% 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_410__3254528_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:37.801

% Could-be-implicit typings (16)
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J_J, type,
    set_op1790085995_state : $tType).
thf(ty_n_t__Option__Ooption_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    option1481361675_state : $tType).
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__Set__Oset_It__Product____Type__Oprod_It__Com__Opname_Mt__Com__Opname_J_J, type,
    set_Pr15211605_pname : $tType).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Com__Opname_Mt__Com__Opname_J_J, type,
    set_Su1011432617_pname : $tType).
thf(ty_n_t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    hoare_958474565_state : $tType).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Com__Opname_J_J, type,
    set_option_pname : $tType).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Com__Ocom_J_J, type,
    set_option_com : $tType).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Com__Opname_J_J, type,
    set_set_pname : $tType).
thf(ty_n_t__Option__Ooption_It__Com__Opname_J, type,
    option_pname : $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__Set__Oset_It__Com__Ocom_J, type,
    set_com : $tType).
thf(ty_n_t__Com__Ostate, type,
    state : $tType).
thf(ty_n_t__Com__Opname, type,
    pname : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).

% Explicit typings (66)
thf(sy_c_Com_OWT, type,
    wt : com > $o).
thf(sy_c_Com_OWT__bodies, type,
    wT_bodies : $o).
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_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_Finite__Set_Ofinite_001t__Option__Ooption_It__Com__Opname_J, type,
    finite1292484209_pname : set_option_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Com__Opname_Mt__Com__Opname_J, type,
    finite1490780830_pname : set_Pr15211605_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Com__Opname_J, type,
    finite505202775_pname : set_set_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Com__Opname_Mt__Com__Opname_J, type,
    finite1993869834_pname : set_Su1011432617_pname > $o).
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_Ostate__not__singleton, type,
    hoare_405891322gleton : $o).
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_Map_Odom_001t__Com__Opname_001t__Com__Ocom, type,
    dom_pname_com : (pname > option_com) > set_pname).
thf(sy_c_Map_Odom_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_001t__Com__Ocom, type,
    dom_Ho256784515te_com : (hoare_958474565_state > option_com) > set_Ho840737317_state).
thf(sy_c_Map_Omap__le_001t__Com__Opname_001t__Com__Ocom, type,
    map_le_pname_com : (pname > option_com) > (pname > option_com) > $o).
thf(sy_c_Natural_Oevalc, type,
    evalc : com > state > state > $o).
thf(sy_c_Option_Ooption_ONone_001t__Com__Ocom, type,
    none_com : option_com).
thf(sy_c_Option_Ooption_ONone_001t__Com__Opname, type,
    none_pname : option_pname).
thf(sy_c_Option_Ooption_ONone_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    none_H2001963430_state : option1481361675_state).
thf(sy_c_Option_Ooption_OSome_001t__Com__Ocom, type,
    some_com : com > option_com).
thf(sy_c_Option_Ooption_OSome_001t__Com__Opname, type,
    some_pname : pname > option_pname).
thf(sy_c_Option_Ooption_OSome_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    some_H1327573290_state : hoare_958474565_state > option1481361675_state).
thf(sy_c_Option_Ooption_Oset__option_001t__Com__Ocom, type,
    set_option_com2 : option_com > set_com).
thf(sy_c_Option_Ooption_Oset__option_001t__Com__Opname, type,
    set_option_pname2 : option_pname > set_pname).
thf(sy_c_Option_Ooption_Oset__option_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    set_op1542142163_state : option1481361675_state > set_Ho840737317_state).
thf(sy_c_Option_Ooption_Othe_001t__Com__Ocom, type,
    the_com : option_com > com).
thf(sy_c_Option_Ooption_Othe_001t__Com__Opname, type,
    the_pname : option_pname > pname).
thf(sy_c_Option_Othese_001t__Com__Ocom, type,
    these_com : set_option_com > set_com).
thf(sy_c_Option_Othese_001t__Com__Opname, type,
    these_pname : set_option_pname > set_pname).
thf(sy_c_Option_Othese_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    these_1282564994_state : set_op1790085995_state > set_Ho840737317_state).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Com__Opname_M_Eo_J, type,
    bot_bot_pname_o : pname > $o).
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__Set__Oset_It__Com__Ocom_J, type,
    bot_bot_set_com : set_com).
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_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Com__Ocom_J_J, type,
    bot_bo697186986on_com : set_option_com).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J_J, type,
    bot_bo472264663_state : set_op1790085995_state).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Com__Opname_J, type,
    top_top_set_pname : set_pname).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    top_to519946101_state : set_Ho840737317_state).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Com__Opname_J_J, type,
    top_to1034942016_pname : set_option_pname).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Com__Opname_Mt__Com__Opname_J_J, type,
    top_to2104644005_pname : set_Pr15211605_pname).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Com__Opname_J_J, type,
    top_to1806476070_pname : set_set_pname).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Com__Opname_Mt__Com__Opname_J_J, type,
    top_to380724057_pname : set_Su1011432617_pname).
thf(sy_c_Set_OCollect_001t__Com__Opname, type,
    collect_pname : (pname > $o) > set_pname).
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__Ocom, type,
    insert_com : com > set_com > set_com).
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_Oinsert_001t__Option__Ooption_It__Com__Ocom_J, type,
    insert_option_com : option_com > set_option_com > set_option_com).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    insert677809883_state : option1481361675_state > set_op1790085995_state > set_op1790085995_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__Com__Opname, type,
    is_singleton_pname : set_pname > $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__Ocom, type,
    member_com : com > set_com > $o).
