% TIMEFORMAT='%3R'; { time (exec 2>&1; /home/martin/bin/satallax -E /home/martin/.isabelle/contrib/e-2.5-1/x86_64-linux/eprover -p tstp -t 5 /home/martin/judgement-day/tptp-thf/tptp/Hoare/prob_356__3253546_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:02.744

% Could-be-implicit typings (8)
thf(ty_n_t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    set_Ho840737317_state : $tType).
thf(ty_n_t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    hoare_958474565_state : $tType).
thf(ty_n_t__Option__Ooption_It__Com__Ocom_J, type,
    option_com : $tType).
thf(ty_n_t__Set__Oset_It__Com__Opname_J, type,
    set_pname : $tType).
thf(ty_n_t__Com__Ostate, type,
    state : $tType).
thf(ty_n_t__Com__Opname, type,
    pname : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).

% Explicit typings (36)
thf(sy_c_Com_Obody, type,
    body : pname > option_com).
thf(sy_c_Com_Ocom_OBODY, type,
    body2 : pname > com).
thf(sy_c_Com_Ocom_OSKIP, type,
    skip : com).
thf(sy_c_Com_Ocom_OSemi, type,
    semi : com > com > com).
thf(sy_c_Finite__Set_Ocard_001t__Com__Opname, type,
    finite_card_pname : set_pname > nat).
thf(sy_c_Finite__Set_Ocard_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    finite1141247341_state : set_Ho840737317_state > nat).
thf(sy_c_Finite__Set_Ofinite_001t__Com__Opname, type,
    finite_finite_pname : set_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    finite1986656878_state : set_Ho840737317_state > $o).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Com__Opname_J, type,
    minus_1937938585_pname : set_pname > set_pname > set_pname).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    minus_1628593292_state : set_Ho840737317_state > set_Ho840737317_state > set_Ho840737317_state).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_OMGT, type,
    hoare_Mirabelle_MGT : com > hoare_958474565_state).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__derivs_001t__Com__Ostate, type,
    hoare_604442164_state : set_Ho840737317_state > set_Ho840737317_state > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__valids_001t__Com__Ostate, type,
    hoare_318887606_state : set_Ho840737317_state > set_Ho840737317_state > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple_Osize__triple_001t__Com__Ostate, type,
    hoare_103296669_state : (state > nat) > hoare_958474565_state > nat).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple_Otriple_001t__Com__Ostate, type,
    hoare_1659279548_state : (state > state > $o) > com > (state > state > $o) > hoare_958474565_state).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple__valid_001t__Com__Ostate, type,
    hoare_364318704_state : nat > hoare_958474565_state > $o).
thf(sy_c_Map_Odom_001t__Com__Opname_001t__Com__Ocom, type,
    dom_pname_com : (pname > option_com) > set_pname).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Nat_Osize__class_Osize_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    size_s1457777073_state : hoare_958474565_state > nat).
thf(sy_c_Natural_Oevalc, type,
    evalc : com > state > state > $o).
thf(sy_c_Natural_Oevaln, type,
    evaln : com > state > nat > state > $o).
thf(sy_c_Option_Ooption_Othe_001t__Com__Ocom, type,
    the_com : option_com > com).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_M_Eo_J, type,
    bot_bo1428770700tate_o : hoare_958474565_state > $o).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat, type,
    bot_bot_nat : nat).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Com__Opname_J, type,
    bot_bot_set_pname : set_pname).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    bot_bo105666705_state : set_Ho840737317_state).
thf(sy_c_Set_OCollect_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    collec305460656_state : (hoare_958474565_state > $o) > set_Ho840737317_state).
thf(sy_c_Set_Oinsert_001t__Com__Opname, type,
    insert_pname : pname > set_pname > set_pname).
thf(sy_c_Set_Oinsert_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    insert776267541_state : hoare_958474565_state > set_Ho840737317_state > set_Ho840737317_state).
thf(sy_c_Set_Ois__empty_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    is_emp266830871_state : set_Ho840737317_state > $o).
thf(sy_c_Set_Ois__singleton_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    is_sin536631353_state : set_Ho840737317_state > $o).
thf(sy_c_Set_Othe__elem_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    the_el1300254266_state : set_Ho840737317_state > hoare_958474565_state).
thf(sy_c_member_001t__Com__Opname, type,
    member_pname : pname > set_pname > $o).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    member109514606_state : hoare_958474565_state > set_Ho840737317_state > $o).
thf(sy_v_G, type,
    g : set_Ho840737317_state).

