirabelle____ddpglwnxwg__O) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 705 (all X all Y (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(fequal1258664663a_bool,X),Y)) | Y != X)) # label(help_fequal_2_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____ddpglwnxwg__Otriple) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 706 (all A all Ba all F_1 all X_1 (hAPP_n1126952044_state(F_1,X_1) = Ba -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,Ba),image_1821565372_state(F_1,A)))))) # label(fact_177_image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 707 (all Ca all A ((A = bot_bot_fun_nat_bool -> bot_bo1181479936a_bool = image_1782127643iple_a(hAPP_H1450464520iple_a(cOMBK_1505147640_a_nat,Ca),A)) & (bot_bot_fun_nat_bool != A -> image_1782127643iple_a(hAPP_H1450464520iple_a(cOMBK_1505147640_a_nat,Ca),A) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ca),bot_bo1181479936a_bool)))) # label(fact_161_image__constant__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 708 (all Ca all A ((bot_bot_fun_nat_bool = A -> image_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,Ca),A) = bot_bot_fun_nat_bool) & (bot_bot_fun_nat_bool != A -> hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) = image_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,Ca),A)))) # label(fact_165_image__constant__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 709 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) | X != Y)) # label(help_fequal_2_1_fequal_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 710 (all B all A_1 all C ((-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),C)) -> hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,B),C) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B)),C)) & (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),C)) -> hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B)),C) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,B),C))))) # label(fact_659_Int__insert__left) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 711 (all A_15 all X_28 all B_11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_28),B_11)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_28),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_15),B_11))))) # label(fact_483_le__supI2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.05 712 (all Ca all Ba all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> ((all X_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Ba)))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_fold_nat_nat(semila972727038up_nat,Ca),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,Ba),Ca)))))) # label(fact_594_fold__sup__le__sup) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 713 (all X_1 all Y_7 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_7)) <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_1),Y_7) = X_1)) # label(fact_635_le__iff__inf) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 714 (all Z_2 all Ba all A all Y_7 (hBOOL(hAPP_nat_bool(finite929467206at_nat(times_times_nat,Ba,A),Y_7)) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),A)) -> hBOOL(hAPP_nat_bool(finite929467206at_nat(times_times_nat,Z_2,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_2),Y_7)))))) # label(fact_415_fold__graph__insert__swap) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 715 (all A_1 all Ba all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ba),A))) -> (A_1 != Ba -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),A))))) # label(fact_18_insertE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 716 (all F_1 all X_1 -hBOOL(hAPP_H1421470952a_bool(finite1537818352iple_a(F_1,bot_bo1181479936a_bool),X_1))) # label(fact_262_empty__fold1SetE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 717 (all B_1_1 all B_2_1 is_bool(hAPP_f540970102l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____ddpglwnxwg__Otriple_It__a_J_Mtc) # label(hypothesis) # label(non_clause). [assumption]. 1.68/2.06 718 (all P all Q all R hAPP_bool_bool(P,hAPP_nat_bool(Q,R)) = hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 719 (all A A = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),bot_bot_fun_nat_bool)) # label(fact_367_Diff__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 720 (all Ba all A_1 all B (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),B)) -> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Ba),B))))) # label(fact_72_insertI2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 721 (all A_1 all B collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_1))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B))) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)) # label(fact_93_insert__compr) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 722 (all X all Y (-hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(fequal_state,X),Y)) | X = Y)) # label(help_fequal_1_1_fequal_000tc__Com__Ostate_T) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 723 (all B all A (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A)) -> hAPP_f921536533e_bool(finite512563852e_bool(hAPP_f531275309e_bool(hAPP_f1470644835e_bool(cOMBB_946224375_state,hAPP_f2049746453e_bool(cOMBC_872426556e_bool,minus_2076558538e_bool)),hAPP_f762269719e_bool(hAPP_f964290431e_bool(cOMBC_488258100e_bool,insert1835143293_state),bot_bo1055319631e_bool)),B),A) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,B),A))) # label(fact_438_minus__fold__remove) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 724 (all Pa (collec268032053iple_a(Pa) = bot_bo1181479936a_bool <-> (all X_2 -hBOOL(hAPP_H1421470952a_bool(Pa,X_2))))) # label(fact_54_empty__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 725 (all X_1 all A all F_1 all F (hBOOL(finite774711482_state(F_1,F)) -> (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A)) -> (-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> hAPP_f2143211163_state(finite202520804_state(F_1,X_1),A) = hAPP_f2143211163_state(F,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),A)))))) # label(fact_430_folding__one_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 726 (all Pa all A_1 ((-hBOOL(hAPP_f54304608l_bool(Pa,A_1)) -> bot_bo1701429464l_bool = collect_fun_nat_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,fconj),hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,fequal_fun_nat_bool),A_1))),Pa))) & (hBOOL(hAPP_f54304608l_bool(Pa,A_1)) -> collect_fun_nat_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,fconj),hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,fequal_fun_nat_bool),A_1))),Pa)) = hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_1),bot_bo1701429464l_bool)))) # label(fact_41_Collect__conv__if) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 727 (all A ((exists X_2 hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_2),A))) <-> A != bot_bo1181479936a_bool)) # label(fact_57_ex__in__conv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 728 (all Y_27 image_306007685iple_a(cOMBI_1218222237iple_a,Y_27) = Y_27) # label(fact_187_image__ident) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 729 (all B_1_1 all B_2_1 is_bool(hAPP_f1637334154l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 730 (all A A = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),bot_bot_fun_nat_bool)) # label(fact_506_Un__empty__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 731 (all A hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),bot_bo1181479936a_bool) = A) # label(fact_369_Diff__empty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 732 (all Ga all F (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F)) | hBOOL(hAPP_f54304608l_bool(finite_finite_nat,Ga)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,F),Ga))))) # label(fact_625_finite__Int) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 733 (all A_1 all Pa hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),collec268032053iple_a(Pa)) = collec268032053iple_a(hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,hAPP_f997599971l_bool(hAPP_f439164429l_bool(cOMBB_1782929690iple_a,fimplies),hAPP_f1591852335a_bool(hAPP_f34030599a_bool(cOMBB_1799513916iple_a,fNot),hAPP_H1190454433a_bool(hAPP_f1371755681a_bool(cOMBC_839220826a_bool,fequal879838495iple_a),A_1)))),Pa))) # label(fact_88_insert__Collect) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 734 (all A all A_1 all B hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool))),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B))) # label(fact_389_Diff__insert2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 735 (all X_1 all A (hBOOL(hAPP_nat_bool(A,X_1)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)))) # label(fact_231_mem__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 736 (all X_1 all A (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> -(all B_15 (A = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),B_15) -> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),B_15)))))) # label(fact_138_Set_Oset__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 737 (all A all A_1 all B hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),bot_bo1055319631e_bool))),B) = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B))) # label(fact_390_Diff__insert2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 738 (all P all Q (is_bool(P) -> P = hAPP_state_bool(hAPP_b2019457360e_bool(cOMBK_bool_state,P),Q))) # label(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Com__Ostate_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 739 (all Pa all F (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F)) -> (hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool)) -> ((all X_2 all F_2 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2)) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),F_2)) -> (hBOOL(hAPP_f54304608l_bool(Pa,F_2)) -> hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),F_2))))))) -> hBOOL(hAPP_f54304608l_bool(Pa,F)))))) # label(fact_331_finite__induct) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 740 (all Com1_1 all Com2_1 all Com1 all Com2 (semi(Com1_1,Com2_1) = semi(Com1,Com2) <-> Com2 = Com2_1 & Com1_1 = Com1)) # label(fact_152_com_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 741 (all A ((exists X_2 exists B_15 (A = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_2),B_15) & -hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_2),B_15)))) <-> bot_bo1055319631e_bool != A)) # label(fact_155_nonempty__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 742 (all Q_1 all Pa all Ga all P_2 all Ca all Q_3 (hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(P_2,Ca,Q_3)),bot_bo1181479936a_bool))) -> ((all Z_7 all S_2 (hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_7),S_2)) -> (all S_3 ((all Z_8 (hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_2,Z_8),S_2)) -> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_3,Z_8),S_3)))) -> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_7),S_3)))))) -> hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(Pa,Ca,Q_1)),bot_bo1181479936a_bool)))))) # label(fact_16_conseq12) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 