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_c_member_001t__Option__Ooption_It__Com__Ocom_J, type,
    member_option_com : option_com > set_option_com > $o).
thf(sy_c_member_001t__Option__Ooption_It__Com__Opname_J, type,
    member_option_pname : option_pname > set_option_pname > $o).
thf(sy_v_c, type,
    c : com).
thf(sy_v_pn, type,
    pn : pname).

% Relevant facts (167)
thf(fact_0_single__stateE, axiom,
    ((hoare_405891322gleton => (![T : state]: (~ ((![S : state]: (S = T)))))))). % single_stateE
thf(fact_1_state__not__singleton__def, axiom,
    ((hoare_405891322gleton = (?[S2 : state]: (?[T2 : state]: (~ ((S2 = T2)))))))). % state_not_singleton_def
thf(fact_2_WT__bodiesD, axiom,
    ((![Pn : pname, B : com]: (wT_bodies => (((body @ Pn) = (some_com @ B)) => (wt @ B)))))). % WT_bodiesD
thf(fact_3_finite__dom__body, axiom,
    ((finite_finite_pname @ (dom_pname_com @ body)))). % finite_dom_body
thf(fact_4_MGF__lemma1, axiom,
    ((![G : set_Ho840737317_state, C : com]: (hoare_405891322gleton => ((![X : pname]: ((member_pname @ X @ (dom_pname_com @ body)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ X)) @ bot_bo105666705_state)))) => ((wt @ C) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_Mirabelle_MGT @ C) @ bot_bo105666705_state)))))))). % MGF_lemma1
thf(fact_5_evalc__elim__cases_I6_J, axiom,
    ((![P : pname, S3 : state, S1 : state]: ((evalc @ (body2 @ P) @ S3 @ S1) => (evalc @ (the_com @ (body @ P)) @ S3 @ S1))))). % evalc_elim_cases(6)
thf(fact_6_evalc_OBody, axiom,
    ((![Pn : pname, S0 : state, S1 : state]: ((evalc @ (the_com @ (body @ Pn)) @ S0 @ S1) => (evalc @ (body2 @ Pn) @ S0 @ S1))))). % evalc.Body
thf(fact_7_WT_OSkip, axiom,
    ((wt @ skip))). % WT.Skip
thf(fact_8_WTs__elim__cases_I4_J, axiom,
    ((![C1 : com, C2 : com]: ((wt @ (semi @ C1 @ C2)) => (~ (((wt @ C1) => (~ ((wt @ C2)))))))))). % WTs_elim_cases(4)
thf(fact_9_WT_OSemi, axiom,
    ((![C0 : com, C1 : com]: ((wt @ C0) => ((wt @ C1) => (wt @ (semi @ C0 @ C1))))))). % WT.Semi
thf(fact_10_WT_OBody, axiom,
    ((![Pn : pname]: ((~ (((body @ Pn) = none_com))) => (wt @ (body2 @ Pn)))))). % WT.Body
thf(fact_11_WTs__elim__cases_I7_J, axiom,
    ((![P : pname]: ((wt @ (body2 @ P)) => (~ ((![Y : com]: (~ (((body @ P) = (some_com @ Y))))))))))). % WTs_elim_cases(7)
thf(fact_12_map__le__def, axiom,
    ((map_le_pname_com = (^[M_1 : pname > option_com]: (^[M_2 : pname > option_com]: (![X2 : pname]: (((member_pname @ X2 @ (dom_pname_com @ M_1))) => (((M_1 @ X2) = (M_2 @ X2)))))))))). % map_le_def
thf(fact_13_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_14_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_15_dom__eq__empty__conv, axiom,
    ((![F : pname > option_com]: (((dom_pname_com @ F) = bot_bot_set_pname) = (F = (^[X2 : pname]: none_com)))))). % dom_eq_empty_conv
thf(fact_16_dom__eq__empty__conv, axiom,
    ((![F : hoare_958474565_state > option_com]: (((dom_Ho256784515te_com @ F) = bot_bo105666705_state) = (F = (^[X2 : hoare_958474565_state]: none_com)))))). % dom_eq_empty_conv
thf(fact_17_com_Odistinct_I41_J, axiom,
    ((![X41 : com, X42 : com, X7 : pname]: (~ (((semi @ X41 @ X42) = (body2 @ X7))))))). % com.distinct(41)
thf(fact_18_com_Odistinct_I11_J, axiom,
    ((![X7 : pname]: (~ ((skip = (body2 @ X7))))))). % com.distinct(11)
thf(fact_19_com_Odistinct_I5_J, axiom,
    ((![X41 : com, X42 : com]: (~ ((skip = (semi @ X41 @ X42))))))). % com.distinct(5)
thf(fact_20_evalc__elim__cases_I4_J, axiom,
    ((![C1 : com, C2 : com, S3 : state, T3 : state]: ((evalc @ (semi @ C1 @ C2) @ S3 @ T3) => (~ ((![S12 : state]: ((evalc @ C1 @ S3 @ S12) => (~ ((evalc @ C2 @ S12 @ T3))))))))))). % evalc_elim_cases(4)
thf(fact_21_evalc__elim__cases_I1_J, axiom,
    ((![S3 : state, T3 : state]: ((evalc @ skip @ S3 @ T3) => (T3 = S3))))). % evalc_elim_cases(1)
thf(fact_22_insert__dom, axiom,
    ((![F : pname > option_com, X3 : pname, Y2 : com]: (((F @ X3) = (some_com @ Y2)) => ((insert_pname @ X3 @ (dom_pname_com @ F)) = (dom_pname_com @ F)))))). % insert_dom
thf(fact_23_insert__dom, axiom,
    ((![F : hoare_958474565_state > option_com, X3 : hoare_958474565_state, Y2 : com]: (((F @ X3) = (some_com @ Y2)) => ((insert776267541_state @ X3 @ (dom_Ho256784515te_com @ F)) = (dom_Ho256784515te_com @ F)))))). % insert_dom