% Relevant facts (147)
thf(fact_0_triple_Oinject, axiom,
    ((![X1 : state > state > $o, X2 : com, X3 : state > state > $o, Y1 : state > state > $o, Y2 : com, Y3 : state > state > $o]: (((hoare_1659279548_state @ X1 @ X2 @ X3) = (hoare_1659279548_state @ Y1 @ Y2 @ Y3)) = (((X1 = Y1)) & ((((X2 = Y2)) & ((X3 = Y3))))))))). % triple.inject
thf(fact_1_conseq1, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o, C : com, Q : state > state > $o, P2 : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state)) => ((![Z : state, S : state]: ((P2 @ Z @ S) => (P @ Z @ S))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ C @ Q) @ bot_bo105666705_state))))))). % conseq1
thf(fact_2_conseq2, axiom,
    ((![G : set_Ho840737317_state, P2 : state > state > $o, C : com, Q2 : state > state > $o, Q : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ C @ Q2) @ bot_bo105666705_state)) => ((![Z : state, S : state]: ((Q2 @ Z @ S) => (Q @ Z @ S))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ C @ Q) @ bot_bo105666705_state))))))). % conseq2
thf(fact_3_conseq12, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o, C : com, Q2 : state > state > $o, P2 : state > state > $o, Q : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q2) @ bot_bo105666705_state)) => ((![Z : state, S : state]: ((P2 @ Z @ S) => (![S2 : state]: ((![Z2 : state]: ((P @ Z2 @ S) => (Q2 @ Z2 @ S2))) => (Q @ Z @ S2))))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ C @ Q) @ bot_bo105666705_state))))))). % conseq12
thf(fact_4_triple_Oinduct, axiom,
    ((![P2 : hoare_958474565_state > $o, Triple : hoare_958474565_state]: ((![X1a : state > state > $o, X2a : com, X3a : state > state > $o]: (P2 @ (hoare_1659279548_state @ X1a @ X2a @ X3a))) => (P2 @ Triple))))). % triple.induct
thf(fact_5_derivs__insertD, axiom,
    ((![G : set_Ho840737317_state, T : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G @ (insert776267541_state @ T @ Ts)) => ((hoare_604442164_state @ G @ (insert776267541_state @ T @ bot_bo105666705_state)) & (hoare_604442164_state @ G @ Ts)))))). % derivs_insertD
thf(fact_6_triple_Oexhaust, axiom,
    ((![Y : hoare_958474565_state]: (~ ((![X12 : state > state > $o, X22 : com, X32 : state > state > $o]: (~ ((Y = (hoare_1659279548_state @ X12 @ X22 @ X32)))))))))). % triple.exhaust
thf(fact_7_cut, axiom,
    ((![G2 : set_Ho840737317_state, Ts : set_Ho840737317_state, G : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ Ts) => ((hoare_604442164_state @ G @ G2) => (hoare_604442164_state @ G @ Ts)))))). % cut
thf(fact_8_hoare__derivs_OSkip, axiom,
    ((![G : set_Ho840737317_state, P2 : state > state > $o]: (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ skip @ P2) @ bot_bo105666705_state))))). % hoare_derivs.Skip
thf(fact_9_empty, axiom,
    ((![G : set_Ho840737317_state]: (hoare_604442164_state @ G @ bot_bo105666705_state)))). % empty
thf(fact_10_conseq, axiom,
    ((![P2 : state > state > $o, G : set_Ho840737317_state, C : com, Q : state > state > $o]: ((![Z : state, S : state]: ((P2 @ Z @ S) => (?[P3 : state > state > $o, Q3 : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P3 @ C @ Q3) @ bot_bo105666705_state)) & (![S2 : state]: ((![Z2 : state]: ((P3 @ Z2 @ S) => (Q3 @ Z2 @ S2))) => (Q @ Z @ S2))))))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ C @ Q) @ bot_bo105666705_state)))))). % conseq
thf(fact_11_hoare__derivs_Oinsert, axiom,
    ((![G : set_Ho840737317_state, T : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G @ (insert776267541_state @ T @ bot_bo105666705_state)) => ((hoare_604442164_state @ G @ Ts) => (hoare_604442164_state @ G @ (insert776267541_state @ T @ Ts))))))). % hoare_derivs.insert
thf(fact_12_singletonI, axiom,
    ((![A : hoare_958474565_state]: (member109514606_state @ A @ (insert776267541_state @ A @ bot_bo105666705_state))))). % singletonI
thf(fact_13_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_14_insertCI, axiom,
    ((![A : hoare_958474565_state, B : set_Ho840737317_state, B2 : hoare_958474565_state]: (((~ ((member109514606_state @ A @ B))) => (A = B2)) => (member109514606_state @ A @ (insert776267541_state @ B2 @ B)))))). % insertCI
thf(fact_15_insert__iff, axiom,
    ((![A : hoare_958474565_state, B2 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ (insert776267541_state @ B2 @ A2)) = (((A = B2)) | ((member109514606_state @ A @ A2))))))). % insert_iff
thf(fact_16_insert__absorb2, axiom,
    ((![X : hoare_958474565_state, A2 : set_Ho840737317_state]: ((insert776267541_state @ X @ (insert776267541_state @ X @ A2)) = (insert776267541_state @ X @ A2))))). % insert_absorb2
thf(fact_17_empty__iff, axiom,
    ((![C : hoare_958474565_state]: (~ ((member109514606_state @ C @ bot_bo105666705_state)))))). % empty_iff
thf(fact_18_all__not__in__conv, axiom,