743 (all P all Q all R hAPP_a849909144l_bool(hAPP_f1509969235l_bool(hAPP_f1178339559l_bool(cOMBB_1355796797bool_a,P),Q),R) = hAPP_f1259673775l_bool(P,hAPP_a2036067514e_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_016) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 744 (all X_1 all A (-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),A)),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),bot_bo1055319631e_bool)) = A)) # label(fact_384_Diff__insert__absorb) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 745 (all Ga all Pa hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(Pa,skip,Pa)),bot_bo1181479936a_bool)))) # label(fact_130_hoare__derivs_OSkip) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 746 (all F_1 all X_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,hAPP_n1126952044_state(F_1,X_1)),image_1821565372_state(F_1,A))))) # label(fact_199_imageI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 747 (all Pa all A_1 ((hBOOL(hAPP_H513860823e_bool(Pa,A_1)) -> hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),bot_bo1055319631e_bool) = collec727977250_state(hAPP_f921536533e_bool(hAPP_f760664097e_bool(cOMBS_1248383340l_bool,hAPP_f1558728829l_bool(hAPP_f1442918689l_bool(cOMBB_1382207997_state,fconj),hAPP_H1645666623e_bool(hAPP_f262880489e_bool(cOMBC_1424981238e_bool,fequal1531560888_state),A_1))),Pa))) & (-hBOOL(hAPP_H513860823e_bool(Pa,A_1)) -> collec727977250_state(hAPP_f921536533e_bool(hAPP_f760664097e_bool(cOMBS_1248383340l_bool,hAPP_f1558728829l_bool(hAPP_f1442918689l_bool(cOMBB_1382207997_state,fconj),hAPP_H1645666623e_bool(hAPP_f262880489e_bool(cOMBC_1424981238e_bool,fequal1531560888_state),A_1))),Pa)) = bot_bo1055319631e_bool))) # label(fact_42_Collect__conv__if) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 748 (all X_1 hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = X_1) # label(fact_608_Sup__fin_Osingleton) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 749 (all X_29 all Y_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_29),Y_23)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_29),Y_23) = Y_23)) # label(fact_482_sup__absorb2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 750 (all Ca all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B))) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B))))) # label(fact_451_UnE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 751 (all X_1 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) != bot_bot_fun_nat_bool -> hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_1),hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))))) & (hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = bot_bot_fun_nat_bool -> X_1 = hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A))))) # label(fact_617_Sup__fin_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 752 (all P all Q all R hAPP_nat_bool(hAPP_f158894502t_bool(hAPP_f1777703707t_bool(cOMBB_955900739ol_nat,P),Q),R) = hAPP_f54304608l_bool(P,hAPP_n1699378549t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 753 (all P all Q all R hAPP_H513860823e_bool(hAPP_f921536533e_bool(hAPP_f760664097e_bool(cOMBS_1248383340l_bool,P),Q),R) = hAPP_bool_bool(hAPP_H226398757l_bool(P,R),hAPP_H513860823e_bool(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_Itc__Com__) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 754 (all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (A != bot_bot_fun_nat_bool -> ((all X_2 all Y_11 hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_2),Y_11)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_11),bot_bot_fun_nat_bool))))) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),A)),A)))))) # label(fact_440_fold1__in) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 755 (all Y (is_vname(Y) -> ((all Glb (is_glb(Glb) -> glb(Glb) != Y)) -> -(all Loc (is_loc(Loc) -> Y != loc(Loc)))))) # label(fact_344_vname_Oexhaust) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 756 (all X_1 all A all F_1 all F (hBOOL(finite774711482_state(F_1,F)) -> (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A)) -> (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> (bot_bo1055319631e_bool != hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),bot_bo1055319631e_bool)) -> hAPP_H563960305_state(hAPP_H521649881_state(F_1,X_1),hAPP_f2143211163_state(F,hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),bot_bo1055319631e_bool)))) = hAPP_f2143211163_state(F,A)) & (hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),bot_bo1055319631e_bool)) = bot_bo1055319631e_bool -> X_1 = hAPP_f2143211163_state(F,A)))))) # label(fact_346_folding__one_Oremove) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 757 (all A hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,bot_bo1055319631e_bool),A) = bot_bo1055319631e_bool) # label(fact_371_empty__Diff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 758 (all A bot_bo1055319631e_bool = hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,A),bot_bo1055319631e_bool)) # label(fact_681_Int__empty__right) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 759 (all Ca all X_1 all A (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),A)) -> image_306007685iple_a(hAPP_H1455657330iple_a(cOMBK_1824972302iple_a,Ca),A) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,Ca),bot_bo1181479936a_bool))) # label(fact_169_image__constant) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 760 (all A all X_1 all B (hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),bot_bo1055319631e_bool))),B)) -> (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),B)))))) # label(fact_586_diff__single__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 761 (all P all Q all R hAPP_f540970102l_bool(P,hAPP_H1190454433a_bool(Q,R)) = hAPP_H1421470952a_bool(hAPP_f772297704a_bool(hAPP_f390613447a_bool(cOMBB_1501709507iple_a,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____ddpglwnxwg__Otriple_I) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 762 (all F_1 all A1 all A2 ((exists A_7 exists A_19 exists X_2 (X_2 = A2 & -hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_7),A_19)) & hBOOL(hAPP_H513860823e_bool(finite1935632226_state(F_1,A_7,A_19),X_2)) & A1 = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_7),A_19))) <-> hBOOL(hAPP_H513860823e_bool(finite9525415_state(F_1,A1),A2)))) # label(fact_300_fold1Set_Osimps) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 763 (all Ba all A_1 (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ba),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),bot_bo1181479936a_bool))) <-> A_1 = Ba)) # label(fact_112_singleton__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 764 (all Pa all Q_1 collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),Pa)),Q_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,collect_nat(Pa)),collect_nat(Q_1))) # label(fact_528_Collect__disj__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 765 (all P all Q all R hAPP_state_bool(hAPP_s1806633685e_bool(P,R),Q) = hAPP_state_bool(hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__Com__Ostate_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 766 (all A hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(minus_469558085a_bool,A),A) = bot_bo1181479936a_bool) # label(fact_366_Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 767 (all B_1_1 all B_2_1 is_bool(big_se275732192ig_nat(B_1_1,B_2_1))) # label(gsy_c_Big__Operators_Osemilattice__big_000tc__Nat__Onat) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 768 (all Pa all A (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A)) -> (hBOOL(hAPP_f1760790145l_bool(Pa,A)) -> ((all A_7 all A_19 (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A_19)) -> (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_7),A_19)) -> (hBOOL(hAPP_f1760790145l_bool(Pa,A_19)) -> hBOOL(hAPP_f1760790145l_bool(Pa,hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A_19),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_7),bot_bo1055319631e_bool)))))))) -> hBOOL(hAPP_f1760790145l_bool(Pa,bot_bo1055319631e_bool)))))) # label(fact_404_finite__empty__induct) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 769 (all A all B ((all X_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) -> (all Xa (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Xa),B)) -> X_2 != Xa)))) <-> hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B) = bot_bot_fun_nat_bool)) # label(fact_677_disjoint__iff__not__equal) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 770 (all Pa all A_1 (hBOOL(hAPP_H1421470952a_bool(Pa,A_1)) -> ((all X_2 (hBOOL(hAPP_H1421470952a_bool(Pa,X_2)) -> X_2 = A_1)) -> hBOOL(hAPP_H1421470952a_bool(Pa,hAPP_f124283079iple_a(the_Ho746640593iple_a,Pa)))))) # label(fact_266_theI) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 771 (all A all B collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A))),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,fNot),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B)))) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)) # label(fact_363_set__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 772 (all P all Q P = hAPP_s1806633685e_bool(hAPP_f1863945078e_bool(cOMBK_631994958_state,P),Q)) # label(help_COMBK_1_1_COMBK_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__Com__O) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 773 (all X_1 all Y_7 (hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila447562797e_bool,X_1),Y_7) = bot_bo1055319631e_bool <-> Y_7 = bot_bo1055319631e_bool & bot_bo1055319631e_bool = X_1)) # label(fact_493_sup__eq__bot__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 774 (all P all Q all R hAPP_H1421470952a_bool(hAPP_f1591852335a_bool(hAPP_f1203760810a_bool(cOMBS_1148211387l_bool,P),Q),R) = hAPP_bool_bool(hAPP_H1017515220l_bool(P,R),hAPP_H1421470952a_bool(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_It__a_J_00) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 775 (all Ca all A all B (hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,A),B))) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Ca),B)))) # label(fact_675_IntD2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 776 (all X_35 hAPP_f2143211163_state(the_el23965208_state,X_35) = hAPP_f2143211163_state(the_Ho10452358_state,hAPP_f1456107715e_bool(hAPP_f1548785833e_bool(cOMBB_1653402815_state,hAPP_f854625363l_bool(fequal1746921144e_bool,X_35)),hAPP_f762269719e_bool(hAPP_f964290431e_bool(cOMBC_488258100e_bool,insert1835143293_state),bot_bo1055319631e_bool)))) # label(fact_151_the__elem__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 777 (all Ca all S_4 all T_1 ((exists N_1 hBOOL(evaln(Ca,S_4,N_1,T_1))) <-> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(Ca),S_4),T_1)))) # label(fact_284_eval__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 778 (all F_1 all A all X_1 (hBOOL(hAPP_H1421470952a_bool(finite1537818352iple_a(F_1,A),X_1)) -> bot_bo1181479936a_bool != A)) # label(fact_265_fold1Set__nonempty) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 779 (all P all Q (is_bool(P) -> P = hAPP_H1421470952a_bool(hAPP_b119575286a_bool(cOMBK_151824839iple_a,P),Q))) # label(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____ddpglwnxwg__) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 780 (all B_1_1 all B_2_1 is_bool(finite950012314iple_a(B_1_1,B_2_1))) # label(gsy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____ddpglwnxwg__O) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 781 (all Ga all Ca all Q_1 all Pa ((all Z_7 all S_2 (hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_7),S_2)) -> hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(hAPP_f762886889e_bool(cOMBK_1458035955bool_a,hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,fequal_state),S_2)),Ca,hAPP_f762886889e_bool(cOMBK_1458035955bool_a,hAPP_a2036067514e_bool(Q_1,Z_7)))),bot_bo1181479936a_bool))))) -> hBOOL(hAPP_f540970102l_bool(hoare_606018542rivs_a(Ga),hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,hoare_1760757500iple_a(Pa,Ca,Q_1)),bot_bo1181479936a_bool))))) # label(fact_10_escape) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 782 (all A_1 (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A_1)) <-> A_1 = bot_bo1055319631e_bool | (exists A_19 exists A_7 (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A_19)) & A_1 = hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_7),A_19))))) # label(fact_335_finite_Osimps) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 783 (all Z_2 all F_1 all A ((exists X_2 (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_2),A)) & Z_2 = hAPP_H1600811558iple_a(F_1,X_2))) <-> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,Z_2),image_129517430iple_a(F_1,A))))) # label(fact_212_image__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 784 (all P all Q all R hAPP_bool_bool(hAPP_f1748468828l_bool(P,R),hAPP_f54304608l_bool(Q,R)) = hAPP_f54304608l_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 785 (all F_1 all Z_2 all X_1 (hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,Z_2,bot_bo1181479936a_bool),X_1)) -> X_1 = Z_2)) # label(fact_287_empty__fold__graphE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 786 (all A_1 all G all F_1 (G = finite1346402327_state(F_1) -> A_1 = hAPP_f2143211163_state(G,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),bot_bo1055319631e_bool)))) # label(fact_423_fold1__singleton__def) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 787 (all A all X_1 all B (hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),B))) <-> (hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),bot_bo1055319631e_bool))),B))) & (-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,A),B))))) # label(fact_589_subset__insert__iff) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 788 (all A_1 all Ba (hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool) -> Ba = A_1)) # label(fact_104_singleton__inject) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 789 (all A all A_1 all B hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B))) # label(fact_392_Diff__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 790 (all A hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),A) = bot_bo1055319631e_bool) # label(fact_365_Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 791 (all X_1 all A all F_1 all F (hBOOL(finite774711482_state(F_1,F)) -> (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A)) -> (-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_1),A)) -> (bot_bo1055319631e_bool != A -> hAPP_f2143211163_state(F,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_1),A)) = hAPP_H563960305_state(hAPP_H521649881_state(F_1,X_1),hAPP_f2143211163_state(F,A))))))) # label(fact_307_folding__one_Oinsert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 792 (all R_1 all S all X_2 (hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),R_1)),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S)),X_2)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,R_1),S))))) # label(fact_546_sup__Un__eq) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 793 (all P all Q P = hAPP_H563960305_state(hAPP_H521649881_state(cOMBK_690957994_state,P),Q)) # label(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____ddpglwnxwg__Otriple_Itc__Com___020) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 794 (all A_1 hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A_1))) # label(fact_548_bot__least) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 795 (all Loc_2 all Fun all Com all Vname all Fun_1 local(Loc_2,Fun,Com) != ass(Vname,Fun_1)) # label(fact_256_com_Osimps_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 796 (all A_1 hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_1),bot_bo1701429464l_bool) = collect_fun_nat_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,A_1))) # label(fact_29_singleton__conv2) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 797 (all P all Q (-hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) | hBOOL(P))) # label(help_fconj_2_1_U) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 798 (all A all F_1 all F (hBOOL(finite774711482_state(F_1,F)) -> (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,A)) -> (bot_bo1055319631e_bool != A -> ((all X_2 all Y_11 hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,hAPP_H563960305_state(hAPP_H521649881_state(F_1,X_2),Y_11)),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_2),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,Y_11),bot_bo1055319631e_bool))))) -> hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,hAPP_f2143211163_state(F,A)),A))))))) # label(fact_326_folding__one_Oclosed) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 