thf(fact_24_com__det, axiom,
    ((![C : com, S3 : state, T3 : state, U : state]: ((evalc @ C @ S3 @ T3) => ((evalc @ C @ S3 @ U) => (U = T3)))))). % com_det
thf(fact_25_evalc_OSemi, axiom,
    ((![C0 : com, S0 : state, S1 : state, C1 : com, S22 : state]: ((evalc @ C0 @ S0 @ S1) => ((evalc @ C1 @ S1 @ S22) => (evalc @ (semi @ C0 @ C1) @ S0 @ S22)))))). % evalc.Semi
thf(fact_26_evalc_OSkip, axiom,
    ((![S3 : state]: (evalc @ skip @ S3 @ S3)))). % evalc.Skip
thf(fact_27_derivs__insertD, axiom,
    ((![G : set_Ho840737317_state, T3 : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G @ (insert776267541_state @ T3 @ Ts)) => ((hoare_604442164_state @ G @ (insert776267541_state @ T3 @ bot_bo105666705_state)) & (hoare_604442164_state @ G @ Ts)))))). % derivs_insertD
thf(fact_28_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_29_empty, axiom,
    ((![G : set_Ho840737317_state]: (hoare_604442164_state @ G @ bot_bo105666705_state)))). % empty
thf(fact_30_hoare__derivs_Oinsert, axiom,
    ((![G : set_Ho840737317_state, T3 : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G @ (insert776267541_state @ T3 @ bot_bo105666705_state)) => ((hoare_604442164_state @ G @ Ts) => (hoare_604442164_state @ G @ (insert776267541_state @ T3 @ Ts))))))). % hoare_derivs.insert
thf(fact_31_MGT__BodyN, axiom,
    ((![Pn : pname, G : set_Ho840737317_state]: ((hoare_604442164_state @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ Pn)) @ G) @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (the_com @ (body @ Pn))) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ Pn)) @ bot_bo105666705_state)))))). % MGT_BodyN
thf(fact_32_domI, axiom,
    ((![M : pname > option_com, A : pname, B : com]: (((M @ A) = (some_com @ B)) => (member_pname @ A @ (dom_pname_com @ M)))))). % domI
thf(fact_33_domD, axiom,
    ((![A : pname, M : pname > option_com]: ((member_pname @ A @ (dom_pname_com @ M)) => (?[B2 : com]: ((M @ A) = (some_com @ B2))))))). % domD
thf(fact_34_domIff, axiom,
    ((![A : pname, M : pname > option_com]: ((member_pname @ A @ (dom_pname_com @ M)) = (~ (((M @ A) = none_com))))))). % domIff
thf(fact_35_option_Ocollapse, axiom,
    ((![Option : option_com]: ((~ ((Option = none_com))) => ((some_com @ (the_com @ Option)) = Option))))). % option.collapse
thf(fact_36_not__None__eq, axiom,
    ((![X3 : option_com]: ((~ ((X3 = none_com))) = (?[Y3 : com]: (X3 = (some_com @ Y3))))))). % not_None_eq
thf(fact_37_not__Some__eq, axiom,
    ((![X3 : option_com]: ((![Y3 : com]: (~ ((X3 = (some_com @ Y3))))) = (X3 = none_com))))). % not_Some_eq
thf(fact_38_finite__insert, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((finite1986656878_state @ (insert776267541_state @ A @ A2)) = (finite1986656878_state @ A2))))). % finite_insert
thf(fact_39_finite__insert, axiom,
    ((![A : pname, A2 : set_pname]: ((finite_finite_pname @ (insert_pname @ A @ A2)) = (finite_finite_pname @ A2))))). % finite_insert
thf(fact_40_singletonI, axiom,
    ((![A : pname]: (member_pname @ A @ (insert_pname @ A @ bot_bot_set_pname))))). % singletonI
thf(fact_41_singletonI, axiom,
    ((![A : hoare_958474565_state]: (member109514606_state @ A @ (insert776267541_state @ A @ bot_bo105666705_state))))). % singletonI
thf(fact_42_option_Oexhaust__sel, axiom,
    ((![Option : option_com]: ((~ ((Option = none_com))) => (Option = (some_com @ (the_com @ Option))))))). % option.exhaust_sel
thf(fact_43_finite_Ocases, axiom,
    ((![A : set_pname]: ((finite_finite_pname @ A) => ((~ ((A = bot_bot_set_pname))) => (~ ((![A3 : set_pname]: ((?[A4 : pname]: (A = (insert_pname @ A4 @ A3))) => (~ ((finite_finite_pname @ A3)))))))))))). % finite.cases
thf(fact_44_finite_Ocases, axiom,
    ((![A : set_Ho840737317_state]: ((finite1986656878_state @ A) => ((~ ((A = bot_bo105666705_state))) => (~ ((![A3 : set_Ho840737317_state]: ((?[A4 : hoare_958474565_state]: (A = (insert776267541_state @ A4 @ A3))) => (~ ((finite1986656878_state @ A3)))))))))))). % finite.cases
thf(fact_45_finite_Osimps, axiom,
    ((finite_finite_pname = (^[A5 : set_pname]: (((A5 = bot_bot_set_pname)) | ((?[A6 : set_pname]: (?[B3 : pname]: (((A5 = (insert_pname @ B3 @ A6))) & ((finite_finite_pname @ A6))))))))))). % finite.simps