    ((![A2 : set_Ho840737317_state]: ((![X4 : hoare_958474565_state]: (~ ((member109514606_state @ X4 @ A2)))) = (A2 = bot_bo105666705_state))))). % all_not_in_conv
thf(fact_19_Collect__empty__eq, axiom,
    ((![P2 : hoare_958474565_state > $o]: (((collec305460656_state @ P2) = bot_bo105666705_state) = (![X4 : hoare_958474565_state]: (~ ((P2 @ X4)))))))). % Collect_empty_eq
thf(fact_20_empty__Collect__eq, axiom,
    ((![P2 : hoare_958474565_state > $o]: ((bot_bo105666705_state = (collec305460656_state @ P2)) = (![X4 : hoare_958474565_state]: (~ ((P2 @ X4)))))))). % empty_Collect_eq
thf(fact_21_evalc__elim__cases_I1_J, axiom,
    ((![S3 : state, T : state]: ((evalc @ skip @ S3 @ T) => (T = S3))))). % evalc_elim_cases(1)
thf(fact_22_evalc_OSkip, axiom,
    ((![S3 : state]: (evalc @ skip @ S3 @ S3)))). % evalc.Skip
thf(fact_23_bot__set__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ bot_bo1428770700tate_o)))). % bot_set_def
thf(fact_24_ex__in__conv, axiom,
    ((![A2 : set_Ho840737317_state]: ((?[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ A2)) = (~ ((A2 = bot_bo105666705_state))))))). % ex_in_conv
thf(fact_25_equals0I, axiom,
    ((![A2 : set_Ho840737317_state]: ((![Y4 : hoare_958474565_state]: (~ ((member109514606_state @ Y4 @ A2)))) => (A2 = bot_bo105666705_state))))). % equals0I
thf(fact_26_equals0D, axiom,
    ((![A2 : set_Ho840737317_state, A : hoare_958474565_state]: ((A2 = bot_bo105666705_state) => (~ ((member109514606_state @ A @ A2))))))). % equals0D
thf(fact_27_emptyE, axiom,
    ((![A : hoare_958474565_state]: (~ ((member109514606_state @ A @ bot_bo105666705_state)))))). % emptyE
thf(fact_28_mk__disjoint__insert, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ A2) => (?[B3 : set_Ho840737317_state]: ((A2 = (insert776267541_state @ A @ B3)) & (~ ((member109514606_state @ A @ B3))))))))). % mk_disjoint_insert
thf(fact_29_insert__commute, axiom,
    ((![X : hoare_958474565_state, Y : hoare_958474565_state, A2 : set_Ho840737317_state]: ((insert776267541_state @ X @ (insert776267541_state @ Y @ A2)) = (insert776267541_state @ Y @ (insert776267541_state @ X @ A2)))))). % insert_commute
thf(fact_30_insert__eq__iff, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state, B2 : hoare_958474565_state, B : set_Ho840737317_state]: ((~ ((member109514606_state @ A @ A2))) => ((~ ((member109514606_state @ B2 @ B))) => (((insert776267541_state @ A @ A2) = (insert776267541_state @ B2 @ B)) = (((((A = B2)) => ((A2 = B)))) & ((((~ ((A = B2)))) => ((?[C2 : set_Ho840737317_state]: (((A2 = (insert776267541_state @ B2 @ C2))) & ((((~ ((member109514606_state @ B2 @ C2)))) & ((((B = (insert776267541_state @ A @ C2))) & ((~ ((member109514606_state @ A @ C2)))))))))))))))))))). % insert_eq_iff
thf(fact_31_insert__absorb, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ A2) => ((insert776267541_state @ A @ A2) = A2))))). % insert_absorb
thf(fact_32_insert__ident, axiom,
    ((![X : hoare_958474565_state, A2 : set_Ho840737317_state, B : set_Ho840737317_state]: ((~ ((member109514606_state @ X @ A2))) => ((~ ((member109514606_state @ X @ B))) => (((insert776267541_state @ X @ A2) = (insert776267541_state @ X @ B)) = (A2 = B))))))). % insert_ident
thf(fact_33_Set_Oset__insert, axiom,
    ((![X : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ X @ A2) => (~ ((![B3 : set_Ho840737317_state]: ((A2 = (insert776267541_state @ X @ B3)) => (member109514606_state @ X @ B3))))))))). % Set.set_insert
thf(fact_34_insertI2, axiom,
    ((![A : hoare_958474565_state, B : set_Ho840737317_state, B2 : hoare_958474565_state]: ((member109514606_state @ A @ B) => (member109514606_state @ A @ (insert776267541_state @ B2 @ B)))))). % insertI2
thf(fact_35_insertI1, axiom,
    ((![A : hoare_958474565_state, B : set_Ho840737317_state]: (member109514606_state @ A @ (insert776267541_state @ A @ B))))). % insertI1
thf(fact_36_insertE, axiom,
    ((![A : hoare_958474565_state, B2 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ (insert776267541_state @ B2 @ A2)) => ((~ ((A = B2))) => (member109514606_state @ A @ A2)))))). % insertE
thf(fact_37_com__det, axiom,
    ((![C : com, S3 : state, T : state, U : state]: ((evalc @ C @ S3 @ T) => ((evalc @ C @ S3 @ U) => (U = T)))))). % com_det
thf(fact_38_singleton__inject, axiom,
    ((![A : hoare_958474565_state, B2 : hoare_958474565_state]: (((insert776267541_state @ A @ bot_bo105666705_state) = (insert776267541_state @ B2 @ bot_bo105666705_state)) => (A = B2))))). % singleton_inject
thf(fact_39_insert__not__empty, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: (~ (((insert776267541_state @ A @ A2) = bot_bo105666705_state)))))). % insert_not_empty
thf(fact_40_doubleton__eq__iff, axiom,
    ((![A : hoare_958474565_state, B2 : hoare_958474565_state, C : hoare_958474565_state, D : hoare_958474565_state]: (((insert776267541_state @ A @ (insert776267541_state @ B2 @ bot_bo105666705_state)) = (insert776267541_state @ C @ (insert776267541_state @ D @ bot_bo105666705_state))) = (((((A = C)) & ((B2 = D)))) | ((((A = D)) & ((B2 = C))))))))). % doubleton_eq_iff