799 (all Z_3 all X_14 all Y_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_14),Y_10)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_14),Z_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,Y_10),Z_3)))))) # label(fact_628_inf__greatest) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 800 (all X_1 all A all F_1 all F (hBOOL(finite988810631ne_nat(F_1,F)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> (hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = bot_bot_fun_nat_bool -> hAPP_f22106695ol_nat(F,A) = X_1) & (bot_bot_fun_nat_bool != hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,X_1),hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) = hAPP_f22106695ol_nat(F,A)))))) # label(fact_345_folding__one_Oremove) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 801 (all B all A_1 all C (-hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_1),C)) -> hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),B)),C) = hAPP_f1591852335a_bool(hAPP_f1026156344a_bool(semila671163144a_bool,B),C))) # label(fact_650_Int__insert__left__if0) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 802 (all A_1 all B hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,A_1),hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,A_1),B)))) # label(fact_98_insertI1) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 803 (all B_1_1 is_bool(finite1956789438t_bool(B_1_1))) # label(gsy_c_Finite__Set_Ocomp__fun__idem_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mt) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 804 (all A all B (hAPP_f921536533e_bool(hAPP_f1583986009e_bool(semila2145357127e_bool,A),B) = bot_bo1055319631e_bool -> hAPP_f921536533e_bool(hAPP_f1583986009e_bool(minus_2076558538e_bool,A),B) = A)) # label(fact_695_Diff__triv) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 805 (all X_1 all A all B (hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,X_1),A)),B)) <-> hBOOL(hAPP_f540970102l_bool(hAPP_f430043647l_bool(ord_le1908022732a_bool,A),B)) & hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,X_1),B)))) # label(fact_467_insert__subset) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 806 (all A_1 (hBOOL(hAPP_f1760790145l_bool(hAPP_f854625363l_bool(ord_le1720872323e_bool,A_1),bot_bo1055319631e_bool)) -> A_1 = bot_bo1055319631e_bool)) # label(fact_559_le__bot) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 807 (all F_1 all A_1 all X_35 all X_1 (hBOOL(hAPP_H1421470952a_bool(finite1537818352iple_a(F_1,hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),X_35)),X_1)) -> -(all A_7 all A_19 (hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_7),A_19) = hAPP_f1591852335a_bool(hAPP_H1641355846a_bool(insert873085594iple_a,A_1),X_35) -> (hBOOL(hAPP_H1421470952a_bool(finite2010942150iple_a(F_1,A_7,A_19),X_1)) -> hBOOL(hAPP_f540970102l_bool(hAPP_H1840393229l_bool(member1713797107iple_a,A_7),A_19))))))) # label(fact_296_insert__fold1SetE) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 808 (all Glb_1 all Loc_1 glb(Glb_1) != loc(Loc_1)) # label(fact_324_vname_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 809 (all A all X_1 all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),B)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B)))))) # label(fact_585_diff__single__insert) # label(axiom) # label(non_clause). [assumption]. 1.68/2.06 810 (all Pa all F (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,F)) -> (hBOOL(hAPP_f1760790145l_bool(Pa,bot_bo1055319631e_bool)) -> ((all X_2 all F_2 (hBOOL(hAPP_f1760790145l_bool(finite784854244_state,F_2)) -> (-hBOOL(hAPP_f1760790145l_bool(hAPP_H248360617l_bool(member1758697444_state,X_2),F_2)) -> (hBOOL(hAPP_f1760790145l_bool(Pa,F_2)) -> hBOOL(hAPP_f1760790145l_bool(Pa,hAPP_f921536533e_bool(hAPP_H727730819e_bool(insert1835143293_state,X_2),F_2))))))) -> hBOOL(hAPP_f1760790145l_bool(Pa,F)))))) # label(fact_332_finite__induct) # label(axiom) # label(non_clause). [assumption]. 2.54/2.78 2.54/2.78 ============================== end of process non-clausal formulas === 2.54/2.78 2.54/2.78 ============================== PROCESS INITIAL CLAUSES =============== 2.54/2.78 2.54/2.78 ============================== PREDICATE ELIMINATION ================= 2.54/2.78 2.54/2.78 ============================== end predicate elimination ============= 2.54/2.78 2.54/2.78 Auto_denials: (non-Horn, no changes). 2.54/2.78 2.54/2.78 Term ordering decisions: 2.54/2.78 Function symbol KB weights: bot_bo1181479936a_bool=1. insert873085594iple_a=1. bot_bo1055319631e_bool=1. insert1835143293_state=1. bot_bot_fun_nat_bool=1. member_nat=1. member1713797107iple_a=1. insert_nat=1. finite_finite_nat=1. ord_less_eq_nat=1. member1758697444_state=1. finite1655202547iple_a=1. ord_le1568362934t_bool=1. finite784854244_state=1. minus_2067140911t_bool=1. ord_le1908022732a_bool=1. semila840543986t_bool=1. semila972727038up_nat=1. minus_469558085a_bool=1. ord_le1720872323e_bool=1. bot_bot_bool=1. semila671163144a_bool=1. minus_2076558538e_bool=1. semila465093516t_bool=1. fconj=1. semila1498788770a_bool=1. big_la43341705in_nat=1. semila2145357127e_bool=1. semila80283416nf_nat=1. cOMBC_1552014468l_bool=1. cOMBC_226598744l_bool=1. fdisj=1. cOMBB_1782929690iple_a=1. cOMBS_1148211387l_bool=1. cOMBB_1015721476ol_nat=1. cOMBS_nat_bool_bool=1. fequal879838495iple_a=1. semila447562797e_bool=1. update=1. the_Ho746640593iple_a=1. cOMBC_839220826a_bool=1. fequal1531560888_state=1. fequal_fun_nat_bool=1. fequal_nat=1. times_times_nat=1. fimplies=1. cOMBC_nat_nat_bool=1. semila1168014