thf(fact_46_finite_Osimps, axiom,
    ((finite1986656878_state = (^[A5 : set_Ho840737317_state]: (((A5 = bot_bo105666705_state)) | ((?[A6 : set_Ho840737317_state]: (?[B3 : hoare_958474565_state]: (((A5 = (insert776267541_state @ B3 @ A6))) & ((finite1986656878_state @ A6))))))))))). % finite.simps
thf(fact_47_mem__Collect__eq, axiom,
    ((![A : pname, P : pname > $o]: ((member_pname @ A @ (collect_pname @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_48_Collect__mem__eq, axiom,
    ((![A2 : set_pname]: ((collect_pname @ (^[X2 : pname]: (member_pname @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_49_finite__induct, axiom,
    ((![F2 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ F2) => ((P @ bot_bot_set_pname) => ((![X : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((~ ((member_pname @ X @ F3))) => ((P @ F3) => (P @ (insert_pname @ X @ F3)))))) => (P @ F2))))))). % finite_induct
thf(fact_50_finite__induct, axiom,
    ((![F2 : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ F2) => ((P @ bot_bo105666705_state) => ((![X : hoare_958474565_state, F3 : set_Ho840737317_state]: ((finite1986656878_state @ F3) => ((~ ((member109514606_state @ X @ F3))) => ((P @ F3) => (P @ (insert776267541_state @ X @ F3)))))) => (P @ F2))))))). % finite_induct
thf(fact_51_finite_Oinducts, axiom,
    ((![X3 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ X3) => ((P @ bot_bot_set_pname) => ((![A3 : set_pname, A4 : pname]: ((finite_finite_pname @ A3) => ((P @ A3) => (P @ (insert_pname @ A4 @ A3))))) => (P @ X3))))))). % finite.inducts
thf(fact_52_finite_Oinducts, axiom,
    ((![X3 : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ X3) => ((P @ bot_bo105666705_state) => ((![A3 : set_Ho840737317_state, A4 : hoare_958474565_state]: ((finite1986656878_state @ A3) => ((P @ A3) => (P @ (insert776267541_state @ A4 @ A3))))) => (P @ X3))))))). % finite.inducts
thf(fact_53_empty__Collect__eq, axiom,
    ((![P : hoare_958474565_state > $o]: ((bot_bo105666705_state = (collec305460656_state @ P)) = (![X2 : hoare_958474565_state]: (~ ((P @ X2)))))))). % empty_Collect_eq
thf(fact_54_Collect__empty__eq, axiom,
    ((![P : hoare_958474565_state > $o]: (((collec305460656_state @ P) = bot_bo105666705_state) = (![X2 : hoare_958474565_state]: (~ ((P @ X2)))))))). % Collect_empty_eq
thf(fact_55_all__not__in__conv, axiom,
    ((![A2 : set_pname]: ((![X2 : pname]: (~ ((member_pname @ X2 @ A2)))) = (A2 = bot_bot_set_pname))))). % all_not_in_conv
thf(fact_56_all__not__in__conv, axiom,
    ((![A2 : set_Ho840737317_state]: ((![X2 : hoare_958474565_state]: (~ ((member109514606_state @ X2 @ A2)))) = (A2 = bot_bo105666705_state))))). % all_not_in_conv
thf(fact_57_empty__iff, axiom,
    ((![C : pname]: (~ ((member_pname @ C @ bot_bot_set_pname)))))). % empty_iff
thf(fact_58_empty__iff, axiom,
    ((![C : hoare_958474565_state]: (~ ((member109514606_state @ C @ bot_bo105666705_state)))))). % empty_iff
thf(fact_59_insert__absorb2, axiom,
    ((![X3 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((insert776267541_state @ X3 @ (insert776267541_state @ X3 @ A2)) = (insert776267541_state @ X3 @ A2))))). % insert_absorb2
thf(fact_60_insert__iff, axiom,
    ((![A : hoare_958474565_state, B : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ (insert776267541_state @ B @ A2)) = (((A = B)) | ((member109514606_state @ A @ A2))))))). % insert_iff
thf(fact_61_insert__iff, axiom,
    ((![A : pname, B : pname, A2 : set_pname]: ((member_pname @ A @ (insert_pname @ B @ A2)) = (((A = B)) | ((member_pname @ A @ A2))))))). % insert_iff
thf(fact_62_insertCI, axiom,
    ((![A : hoare_958474565_state, B4 : set_Ho840737317_state, B : hoare_958474565_state]: (((~ ((member109514606_state @ A @ B4))) => (A = B)) => (member109514606_state @ A @ (insert776267541_state @ B @ B4)))))). % insertCI
thf(fact_63_insertCI, axiom,
    ((![A : pname, B4 : set_pname, B : pname]: (((~ ((member_pname @ A @ B4))) => (A = B)) => (member_pname @ A @ (insert_pname @ B @ B4)))))). % insertCI
thf(fact_64_option_Oinject, axiom,
    ((![X22 : com, Y22 : com]: (((some_com @ X22) = (some_com @ Y22)) = (X22 = Y22))))). % option.inject