thf(fact_41_singleton__iff, axiom,
    ((![B2 : hoare_958474565_state, A : hoare_958474565_state]: ((member109514606_state @ B2 @ (insert776267541_state @ A @ bot_bo105666705_state)) = (B2 = A))))). % singleton_iff
thf(fact_42_singletonD, axiom,
    ((![B2 : hoare_958474565_state, A : hoare_958474565_state]: ((member109514606_state @ B2 @ (insert776267541_state @ A @ bot_bo105666705_state)) => (B2 = A))))). % singletonD
thf(fact_43_com_Odistinct_I11_J, axiom,
    ((![X7 : pname]: (~ ((skip = (body2 @ X7))))))). % com.distinct(11)
thf(fact_44_the__elem__eq, axiom,
    ((![X : hoare_958474565_state]: ((the_el1300254266_state @ (insert776267541_state @ X @ bot_bo105666705_state)) = X)))). % the_elem_eq
thf(fact_45_weak__Body, axiom,
    ((![G : set_Ho840737317_state, P2 : state > state > $o, Pn : pname, Q : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ (the_com @ (body @ Pn)) @ Q) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ (body2 @ Pn) @ Q) @ bot_bo105666705_state)))))). % weak_Body
thf(fact_46_BodyN, axiom,
    ((![P2 : state > state > $o, Pn : pname, Q : state > state > $o, G : set_Ho840737317_state]: ((hoare_604442164_state @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ (body2 @ Pn) @ Q) @ G) @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ (the_com @ (body @ Pn)) @ Q) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ (body2 @ Pn) @ Q) @ bot_bo105666705_state)))))). % BodyN
thf(fact_47_is__singletonI, axiom,
    ((![X : hoare_958474565_state]: (is_sin536631353_state @ (insert776267541_state @ X @ bot_bo105666705_state))))). % is_singletonI
thf(fact_48_MGF__complete, axiom,
    ((![C : com, P2 : state > state > $o, Q : state > state > $o]: ((hoare_604442164_state @ bot_bo105666705_state @ (insert776267541_state @ (hoare_Mirabelle_MGT @ C) @ bot_bo105666705_state)) => ((hoare_318887606_state @ bot_bo105666705_state @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ C @ Q) @ bot_bo105666705_state)) => (hoare_604442164_state @ bot_bo105666705_state @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ C @ Q) @ bot_bo105666705_state))))))). % MGF_complete
thf(fact_49_Comp, axiom,
    ((![G : set_Ho840737317_state, P2 : state > state > $o, C : com, Q : state > state > $o, D : com, R : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ C @ Q) @ bot_bo105666705_state)) => ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ Q @ D @ R) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ (semi @ C @ D) @ R) @ bot_bo105666705_state))))))). % Comp
thf(fact_50_Set_Ois__empty__def, axiom,
    ((is_emp266830871_state = (^[A3 : set_Ho840737317_state]: (A3 = bot_bo105666705_state))))). % Set.is_empty_def
thf(fact_51_is__singleton__def, axiom,
    ((is_sin536631353_state = (^[A3 : set_Ho840737317_state]: (?[X4 : hoare_958474565_state]: (A3 = (insert776267541_state @ X4 @ bot_bo105666705_state))))))). % is_singleton_def
thf(fact_52_com_Oinject_I3_J, axiom,
    ((![X41 : com, X42 : com, Y41 : com, Y42 : com]: (((semi @ X41 @ X42) = (semi @ Y41 @ Y42)) = (((X41 = Y41)) & ((X42 = Y42))))))). % com.inject(3)
thf(fact_53_com_Odistinct_I41_J, axiom,
    ((![X41 : com, X42 : com, X7 : pname]: (~ (((semi @ X41 @ X42) = (body2 @ X7))))))). % com.distinct(41)
thf(fact_54_evalc__elim__cases_I4_J, axiom,
    ((![C1 : com, C22 : com, S3 : state, T : state]: ((evalc @ (semi @ C1 @ C22) @ S3 @ T) => (~ ((![S1 : state]: ((evalc @ C1 @ S3 @ S1) => (~ ((evalc @ C22 @ S1 @ T))))))))))). % evalc_elim_cases(4)
thf(fact_55_evalc_OSemi, axiom,
    ((![C0 : com, S0 : state, S12 : state, C1 : com, S22 : state]: ((evalc @ C0 @ S0 @ S12) => ((evalc @ C1 @ S12 @ S22) => (evalc @ (semi @ C0 @ C1) @ S0 @ S22)))))). % evalc.Semi
thf(fact_56_com_Odistinct_I5_J, axiom,
    ((![X41 : com, X42 : com]: (~ ((skip = (semi @ X41 @ X42))))))). % com.distinct(5)
thf(fact_57_is__singleton__the__elem, axiom,
    ((is_sin536631353_state = (^[A3 : set_Ho840737317_state]: (A3 = (insert776267541_state @ (the_el1300254266_state @ A3) @ bot_bo105666705_state)))))). % is_singleton_the_elem
thf(fact_58_is__singletonI_H, axiom,
    ((![A2 : set_Ho840737317_state]: ((~ ((A2 = bot_bo105666705_state))) => ((![X5 : hoare_958474565_state, Y4 : hoare_958474565_state]: ((member109514606_state @ X5 @ A2) => ((member109514606_state @ Y4 @ A2) => (X5 = Y4)))) => (is_sin536631353_state @ A2)))))). % is_singletonI'
thf(fact_59_hoare__sound, axiom,
    ((![G : set_Ho840737317_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G @ Ts) => (hoare_318887606_state @ G @ Ts))))). % hoare_sound
thf(fact_60_evalc__elim__cases_I6_J, axiom,
    ((![P2 : pname, S3 : state, S12 : state]: ((evalc @ (body2 @ P2) @ S3 @ S12) => (evalc @ (the_com @ (body @ P2)) @ S3 @ S12))))). % evalc_elim_cases(6)