441p_bool=1. bot_bo1701429464l_bool=1. fNot=1. cOMBB_1382207997_state=1. cOMBB_444170502t_bool=1. cOMBC_1424981238e_bool=1. cOMBC_1693257480l_bool=1. cOMBS_1187019125l_bool=1. cOMBS_1248383340l_bool=1. bot_bot_nat=1. cOMBB_1799513916iple_a=1. cOMBB_188601460_state=1. cOMBB_844853809_state=1. cOMBC_1193272608_state=1. fequal_state=1. insert_fun_nat_bool=1. ord_less_eq_bool=1. fFalse=1. skip=1. cOMBC_231445413l_bool=1. cOMBS_777315357_state=1. cOMBB_1941618714_state=1. cOMBB_2139825703bool_a=1. cOMBC_178881787t_bool=1. cOMBC_488258100e_bool=1. cOMBC_524597097e_bool=1. cOMBC_708118077a_bool=1. cOMBC_764456866l_bool=1. cOMBC_867582640e_bool=1. cOMBK_1505147640_a_nat=1. cOMBK_1824972302iple_a=1. cOMBK_2022976521_state=1. cOMBK_547979437iple_a=1. cOMBK_690957994_state=1. cOMBK_944981412iple_a=1. cOMBK_nat_nat=1. semila310582991f_bool=1. cOMBB_1348041619bool_a=1. cOMBB_1355796797bool_a=1. cOMBB_1654519265ol_nat=1. cOMBB_1757942702_state=1. cOMBB_1759179140_state=1. cOMBB_2080540641iple_a=1. cOMBB_946224375_state=1. cOMBB_bool_bool_nat=1. cOMBC_1245412066l_bool=1. cOMBC_1777403949_state=1. cOMBC_2027030106e_bool=1. cOMBC_314077933a_bool=1. cOMBC_41962815e_bool=1. cOMBC_471052088e_bool=1. cOMBC_646315179t_bool=1. cOMBC_872426556e_bool=1. cOMBK_1458035955bool_a=1. cOMBK_631994958_state=1. cOMBS_1378840469l_bool=1. fequal1258664663a_bool=1. fequal1746921144e_bool=1. cOMBB_1501709507iple_a=1. cOMBB_1653402815_state=1. cOMBB_237455441bool_a=1. cOMBB_238756964t_bool=1. cOMBB_416661851_state=1. cOMBB_572666224_state=1. cOMBB_800536526ol_nat=1. cOMBB_955900739ol_nat=1. cOMBI_1218222237iple_a=1. cOMBI_1367665338_state=1. cOMBI_nat=1. cOMBK_1079618832_state=1. cOMBK_151824839iple_a=1. cOMBK_1950023923_state=1. cOMBK_1994329625t_bool=1. cOMBK_bool_nat=1. fTrue=1. member_fun_nat_bool=1. the_Ho10452358_state=1. the_el1436340927iple_a=1. the_el23965208_state=1. the_elem_nat=1. the_nat=1. cOMBB_145932198bool_a=1. cOMBB_160679318_state=1. cOMBC_892787026e_bool=1. cOMBK_bool_state=1. finite2012431853t_bool=1. glb_1=1. loc_1=1. b=1. c=1. g=1. p=1. hAPP_f54304608l_bool=1. hAPP_f540970102l_bool=1. hAPP_f1591852335a_bool=1. hAPP_f800510211t_bool=1. hAPP_f921536533e_bool=1. hAPP_f1760790145l_bool=1. hAPP_H1641355846a_bool=1. hAPP_H727730819e_bool=1. hAPP_n1512601776t_bool=1. hAPP_n215258509l_bool=1. hAPP_H1840393229l_bool=1. hAPP_nat_bool=1. hAPP_f1730770594t_bool=1. hAPP_n1699378549t_bool=1. hAPP_f1026156344a_bool=1. hAPP_H1421470952a_bool=1. hAPP_H248360617l_bool=1. hAPP_nat_nat=1. hAPP_f1583986009e_bool=1. hAPP_nat_fun_nat_nat=1. hAPP_f22106695ol_nat=1. hAPP_f103356543l_bool=1. hAPP_state_bool=1. hAPP_f430043647l_bool=1. hAPP_s1806633685e_bool=1. hAPP_f124283079iple_a=1. hAPP_H513860823e_bool=1. hAPP_f854625363l_bool=1. hAPP_bool_bool=1. hAPP_b589554111l_bool=1. hAPP_f2143211163_state=1. hAPP_H678412245iple_a=1. hAPP_a2036067514e_bool=1. hAPP_H1190454433a_bool=1. hAPP_f1722879237t_bool=1. hAPP_f672239281a_bool=1. hAPP_H1455657330iple_a=1. hAPP_H563960305_state=1. hAPP_f1203760810a_bool=1. hAPP_f1246832597l_bool=1. hAPP_f439164429l_bool=1. hAPP_f997599971l_bool=1. hAPP_f1080886329l_bool=1. hAPP_f1146629647l_bool=1. hAPP_f561022312t_bool=1. image_1782127643iple_a=1. hAPP_H1645666623e_bool=1. image_129517430iple_a=1. image_1604514514_state=1. image_194810223_a_nat=1. hAPP_nat_state=1. semi=1. hAPP_H521649881_state=1. hAPP_f1371755681a_bool=1. hAPP_s712361723_state=1. hAPP_v594194232_state=1. image_306007685iple_a=1. ass=1. image_nat_nat=1. finite1537818352iple_a=1. image_650584225_state=1. finite1669978781iple_a=1. finite9525415_state=1. finite988810631ne_nat=1. finite_fold1Set_nat=1. finite_fold_nat_nat=1. hAPP_f229349961t_bool=1. finite774711482_state=1. hAPP_state_nat=1. hAPP_f1442918689l_bool=1. hAPP_f1558728829l_bool=1. hAPP_f1743029098l_bool=1. hAPP_f1974927549l_bool=1. hAPP_f262880489e_bool=1. hAPP_f643944041l_bool=1. hAPP_f760664097e_bool=1. hAPP_f857404385l_bool=1. hAPP_n178040171iple_a=1. finite795500164em_nat=1. finite950012314iple_a=1. hAPP_H1600811558iple_a=1. hAPP_H738206399_a_nat=1. hAPP_H928324994_state=1. hAPP_f1542232213_state=1. hAPP_f1561913689l_bool=1. hAPP_f34030599a_bool=1. hAPP_f633452666l_bool=1. hAPP_v365393659_state=1. finite506823037_state=1. getlocs=1. finite978536264iple_a=1. hAPP_f1259673775l_bool=1. hAPP_f1759915619e_bool=1. hAPP_f817621513e_bool=1. hAPP_f851239890_state=1. hAPP_f871651461_state=1. hAPP_H1450464520iple_a=1. hAPP_H377435237iple_a=1. hAPP_H558669354_state=1. hAPP_H716259088te_nat=1. hAPP_b2019457360e_bool=1. hAPP_f100967412e_bool=1. hAPP_f1151843515e_bool=1. hAPP_f1262649863e_bool=1. hAPP_f1406200875e_bool=1. hAPP_f1460451647e_bool=1. hAPP_f1570313510t_bool=1. hAPP_f1665407592a_bool=1. hAPP_f1706273077e_bool=1. hAPP_f1838002347e_bool=1. hAPP_f289738463e_bool=1. hAPP_f549683569a_bool=1. hAPP_f635443597e_bool=1. hAPP_f644196280e_bool=1. hAPP_f688831301t_bool=1. hAPP_f762269719e_bool=1. hAPP_f762886889e_bool=1. hAPP_f964290431e_bool=1. hAPP_n1800114674_a_nat=1. image_1410872416te_nat=1. hAPP_a849909144l_bool=1. hAPP_b1095269219e_bool=1. hAPP_b540892988e_bool=1. hAPP_f1006724181e_bool=1. hAPP_f10625010t_bool=1. hAPP_f1115950719t_bool=1. hAPP_f1178339559l_bool=1. hAPP_f1283379615l_bool=1. hAPP_f1311642927t_bool=1. hAPP_f1345202233l_bool=1. hAPP_f1398071125e_bool=1. hAPP_f1443436725l_bool=1. hAPP_f1470644835e_bool=1. hAPP_f1509969235l_bool=1. hAPP_f162060345e_bool=1. hAPP_f1637334154l_bool=1. hAPP_f167292325e_bool=1. hAPP_f1824947087e_bool=1. hAPP_f1848060885_state=1. hAPP_f1863945078e_bool=1. hAPP_f2049746453e_bool=1. hAPP_f2085120383a_bool=1. hAPP_f340725611e_bool=1. hAPP_f450029403a_