thf(fact_65_ex__in__conv, axiom,
    ((![A2 : set_pname]: ((?[X2 : pname]: (member_pname @ X2 @ A2)) = (~ ((A2 = bot_bot_set_pname))))))). % ex_in_conv
thf(fact_66_ex__in__conv, axiom,
    ((![A2 : set_Ho840737317_state]: ((?[X2 : hoare_958474565_state]: (member109514606_state @ X2 @ A2)) = (~ ((A2 = bot_bo105666705_state))))))). % ex_in_conv
thf(fact_67_equals0I, axiom,
    ((![A2 : set_pname]: ((![Y : pname]: (~ ((member_pname @ Y @ A2)))) => (A2 = bot_bot_set_pname))))). % equals0I
thf(fact_68_equals0I, axiom,
    ((![A2 : set_Ho840737317_state]: ((![Y : hoare_958474565_state]: (~ ((member109514606_state @ Y @ A2)))) => (A2 = bot_bo105666705_state))))). % equals0I
thf(fact_69_equals0D, axiom,
    ((![A2 : set_pname, A : pname]: ((A2 = bot_bot_set_pname) => (~ ((member_pname @ A @ A2))))))). % equals0D
thf(fact_70_equals0D, axiom,
    ((![A2 : set_Ho840737317_state, A : hoare_958474565_state]: ((A2 = bot_bo105666705_state) => (~ ((member109514606_state @ A @ A2))))))). % equals0D
thf(fact_71_emptyE, axiom,
    ((![A : pname]: (~ ((member_pname @ A @ bot_bot_set_pname)))))). % emptyE
thf(fact_72_emptyE, axiom,
    ((![A : hoare_958474565_state]: (~ ((member109514606_state @ A @ bot_bo105666705_state)))))). % emptyE
thf(fact_73_mk__disjoint__insert, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ A2) => (?[B5 : set_Ho840737317_state]: ((A2 = (insert776267541_state @ A @ B5)) & (~ ((member109514606_state @ A @ B5))))))))). % mk_disjoint_insert
thf(fact_74_mk__disjoint__insert, axiom,
    ((![A : pname, A2 : set_pname]: ((member_pname @ A @ A2) => (?[B5 : set_pname]: ((A2 = (insert_pname @ A @ B5)) & (~ ((member_pname @ A @ B5))))))))). % mk_disjoint_insert
thf(fact_75_insert__commute, axiom,
    ((![X3 : hoare_958474565_state, Y2 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((insert776267541_state @ X3 @ (insert776267541_state @ Y2 @ A2)) = (insert776267541_state @ Y2 @ (insert776267541_state @ X3 @ A2)))))). % insert_commute
thf(fact_76_insert__eq__iff, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state, B : hoare_958474565_state, B4 : set_Ho840737317_state]: ((~ ((member109514606_state @ A @ A2))) => ((~ ((member109514606_state @ B @ B4))) => (((insert776267541_state @ A @ A2) = (insert776267541_state @ B @ B4)) = (((((A = B)) => ((A2 = B4)))) & ((((~ ((A = B)))) => ((?[C3 : set_Ho840737317_state]: (((A2 = (insert776267541_state @ B @ C3))) & ((((~ ((member109514606_state @ B @ C3)))) & ((((B4 = (insert776267541_state @ A @ C3))) & ((~ ((member109514606_state @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_77_insert__eq__iff, axiom,
    ((![A : pname, A2 : set_pname, B : pname, B4 : set_pname]: ((~ ((member_pname @ A @ A2))) => ((~ ((member_pname @ B @ B4))) => (((insert_pname @ A @ A2) = (insert_pname @ B @ B4)) = (((((A = B)) => ((A2 = B4)))) & ((((~ ((A = B)))) => ((?[C3 : set_pname]: (((A2 = (insert_pname @ B @ C3))) & ((((~ ((member_pname @ B @ C3)))) & ((((B4 = (insert_pname @ A @ C3))) & ((~ ((member_pname @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_78_insert__absorb, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ A2) => ((insert776267541_state @ A @ A2) = A2))))). % insert_absorb
thf(fact_79_insert__absorb, axiom,
    ((![A : pname, A2 : set_pname]: ((member_pname @ A @ A2) => ((insert_pname @ A @ A2) = A2))))). % insert_absorb
thf(fact_80_insert__ident, axiom,
    ((![X3 : hoare_958474565_state, A2 : set_Ho840737317_state, B4 : set_Ho840737317_state]: ((~ ((member109514606_state @ X3 @ A2))) => ((~ ((member109514606_state @ X3 @ B4))) => (((insert776267541_state @ X3 @ A2) = (insert776267541_state @ X3 @ B4)) = (A2 = B4))))))). % insert_ident
thf(fact_81_insert__ident, axiom,
    ((![X3 : pname, A2 : set_pname, B4 : set_pname]: ((~ ((member_pname @ X3 @ A2))) => ((~ ((member_pname @ X3 @ B4))) => (((insert_pname @ X3 @ A2) = (insert_pname @ X3 @ B4)) = (A2 = B4))))))). % insert_ident
thf(fact_82_Set_Oset__insert, axiom,
    ((![X3 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ X3 @ A2) => (~ ((![B5 : set_Ho840737317_state]: ((A2 = (insert776267541_state @ X3 @ B5)) => (member109514606_state @ X3 @ B5))))))))). % Set.set_insert
thf(fact_83_Set_Oset__insert, axiom,