thf(fact_61_evalc_OBody, axiom,
    ((![Pn : pname, S0 : state, S12 : state]: ((evalc @ (the_com @ (body @ Pn)) @ S0 @ S12) => (evalc @ (body2 @ Pn) @ S0 @ S12))))). % evalc.Body
thf(fact_62_is__singletonE, axiom,
    ((![A2 : set_Ho840737317_state]: ((is_sin536631353_state @ A2) => (~ ((![X5 : hoare_958474565_state]: (~ ((A2 = (insert776267541_state @ X5 @ bot_bo105666705_state))))))))))). % is_singletonE
thf(fact_63_bot__empty__eq, axiom,
    ((bot_bo1428770700tate_o = (^[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ bot_bo105666705_state))))). % bot_empty_eq
thf(fact_64_Collect__empty__eq__bot, axiom,
    ((![P2 : hoare_958474565_state > $o]: (((collec305460656_state @ P2) = bot_bo105666705_state) = (P2 = bot_bo1428770700tate_o))))). % Collect_empty_eq_bot
thf(fact_65_finite__dom__body, axiom,
    ((finite_finite_pname @ (dom_pname_com @ body)))). % finite_dom_body
thf(fact_66_Body__triple__valid__Suc, axiom,
    ((![N : nat, P2 : state > state > $o, Pn : pname, Q : state > state > $o]: ((hoare_364318704_state @ N @ (hoare_1659279548_state @ P2 @ (the_com @ (body @ Pn)) @ Q)) = (hoare_364318704_state @ (suc @ N) @ (hoare_1659279548_state @ P2 @ (body2 @ Pn) @ Q)))))). % Body_triple_valid_Suc
thf(fact_67_finite__insert, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((finite1986656878_state @ (insert776267541_state @ A @ A2)) = (finite1986656878_state @ A2))))). % finite_insert
thf(fact_68_finite__insert, axiom,
    ((![A : pname, A2 : set_pname]: ((finite_finite_pname @ (insert_pname @ A @ A2)) = (finite_finite_pname @ A2))))). % finite_insert
thf(fact_69_infinite__finite__induct, axiom,
    ((![P2 : set_pname > $o, A2 : set_pname]: ((![A4 : set_pname]: ((~ ((finite_finite_pname @ A4))) => (P2 @ A4))) => ((P2 @ bot_bot_set_pname) => ((![X5 : pname, F : set_pname]: ((finite_finite_pname @ F) => ((~ ((member_pname @ X5 @ F))) => ((P2 @ F) => (P2 @ (insert_pname @ X5 @ F)))))) => (P2 @ A2))))))). % infinite_finite_induct
thf(fact_70_infinite__finite__induct, axiom,
    ((![P2 : set_Ho840737317_state > $o, A2 : set_Ho840737317_state]: ((![A4 : set_Ho840737317_state]: ((~ ((finite1986656878_state @ A4))) => (P2 @ A4))) => ((P2 @ bot_bo105666705_state) => ((![X5 : hoare_958474565_state, F : set_Ho840737317_state]: ((finite1986656878_state @ F) => ((~ ((member109514606_state @ X5 @ F))) => ((P2 @ F) => (P2 @ (insert776267541_state @ X5 @ F)))))) => (P2 @ A2))))))). % infinite_finite_induct
thf(fact_71_finite__ne__induct, axiom,
    ((![F2 : set_pname, P2 : set_pname > $o]: ((finite_finite_pname @ F2) => ((~ ((F2 = bot_bot_set_pname))) => ((![X5 : pname]: (P2 @ (insert_pname @ X5 @ bot_bot_set_pname))) => ((![X5 : pname, F : set_pname]: ((finite_finite_pname @ F) => ((~ ((F = bot_bot_set_pname))) => ((~ ((member_pname @ X5 @ F))) => ((P2 @ F) => (P2 @ (insert_pname @ X5 @ F))))))) => (P2 @ F2)))))))). % finite_ne_induct
thf(fact_72_finite__ne__induct, axiom,
    ((![F2 : set_Ho840737317_state, P2 : set_Ho840737317_state > $o]: ((finite1986656878_state @ F2) => ((~ ((F2 = bot_bo105666705_state))) => ((![X5 : hoare_958474565_state]: (P2 @ (insert776267541_state @ X5 @ bot_bo105666705_state))) => ((![X5 : hoare_958474565_state, F : set_Ho840737317_state]: ((finite1986656878_state @ F) => ((~ ((F = bot_bo105666705_state))) => ((~ ((member109514606_state @ X5 @ F))) => ((P2 @ F) => (P2 @ (insert776267541_state @ X5 @ F))))))) => (P2 @ F2)))))))). % finite_ne_induct
thf(fact_73_finite_Oinducts, axiom,
    ((![X : set_pname, P2 : set_pname > $o]: ((finite_finite_pname @ X) => ((P2 @ bot_bot_set_pname) => ((![A4 : set_pname, A5 : pname]: ((finite_finite_pname @ A4) => ((P2 @ A4) => (P2 @ (insert_pname @ A5 @ A4))))) => (P2 @ X))))))). % finite.inducts
thf(fact_74_finite_Oinducts, axiom,
    ((![X : set_Ho840737317_state, P2 : set_Ho840737317_state > $o]: ((finite1986656878_state @ X) => ((P2 @ bot_bo105666705_state) => ((![A4 : set_Ho840737317_state, A5 : hoare_958474565_state]: ((finite1986656878_state @ A4) => ((P2 @ A4) => (P2 @ (insert776267541_state @ A5 @ A4))))) => (P2 @ X))))))). % finite.inducts
thf(fact_75_finite_OemptyI, axiom,
    ((finite_finite_pname @ bot_bot_set_pname))). % finite.emptyI
thf(fact_76_finite_OemptyI, axiom,
    ((finite1986656878_state @ bot_bo105666705_state))). % finite.emptyI
thf(fact_77_infinite__imp__nonempty, axiom,
    ((![S4 : set_pname]: ((~ ((finite_finite_pname @ S4))) => (~ ((S4 = bot_bot_set_pname))))))). % infinite_imp_nonempty
thf(fact_78_infinite__imp__nonempty, axiom,
    ((![S4 : set_Ho840737317_state]: ((~ ((finite1986656878_state @ S4))) => (~ ((S4 = bot_bo105666705_state))))))). % infinite_imp_nonempty
thf(fact_79_finite_OinsertI, axiom,
    ((![A2 : set_Ho840737317_state, A : hoare_958474565_state]: ((finite1986656878_state @ A2) => (finite1986656878_state @ (insert776267541_state @ A @ A2)))))). % finite.insertI