bool=1. hAPP_f485051996a_bool=1. hAPP_f531275309e_bool=1. hAPP_f701449317e_bool=1. hAPP_f746301080e_bool=1. hAPP_f873506917e_bool=1. hAPP_f894608603t_bool=1. hAPP_n1126952044_state=1. hAPP_n1547241352_state=1. hAPP_s1892499976_state=1. hAPP_s58564346l_bool=1. hAPP_state_state=1. image_1821565372_state=1. big_se275732192ig_nat=1. finite1578363458t_bool=1. finite1979045230a_bool=1. finite202520804_state=1. finite512563852e_bool=1. hAPP_H1017515220l_bool=1. hAPP_H226398757l_bool=1. hAPP_a1200519163e_bool=1. hAPP_a1224971408e_bool=1. hAPP_a723219176e_bool=1. hAPP_b1013836512t_bool=1. hAPP_b119575286a_bool=1. hAPP_b1245957081e_bool=1. hAPP_b1630757474l_bool=1. hAPP_f1037965299e_bool=1. hAPP_f1159960589e_bool=1. hAPP_f1164891443l_bool=1. hAPP_f1456107715e_bool=1. hAPP_f1505651103t_bool=1. hAPP_f1548785833e_bool=1. hAPP_f158894502t_bool=1. hAPP_f1748468828l_bool=1. hAPP_f1777703707t_bool=1. hAPP_f390613447a_bool=1. hAPP_f512427579e_bool=1. hAPP_f618557131t_bool=1. hAPP_f769584981e_bool=1. hAPP_f772297704a_bool=1. hAPP_n1006566506l_bool=1. hAPP_n436597305te_nat=1. hAPP_s1874344717e_bool=1. hAPP_s2001034685l_bool=1. hAPP_s286259371e_bool=1. hAPP_f1261923407e_bool=1. hAPP_f1951378235l_bool=1. hAPP_f2073279419e_bool=1. hAPP_f375255701e_bool=1. hAPP_f963367678e_bool=1. f3=1. f4=1. f11=1. f12=1. f13=1. f21=1. f22=1. f23=1. f26=1. f30=1. f31=1. f44=1. f53=1. f59=1. f60=1. f61=1. f67=1. f68=1. f69=1. f70=1. f73=1. f74=1. f87=1. f88=1. f95=1. f96=1. f97=1. f102=1. f112=1. f113=1. f127=1. f128=1. f130=1. f135=1. f136=1. f141=1. f142=1. f143=1. f156=1. f157=1. f158=1. f159=1. f160=1. f171=1. f172=1. hoare_606018542rivs_a=1. hoare_659004819_state=1. collect_nat=1. collec268032053iple_a=1. evalc=1. loc=1. collec727977250_state=1. collect_fun_nat_bool=1. finite_fold1_nat=1. finite1200705745iple_a=1. glb=1. finite1382394752iple_a=1. hoare_560051114alid_a=1. hoare_592710359_state=1. finite1317819144e_bool=1. finite1956789438t_bool=1. finite419198954a_bool=1. finite1346402327_state=1. partia1866111638iple_a=1. partia415982977_state=1. partial_flat_lub_nat=1. hoare_Mirabelle_MGT=1. undefined_glb=1. undefined_loc=1. f10=1. f16=1. f19=1. f20=1. f25=1. f27=1. f32=1. f37=1. f38=1. f45=1. f46=1. f47=1. f54=1. f77=1. f79=1. f80=1. f81=1. f84=1. f89=1. f90=1. f94=1. f101=1. f104=1. f105=1. f106=1. f107=1. f108=1. f114=1. f115=1. f116=1. f131=1. f137=1. f139=1. f140=1. f144=1. f145=1. f149=1. f150=1. f151=1. f152=1. f164=1. f165=1. hoare_1575745797_state=1. hoare_1760757500iple_a=1. finite2010942150iple_a=1. local=1. finite929467206at_nat=1. finite1935632226_state=1. f1=1. f2=1. f17=1. f18=1. f24=1. f29=1. f48=1. f49=1. f50=1. f55=1. f62=1. f63=1. f78=1. f82=1. f83=1. f85=1. f86=1. f91=1. f103=1. f121=1. f122=1. f123=1. f124=1. f125=1. f129=1. f138=1. f153=1. f154=1. f155=1. f161=1. f166=1. f167=1. f168=1. evaln=1. f5=1. f6=1. f7=1. f8=1. f9=1. f14=1. f15=1. f34=1. f35=1. f36=1. f39=1. f40=1. f42=1. f43=1. f58=1. f64=1. f65=1. f75=1. f76=1. f92=1. f93=1. f117=1. f118=1. f119=1. f132=1. f133=1. f162=1. f163=1. f169=1. f170=1. f28=1. f33=1. f41=1. f51=1. f52=1. f56=1. f57=1. f71=1. f72=1. f109=1. f110=1. f111=1. f66=1. f98=1. f99=1. f100=1. f126=1. f134=1. f146=1. f147=1. f148=1. f120=1. 14.19/14.50 14.19/14.50 ============================== end of process initial clauses ======== 14.19/14.50 14.19/14.50 ============================== CLAUSES FOR SEARCH ==================== 14.19/14.50 14.19/14.50 ============================== end of clauses for search ============= 14.19/14.50 14.19/14.50 ============================== SEARCH ================================ 14.19/14.50 14.19/14.50 % Starting search at 0.86 seconds. 14.19/14.50 14.19/14.50 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 47 (0.00 of 2.15 sec). 14.19/14.50 14.19/14.50 Low Water (keep): wt=37.000, iters=3582 14.19/14.50 14.19/14.50 Low Water (keep): wt=35.000, iters=3503 14.19/14.50 14.19/14.50 Low Water (keep): wt=32.000, iters=3337 14.19/14.50 14.19/14.50 Low Water (keep): wt=31.000, iters=3357 14.19/14.50 14.19/14.50 Low Water (keep): wt=30.000, iters=3411 14.19/14.50 14.19/14.50 Low Water (keep): wt=29.000, iters=3451 14.19/14.50 14.19/14.50 Low Water (keep): wt=28.000, iters=3340 14.19/14.50 14.19/14.50 Low Water (keep): wt=27.000, iters=3375 14.19/14.50 14.19/14.50 Low Water (keep): wt=26.000, iters=3423 14.19/14.50 14.19/14.50 Low Water (keep): wt=25.000, iters=3339 14.19/14.50 14.19/14.50 Low Water (keep): wt=24.000, iters=3335 14.19/14.50 14.19/14.50 Low Water (keep): wt=23.000, iters=3333 14.19/14.50 14.19/14.50 Low Water (keep): wt=22.000, iters=3334 14.19/14.50 14.19/14.50 Low Water (keep): wt=21.000, iters=3365 14.19/14.50 14.19/14.50 Low Water (keep): wt=20.000, iters=3357 14.19/14.50 14.19/14.50 Low Water (keep): wt=19.000, iters=3407 14.19/14.50 14.19/14.50 Low Water (keep): wt=18.000, iters=3354 14.19/14.50 14.19/14.50 Low Water (keep): wt=17.000, iters=3376 14.19/14.50 14.19/14.50 Low Water (keep): wt=16.000, iters=3357 14.19/14.50 14.19/14.50 Low Water (keep): wt=15.000, iters=3489 14.19/14.50 14.19/14.50 Low Water (keep): wt=14.000, iters=3339 14.19/14.50 14.19/14.50 Low Water (keep): wt=13.000, iters=3339 14.19/14.50 14.19/14.50 Low Water (displace): id=3943, wt=82.000 14.19/14.50 14.19/14.50 Low Water (displace): id=4159, wt=75.000 14.19/14.50 14.19/14.50 Low Water (displace): id=3113, wt=73.000 14.19/14.50 14.19/14.50 Low Water (displace): id=3944, wt=68.000 14.19/14.50 14.19/14.50 Low Water (displace): id=4156, wt=65.000 14.19/14.50 14.19/14.50 Low Water (displace): id=3465, Alarm clock 179.57/180.09 Prover9 interrupted 179.57/180.10 EOF