    ((![X3 : pname, A2 : set_pname]: ((member_pname @ X3 @ A2) => (~ ((![B5 : set_pname]: ((A2 = (insert_pname @ X3 @ B5)) => (member_pname @ X3 @ B5))))))))). % Set.set_insert
thf(fact_84_insertI2, axiom,
    ((![A : hoare_958474565_state, B4 : set_Ho840737317_state, B : hoare_958474565_state]: ((member109514606_state @ A @ B4) => (member109514606_state @ A @ (insert776267541_state @ B @ B4)))))). % insertI2
thf(fact_85_insertI2, axiom,
    ((![A : pname, B4 : set_pname, B : pname]: ((member_pname @ A @ B4) => (member_pname @ A @ (insert_pname @ B @ B4)))))). % insertI2
thf(fact_86_insertI1, axiom,
    ((![A : hoare_958474565_state, B4 : set_Ho840737317_state]: (member109514606_state @ A @ (insert776267541_state @ A @ B4))))). % insertI1
thf(fact_87_insertI1, axiom,
    ((![A : pname, B4 : set_pname]: (member_pname @ A @ (insert_pname @ A @ B4))))). % insertI1
thf(fact_88_insertE, axiom,
    ((![A : hoare_958474565_state, B : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ (insert776267541_state @ B @ A2)) => ((~ ((A = B))) => (member109514606_state @ A @ A2)))))). % insertE
thf(fact_89_insertE, axiom,
    ((![A : pname, B : pname, A2 : set_pname]: ((member_pname @ A @ (insert_pname @ B @ A2)) => ((~ ((A = B))) => (member_pname @ A @ A2)))))). % insertE
thf(fact_90_infinite__imp__nonempty, axiom,
    ((![S4 : set_pname]: ((~ ((finite_finite_pname @ S4))) => (~ ((S4 = bot_bot_set_pname))))))). % infinite_imp_nonempty
thf(fact_91_infinite__imp__nonempty, axiom,
    ((![S4 : set_Ho840737317_state]: ((~ ((finite1986656878_state @ S4))) => (~ ((S4 = bot_bo105666705_state))))))). % infinite_imp_nonempty
thf(fact_92_finite_OemptyI, axiom,
    ((finite_finite_pname @ bot_bot_set_pname))). % finite.emptyI
thf(fact_93_finite_OemptyI, axiom,
    ((finite1986656878_state @ bot_bo105666705_state))). % finite.emptyI
thf(fact_94_singleton__inject, axiom,
    ((![A : hoare_958474565_state, B : hoare_958474565_state]: (((insert776267541_state @ A @ bot_bo105666705_state) = (insert776267541_state @ B @ bot_bo105666705_state)) => (A = B))))). % singleton_inject
thf(fact_95_insert__not__empty, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: (~ (((insert776267541_state @ A @ A2) = bot_bo105666705_state)))))). % insert_not_empty
thf(fact_96_doubleton__eq__iff, axiom,
    ((![A : hoare_958474565_state, B : hoare_958474565_state, C : hoare_958474565_state, D : hoare_958474565_state]: (((insert776267541_state @ A @ (insert776267541_state @ B @ bot_bo105666705_state)) = (insert776267541_state @ C @ (insert776267541_state @ D @ bot_bo105666705_state))) = (((((A = C)) & ((B = D)))) | ((((A = D)) & ((B = C))))))))). % doubleton_eq_iff
thf(fact_97_singleton__iff, axiom,
    ((![B : pname, A : pname]: ((member_pname @ B @ (insert_pname @ A @ bot_bot_set_pname)) = (B = A))))). % singleton_iff
thf(fact_98_singleton__iff, axiom,
    ((![B : hoare_958474565_state, A : hoare_958474565_state]: ((member109514606_state @ B @ (insert776267541_state @ A @ bot_bo105666705_state)) = (B = A))))). % singleton_iff
thf(fact_99_singletonD, axiom,
    ((![B : pname, A : pname]: ((member_pname @ B @ (insert_pname @ A @ bot_bot_set_pname)) => (B = A))))). % singletonD
thf(fact_100_singletonD, axiom,
    ((![B : hoare_958474565_state, A : hoare_958474565_state]: ((member109514606_state @ B @ (insert776267541_state @ A @ bot_bo105666705_state)) => (B = A))))). % singletonD
thf(fact_101_finite_OinsertI, axiom,
    ((![A2 : set_Ho840737317_state, A : hoare_958474565_state]: ((finite1986656878_state @ A2) => (finite1986656878_state @ (insert776267541_state @ A @ A2)))))). % finite.insertI
thf(fact_102_finite_OinsertI, axiom,
    ((![A2 : set_pname, A : pname]: ((finite_finite_pname @ A2) => (finite_finite_pname @ (insert_pname @ A @ A2)))))). % finite.insertI
thf(fact_103_combine__options__cases, axiom,
    ((![X3 : option_com, P : option_com > option_com > $o, Y2 : option_com]: (((X3 = none_com) => (P @ X3 @ Y2)) => (((Y2 = none_com) => (P @ X3 @ Y2)) => ((![A4 : com, B2 : com]: ((X3 = (some_com @ A4)) => ((Y2 = (some_com @ B2)) => (P @ X3 @ Y2)))) => (P @ X3 @ Y2))))))). % combine_options_cases
thf(fact_104_split__option__all, axiom,