thf(fact_80_finite_OinsertI, axiom,
    ((![A2 : set_pname, A : pname]: ((finite_finite_pname @ A2) => (finite_finite_pname @ (insert_pname @ A @ A2)))))). % finite.insertI
thf(fact_81_finite_Ocases, axiom,
    ((![A : set_pname]: ((finite_finite_pname @ A) => ((~ ((A = bot_bot_set_pname))) => (~ ((![A4 : set_pname]: ((?[A5 : pname]: (A = (insert_pname @ A5 @ A4))) => (~ ((finite_finite_pname @ A4)))))))))))). % finite.cases
thf(fact_82_finite_Ocases, axiom,
    ((![A : set_Ho840737317_state]: ((finite1986656878_state @ A) => ((~ ((A = bot_bo105666705_state))) => (~ ((![A4 : set_Ho840737317_state]: ((?[A5 : hoare_958474565_state]: (A = (insert776267541_state @ A5 @ A4))) => (~ ((finite1986656878_state @ A4)))))))))))). % finite.cases
thf(fact_83_finite_Osimps, axiom,
    ((finite_finite_pname = (^[A6 : set_pname]: (((A6 = bot_bot_set_pname)) | ((?[A3 : set_pname]: (?[B4 : pname]: (((A6 = (insert_pname @ B4 @ A3))) & ((finite_finite_pname @ A3))))))))))). % finite.simps
thf(fact_84_finite_Osimps, axiom,
    ((finite1986656878_state = (^[A6 : set_Ho840737317_state]: (((A6 = bot_bo105666705_state)) | ((?[A3 : set_Ho840737317_state]: (?[B4 : hoare_958474565_state]: (((A6 = (insert776267541_state @ B4 @ A3))) & ((finite1986656878_state @ A3))))))))))). % finite.simps
thf(fact_85_finite__induct, axiom,
    ((![F2 : set_pname, P2 : set_pname > $o]: ((finite_finite_pname @ F2) => ((P2 @ bot_bot_set_pname) => ((![X5 : pname, F : set_pname]: ((finite_finite_pname @ F) => ((~ ((member_pname @ X5 @ F))) => ((P2 @ F) => (P2 @ (insert_pname @ X5 @ F)))))) => (P2 @ F2))))))). % finite_induct
thf(fact_86_finite__induct, axiom,
    ((![F2 : set_Ho840737317_state, P2 : set_Ho840737317_state > $o]: ((finite1986656878_state @ F2) => ((P2 @ bot_bo105666705_state) => ((![X5 : hoare_958474565_state, F : set_Ho840737317_state]: ((finite1986656878_state @ F) => ((~ ((member109514606_state @ X5 @ F))) => ((P2 @ F) => (P2 @ (insert776267541_state @ X5 @ F)))))) => (P2 @ F2))))))). % finite_induct
thf(fact_87_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_88_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_89_evaln__elim__cases_I6_J, axiom,
    ((![P2 : pname, S3 : state, N : nat, S12 : state]: ((evaln @ (body2 @ P2) @ S3 @ N @ S12) => (~ ((![N2 : nat]: ((N = (suc @ N2)) => (~ ((evaln @ (the_com @ (body @ P2)) @ S3 @ N2 @ S12))))))))))). % evaln_elim_cases(6)
thf(fact_90_eval__eq, axiom,
    ((evalc = (^[C3 : com]: (^[S5 : state]: (^[T2 : state]: (?[N3 : nat]: (evaln @ C3 @ S5 @ N3 @ T2)))))))). % eval_eq
thf(fact_91_evalc__evaln, axiom,
    ((![C : com, S3 : state, T : state]: ((evalc @ C @ S3 @ T) => (?[N2 : nat]: (evaln @ C @ S3 @ N2 @ T)))))). % evalc_evaln
thf(fact_92_evaln__evalc, axiom,
    ((![C : com, S3 : state, N : nat, T : state]: ((evaln @ C @ S3 @ N @ T) => (evalc @ C @ S3 @ T))))). % evaln_evalc
thf(fact_93_evaln__max2, axiom,
    ((![C1 : com, S12 : state, N1 : nat, T1 : state, C22 : com, S22 : state, N22 : nat, T22 : state]: ((evaln @ C1 @ S12 @ N1 @ T1) => ((evaln @ C22 @ S22 @ N22 @ T22) => (?[N2 : nat]: ((evaln @ C1 @ S12 @ N2 @ T1) & (evaln @ C22 @ S22 @ N2 @ T22)))))))). % evaln_max2
thf(fact_94_evaln__Suc, axiom,
    ((![C : com, S3 : state, N : nat, S6 : state]: ((evaln @ C @ S3 @ N @ S6) => (evaln @ C @ S3 @ (suc @ N) @ S6))))). % evaln_Suc
thf(fact_95_evaln_OSemi, axiom,
    ((![C0 : com, S0 : state, N : nat, S12 : state, C1 : com, S22 : state]: ((evaln @ C0 @ S0 @ N @ S12) => ((evaln @ C1 @ S12 @ N @ S22) => (evaln @ (semi @ C0 @ C1) @ S0 @ N @ S22)))))). % evaln.Semi
thf(fact_96_evaln__elim__cases_I4_J, axiom,
    ((![C1 : com, C22 : com, S3 : state, N : nat, T : state]: ((evaln @ (semi @ C1 @ C22) @ S3 @ N @ T) => (~ ((![S1 : state]: ((evaln @ C1 @ S3 @ N @ S1) => (~ ((evaln @ C22 @ S1 @ N @ T))))))))))). % evaln_elim_cases(4)
thf(fact_97_evaln_OSkip, axiom,
    ((![S3 : state, N : nat]: (evaln @ skip @ S3 @ N @ S3)))). % evaln.Skip
thf(fact_98_evaln__elim__cases_I1_J, axiom,
    ((![S3 : state, N : nat, T : state]: ((evaln @ skip @ S3 @ N @ T) => (T = S3))))). % evaln_elim_cases(1)
thf(fact_99_triple__valid__def2, axiom,
    ((![N : nat, P2 : state > state > $o, C : com, Q : state > state > $o]: ((hoare_364318704_state @ N @ (hoare_1659279548_state @ P2 @ C @ Q)) = (![Z3 : state]: (![S5 : state]: (((P2 @ Z3 @ S5)) => ((![S7 : state]: (((evaln @ C @ S5 @ N @ S7)) => ((Q @ Z3 @ S7)))))))))))). % triple_valid_def2
thf(fact_100_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_101_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_102_evaln_OBody, axiom,
    ((![Pn : pname, S0 : state, N : nat, S12 : state]: ((evaln @ (the_com @ (body @ Pn)) @ S0 @ N @ S12) => (evaln @ (body2 @ Pn) @ S0 @ (suc @ N) @ S12))))). % evaln.Body
thf(fact_103_Body__triple__valid__0, axiom,