    (((^[P2 : option_com > $o]: (![X4 : option_com]: (P2 @ X4))) = (^[P3 : option_com > $o]: (((P3 @ none_com)) & ((![X2 : com]: (P3 @ (some_com @ X2))))))))). % split_option_all
thf(fact_105_split__option__ex, axiom,
    (((^[P2 : option_com > $o]: (?[X4 : option_com]: (P2 @ X4))) = (^[P3 : option_com > $o]: (((P3 @ none_com)) | ((?[X2 : com]: (P3 @ (some_com @ X2))))))))). % split_option_ex
thf(fact_106_option_Oinducts, axiom,
    ((![P : option_com > $o, Option : option_com]: ((P @ none_com) => ((![X : com]: (P @ (some_com @ X))) => (P @ Option)))))). % option.inducts
thf(fact_107_option_Oexhaust, axiom,
    ((![Y2 : option_com]: ((~ ((Y2 = none_com))) => (~ ((![X23 : com]: (~ ((Y2 = (some_com @ X23))))))))))). % option.exhaust
thf(fact_108_option_OdiscI, axiom,
    ((![Option : option_com, X22 : com]: ((Option = (some_com @ X22)) => (~ ((Option = none_com))))))). % option.discI
thf(fact_109_option_Odistinct_I1_J, axiom,
    ((![X22 : com]: (~ ((none_com = (some_com @ X22))))))). % option.distinct(1)
thf(fact_110_option_Oexpand, axiom,
    ((![Option : option_com, Option2 : option_com]: (((Option = none_com) = (Option2 = none_com)) => (((~ ((Option = none_com))) => ((~ ((Option2 = none_com))) => ((the_com @ Option) = (the_com @ Option2)))) => (Option = Option2)))))). % option.expand
thf(fact_111_option_Osel, axiom,
    ((![X22 : com]: ((the_com @ (some_com @ X22)) = X22)))). % option.sel
thf(fact_112_infinite__finite__induct, axiom,
    ((![P : set_pname > $o, A2 : set_pname]: ((![A3 : set_pname]: ((~ ((finite_finite_pname @ A3))) => (P @ A3))) => ((P @ bot_bot_set_pname) => ((![X : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((~ ((member_pname @ X @ F3))) => ((P @ F3) => (P @ (insert_pname @ X @ F3)))))) => (P @ A2))))))). % infinite_finite_induct
thf(fact_113_infinite__finite__induct, axiom,
    ((![P : set_Ho840737317_state > $o, A2 : set_Ho840737317_state]: ((![A3 : set_Ho840737317_state]: ((~ ((finite1986656878_state @ A3))) => (P @ A3))) => ((P @ bot_bo105666705_state) => ((![X : hoare_958474565_state, F3 : set_Ho840737317_state]: ((finite1986656878_state @ F3) => ((~ ((member109514606_state @ X @ F3))) => ((P @ F3) => (P @ (insert776267541_state @ X @ F3)))))) => (P @ A2))))))). % infinite_finite_induct
thf(fact_114_finite__ne__induct, axiom,
    ((![F2 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ F2) => ((~ ((F2 = bot_bot_set_pname))) => ((![X : pname]: (P @ (insert_pname @ X @ bot_bot_set_pname))) => ((![X : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((~ ((F3 = bot_bot_set_pname))) => ((~ ((member_pname @ X @ F3))) => ((P @ F3) => (P @ (insert_pname @ X @ F3))))))) => (P @ F2)))))))). % finite_ne_induct
thf(fact_115_finite__ne__induct, axiom,
    ((![F2 : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ F2) => ((~ ((F2 = bot_bo105666705_state))) => ((![X : hoare_958474565_state]: (P @ (insert776267541_state @ X @ bot_bo105666705_state))) => ((![X : hoare_958474565_state, F3 : set_Ho840737317_state]: ((finite1986656878_state @ F3) => ((~ ((F3 = bot_bo105666705_state))) => ((~ ((member109514606_state @ X @ F3))) => ((P @ F3) => (P @ (insert776267541_state @ X @ F3))))))) => (P @ F2)))))))). % finite_ne_induct
thf(fact_116_the__elem__eq, axiom,
    ((![X3 : hoare_958474565_state]: ((the_el1300254266_state @ (insert776267541_state @ X3 @ bot_bo105666705_state)) = X3)))). % the_elem_eq
thf(fact_117_is__singletonI, axiom,
    ((![X3 : hoare_958474565_state]: (is_sin536631353_state @ (insert776267541_state @ X3 @ bot_bo105666705_state))))). % is_singletonI
thf(fact_118_BodyN, axiom,
    ((![P : state > state > $o, Pn : pname, Q : state > state > $o, G : set_Ho840737317_state]: ((hoare_604442164_state @ (insert776267541_state @ (hoare_1659279548_state @ P @ (body2 @ Pn) @ Q) @ G) @ (insert776267541_state @ (hoare_1659279548_state @ P @ (the_com @ (body @ Pn)) @ Q) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ (body2 @ Pn) @ Q) @ bot_bo105666705_state)))))). % BodyN
thf(fact_119_weak__Body, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o, Pn : pname, Q : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ (the_com @ (body @ Pn)) @ Q) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ (body2 @ Pn) @ Q) @ bot_bo105666705_state)))))). % weak_Body