    ((![P2 : state > state > $o, Pn : pname, Q : state > state > $o]: (hoare_364318704_state @ zero_zero_nat @ (hoare_1659279548_state @ P2 @ (body2 @ Pn) @ Q))))). % Body_triple_valid_0
thf(fact_104_card__insert__disjoint, axiom,
    ((![A2 : set_Ho840737317_state, X : hoare_958474565_state]: ((finite1986656878_state @ A2) => ((~ ((member109514606_state @ X @ A2))) => ((finite1141247341_state @ (insert776267541_state @ X @ A2)) = (suc @ (finite1141247341_state @ A2)))))))). % card_insert_disjoint
thf(fact_105_card__insert__disjoint, axiom,
    ((![A2 : set_pname, X : pname]: ((finite_finite_pname @ A2) => ((~ ((member_pname @ X @ A2))) => ((finite_card_pname @ (insert_pname @ X @ A2)) = (suc @ (finite_card_pname @ A2)))))))). % card_insert_disjoint
thf(fact_106_card_Oempty, axiom,
    (((finite1141247341_state @ bot_bo105666705_state) = zero_zero_nat))). % card.empty
thf(fact_107_card_Oinfinite, axiom,
    ((![A2 : set_pname]: ((~ ((finite_finite_pname @ A2))) => ((finite_card_pname @ A2) = zero_zero_nat))))). % card.infinite
thf(fact_108_card__0__eq, axiom,
    ((![A2 : set_pname]: ((finite_finite_pname @ A2) => (((finite_card_pname @ A2) = zero_zero_nat) = (A2 = bot_bot_set_pname)))))). % card_0_eq
thf(fact_109_card__0__eq, axiom,
    ((![A2 : set_Ho840737317_state]: ((finite1986656878_state @ A2) => (((finite1141247341_state @ A2) = zero_zero_nat) = (A2 = bot_bo105666705_state)))))). % card_0_eq
thf(fact_110_bot__nat__def, axiom,
    ((bot_bot_nat = zero_zero_nat))). % bot_nat_def
thf(fact_111_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M : nat]: (N = (suc @ M))))))). % not0_implies_Suc
thf(fact_112_old_Onat_Oinducts, axiom,
    ((![P2 : nat > $o, Nat : nat]: ((P2 @ zero_zero_nat) => ((![Nat3 : nat]: ((P2 @ Nat3) => (P2 @ (suc @ Nat3)))) => (P2 @ Nat)))))). % old.nat.inducts
thf(fact_113_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_114_Zero__not__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_not_Suc
thf(fact_115_Zero__neq__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_neq_Suc
thf(fact_116_Suc__neq__Zero, axiom,
    ((![M2 : nat]: (~ (((suc @ M2) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_117_zero__induct, axiom,
    ((![P2 : nat > $o, K : nat]: ((P2 @ K) => ((![N2 : nat]: ((P2 @ (suc @ N2)) => (P2 @ N2))) => (P2 @ zero_zero_nat)))))). % zero_induct
thf(fact_118_diff__induct, axiom,
    ((![P2 : nat > nat > $o, M2 : nat, N : nat]: ((![X5 : nat]: (P2 @ X5 @ zero_zero_nat)) => ((![Y4 : nat]: (P2 @ zero_zero_nat @ (suc @ Y4))) => ((![X5 : nat, Y4 : nat]: ((P2 @ X5 @ Y4) => (P2 @ (suc @ X5) @ (suc @ Y4)))) => (P2 @ M2 @ N))))))). % diff_induct
thf(fact_119_nat__induct, axiom,
    ((![P2 : nat > $o, N : nat]: ((P2 @ zero_zero_nat) => ((![N2 : nat]: ((P2 @ N2) => (P2 @ (suc @ N2)))) => (P2 @ N)))))). % nat_induct
thf(fact_120_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_121_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_122_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_123_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_124_card__Suc__eq, axiom,
    ((![A2 : set_Ho840737317_state, K : nat]: (((finite1141247341_state @ A2) = (suc @ K)) = (?[B4 : hoare_958474565_state]: (?[B5 : set_Ho840737317_state]: (((A2 = (insert776267541_state @ B4 @ B5))) & ((((~ ((member109514606_state @ B4 @ B5)))) & (((((finite1141247341_state @ B5) = K)) & ((((K = zero_zero_nat)) => ((B5 = bot_bo105666705_state))))))))))))))). % card_Suc_eq
thf(fact_125_card__eq__SucD, axiom,
    ((![A2 : set_Ho840737317_state, K : nat]: (((finite1141247341_state @ A2) = (suc @ K)) => (?[B6 : hoare_958474565_state, B3 : set_Ho840737317_state]: ((A2 = (insert776267541_state @ B6 @ B3)) & ((~ ((member109514606_state @ B6 @ B3))) & (((finite1141247341_state @ B3) = K) & ((K = zero_zero_nat) => (B3 = bot_bo105666705_state)))))))))). % card_eq_SucD
thf(fact_126_card__1__singleton__iff, axiom,
    ((![A2 : set_Ho840737317_state]: (((finite1141247341_state @ A2) = (suc @ zero_zero_nat)) = (?[X4 : hoare_958474565_state]: (A2 = (insert776267541_state @ X4 @ bot_bo105666705_state))))))). % card_1_singleton_iff
thf(fact_127_card__eq__0__iff, axiom,
    ((![A2 : set_pname]: (((finite_card_pname @ A2) = zero_zero_nat) = (((A2 = bot_bot_set_pname)) | ((~ ((finite_finite_pname @ A2))))))))). % card_eq_0_iff
thf(fact_128_card__eq__0__iff, axiom,
    ((![A2 : set_Ho840737317_state]: (((finite1141247341_state @ A2) = zero_zero_nat) = (((A2 = bot_bo105666705_state)) | ((~ ((finite1986656878_state @ A2))))))))). % card_eq_0_iff
thf(fact_129_card__insert__if, axiom,
    ((![A2 : set_Ho840737317_state, X : hoare_958474565_state]: ((finite1986656878_state @ A2) => (((member109514606_state @ X @ A2) => ((finite1141247341_state @ (insert776267541_state @ X @ A2)) = (finite1141247341_state @ A2))) & ((~ ((member109514606_state @ X @ A2))) => ((finite1141247341_state @ (insert776267541_state @ X @ A2)) = (suc @ (finite1141247341_state @ A2))))))))). % card_insert_if
thf(fact_130_card__insert__if, axiom,
    ((![A2 : set_pname, X : pname]: ((finite_finite_pname @ A2) => (((member_pname @ X @ A2) => ((finite_card_pname @ (insert_pname @ X @ A2)) = (finite_card_pname @ A2))) & ((~ ((member_pname @ X @ A2))) => ((finite_card_pname @ (insert_pname @ X @ A2)) = (suc @ (finite_card_pname @ A2))))))))). % card_insert_if
thf(fact_131_triple_Osize__gen, axiom,
    ((![X : state > nat, X1 : state > state > $o, X2 : com, X3 : state > state > $o]: ((hoare_103296669_state @ X @ (hoare_1659279548_state @ X1 @ X2 @ X3)) = (suc @ zero_zero_nat))))). % triple.size_gen
thf(fact_132_triple_Osize_I2_J, axiom,
    ((![X1 : state > state > $o, X2 : com, X3 : state > state > $o]: ((size_s1457777073_state @ (hoare_1659279548_state @ X1 @ X2 @ X3)) = (suc @ zero_zero_nat))))). % triple.size(2)
thf(fact_133_card_Oremove, axiom,
    ((![A2 : set_pname, X : pname]: ((finite_finite_pname @ A2) => ((member_pname @ X @ A2) => ((finite_card_pname @ A2) = (suc @ (finite_card_pname @ (minus_1937938585_pname @ A2 @ (insert_pname @ X @ bot_bot_set_pname)))))))))). % card.remove
thf(fact_134_card_Oremove, axiom,
    ((![A2 : set_Ho840737317_state, X : hoare_958474565_state]: ((finite1986656878_state @ A2) => ((member109514606_state @ X @ A2) => ((finite1141247341_state @ A2) = (suc @ (finite1141247341_state @ (minus_1628593292_state @ A2 @ (insert776267541_state @ X @ bot_bo105666705_state)))))))))). % card.remove
thf(fact_135_card__Suc__Diff1, axiom,
    ((![A2 : set_pname, X : pname]: ((finite_finite_pname @ A2) => ((member_pname @ X @ A2) => ((suc @ (finite_card_pname @ (minus_1937938585_pname @ A2 @ (insert_pname @ X @ bot_bot_set_pname)))) = (finite_card_pname @ A2))))))). % card_Suc_Diff1
thf(fact_136_card__Suc__Diff1, axiom,
    ((![A2 : set_Ho840737317_state, X : hoare_958474565_state]: ((finite1986656878_state @ A2) => ((member109514606_state @ X @ A2) => ((suc @ (finite1141247341_state @ (minus_1628593292_state @ A2 @ (insert776267541_state @ X @ bot_bo105666705_state)))) = (finite1141247341_state @ A2))))))). % card_Suc_Diff1
thf(fact_137_Diff__empty, axiom,
    ((![A2 : set_Ho840737317_state]: ((minus_1628593292_state @ A2 @ bot_bo105666705_state) = A2)))). % Diff_empty
thf(fact_138_empty__Diff, axiom,
    ((![A2 : set_Ho840737317_state]: ((minus_1628593292_state @ bot_bo105666705_state @ A2) = bot_bo105666705_state)))). % empty_Diff
thf(fact_139_Diff__cancel, axiom,
    ((![A2 : set_Ho840737317_state]: ((minus_1628593292_state @ A2 @ A2) = bot_bo105666705_state)))). % Diff_cancel
thf(fact_140_finite__Diff2, axiom,
    ((![B : set_pname, A2 : set_pname]: ((finite_finite_pname @ B) => ((finite_finite_pname @ (minus_1937938585_pname @ A2 @ B)) = (finite_finite_pname @ A2)))))). % finite_Diff2
thf(fact_141_finite__Diff, axiom,
    ((![A2 : set_pname, B : set_pname]: ((finite_finite_pname @ A2) => (finite_finite_pname @ (minus_1937938585_pname @ A2 @ B)))))). % finite_Diff
thf(fact_142_insert__Diff1, axiom,
    ((![X : hoare_958474565_state, B : set_Ho840737317_state, A2 : set_Ho840737317_state]: ((member109514606_state @ X @ B) => ((minus_1628593292_state @ (insert776267541_state @ X @ A2) @ B) = (minus_1628593292_state @ A2 @ B)))))). % insert_Diff1
thf(fact_143_Diff__insert0, axiom,
    ((![X : hoare_958474565_state, A2 : set_Ho840737317_state, B : set_Ho840737317_state]: ((~ ((member109514606_state @ X @ A2))) => ((minus_1628593292_state @ A2 @ (insert776267541_state @ X @ B)) = (minus_1628593292_state @ A2 @ B)))))). % Diff_insert0
thf(fact_144_insert__Diff__single, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((insert776267541_state @ A @ (minus_1628593292_state @ A2 @ (insert776267541_state @ A @ bot_bo105666705_state))) = (insert776267541_state @ A @ A2))))). % insert_Diff_single
thf(fact_145_finite__Diff__insert, axiom,
    ((![A2 : set_Ho840737317_state, A : hoare_958474565_state, B : set_Ho840737317_state]: ((finite1986656878_state @ (minus_1628593292_state @ A2 @ (insert776267541_state @ A @ B))) = (finite1986656878_state @ (minus_1628593292_state @ A2 @ B)))))). % finite_Diff_insert
thf(fact_146_finite__Diff__insert, axiom,
    ((![A2 : set_pname, A : pname, B : set_pname]: ((finite_finite_pname @ (minus_1937938585_pname @ A2 @ (insert_pname @ A @ B))) = (finite_finite_pname @ (minus_1937938585_pname @ A2 @ B)))))). % finite_Diff_insert

% Conjectures (2)
thf(conj_0, hypothesis,
    ((![X6 : pname]: ((member_pname @ X6 @ (dom_pname_com @ body)) => (hoare_604442164_state @ g @ (insert776267541_state @ (hoare_1659279548_state @ (^[Y5 : state]: (^[Z4 : state]: (Y5 = Z4))) @ (body2 @ X6) @ (evalc @ (body2 @ X6))) @ bot_bo105666705_state)))))).
thf(conj_1, conjecture,
    ((![Z : state, S : state]: ((~ ((Z = S))) | (evalc @ skip @ Z @ S))))).