thf(fact_120_these__insert__Some, axiom,
    ((![X3 : hoare_958474565_state, A2 : set_op1790085995_state]: ((these_1282564994_state @ (insert677809883_state @ (some_H1327573290_state @ X3) @ A2)) = (insert776267541_state @ X3 @ (these_1282564994_state @ A2)))))). % these_insert_Some
thf(fact_121_these__insert__Some, axiom,
    ((![X3 : com, A2 : set_option_com]: ((these_com @ (insert_option_com @ (some_com @ X3) @ A2)) = (insert_com @ X3 @ (these_com @ A2)))))). % these_insert_Some
thf(fact_122_triple_Oinject, axiom,
    ((![X1 : state > state > $o, X22 : com, X32 : state > state > $o, Y1 : state > state > $o, Y22 : com, Y32 : state > state > $o]: (((hoare_1659279548_state @ X1 @ X22 @ X32) = (hoare_1659279548_state @ Y1 @ Y22 @ Y32)) = (((X1 = Y1)) & ((((X22 = Y22)) & ((X32 = Y32))))))))). % triple.inject
thf(fact_123_these__empty, axiom,
    (((these_1282564994_state @ bot_bo472264663_state) = bot_bo105666705_state))). % these_empty
thf(fact_124_these__insert__None, axiom,
    ((![A2 : set_option_com]: ((these_com @ (insert_option_com @ none_com @ A2)) = (these_com @ A2))))). % these_insert_None
thf(fact_125_triple_Oinduct, axiom,
    ((![P : hoare_958474565_state > $o, Triple : hoare_958474565_state]: ((![X1a : state > state > $o, X2a : com, X3a : state > state > $o]: (P @ (hoare_1659279548_state @ X1a @ X2a @ X3a))) => (P @ Triple))))). % triple.induct
thf(fact_126_triple_Oexhaust, axiom,
    ((![Y2 : hoare_958474565_state]: (~ ((![X12 : state > state > $o, X23 : com, X33 : state > state > $o]: (~ ((Y2 = (hoare_1659279548_state @ X12 @ X23 @ X33)))))))))). % triple.exhaust
thf(fact_127_bot__set__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ bot_bo1428770700tate_o)))). % bot_set_def
thf(fact_128_in__these__eq, axiom,
    ((![X3 : pname, A2 : set_option_pname]: ((member_pname @ X3 @ (these_pname @ A2)) = (member_option_pname @ (some_pname @ X3) @ A2))))). % in_these_eq
thf(fact_129_in__these__eq, axiom,
    ((![X3 : com, A2 : set_option_com]: ((member_com @ X3 @ (these_com @ A2)) = (member_option_com @ (some_com @ X3) @ A2))))). % in_these_eq
thf(fact_130_is__singleton__the__elem, axiom,
    ((is_sin536631353_state = (^[A6 : set_Ho840737317_state]: (A6 = (insert776267541_state @ (the_el1300254266_state @ A6) @ bot_bo105666705_state)))))). % is_singleton_the_elem
thf(fact_131_is__singletonI_H, axiom,
    ((![A2 : set_pname]: ((~ ((A2 = bot_bot_set_pname))) => ((![X : pname, Y : pname]: ((member_pname @ X @ A2) => ((member_pname @ Y @ A2) => (X = Y)))) => (is_singleton_pname @ A2)))))). % is_singletonI'
thf(fact_132_is__singletonI_H, axiom,
    ((![A2 : set_Ho840737317_state]: ((~ ((A2 = bot_bo105666705_state))) => ((![X : hoare_958474565_state, Y : hoare_958474565_state]: ((member109514606_state @ X @ A2) => ((member109514606_state @ Y @ A2) => (X = Y)))) => (is_sin536631353_state @ A2)))))). % is_singletonI'
thf(fact_133_conseq1, axiom,
    ((![G : set_Ho840737317_state, P4 : state > state > $o, C : com, Q : state > state > $o, P : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P4 @ C @ Q) @ bot_bo105666705_state)) => ((![Z : state, S : state]: ((P @ Z @ S) => (P4 @ Z @ S))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))))))). % conseq1
thf(fact_134_conseq2, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o, C : com, Q2 : 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]: ((Q2 @ Z @ S) => (Q @ Z @ S))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))))))). % conseq2
thf(fact_135_conseq12, axiom,
    ((![G : set_Ho840737317_state, P4 : state > state > $o, C : com, Q2 : state > state > $o, P : state > state > $o, Q : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P4 @ C @ Q2) @ bot_bo105666705_state)) => ((![Z : state, S : state]: ((P @ Z @ S) => (![S5 : state]: ((![Z2 : state]: ((P4 @ Z2 @ S) => (Q2 @ Z2 @ S5))) => (Q @ Z @ S5))))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))))))). % conseq12
thf(fact_136_conseq, axiom,
    ((![P : state > state > $o, G : set_Ho840737317_state, C : com, Q : state > state > $o]: ((![Z : state, S : state]: ((P @ Z @ S) => (?[P5 : state > state > $o, Q3 : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P5 @ C @ Q3) @ bot_bo105666705_state)) & (![S5 : state]: ((![Z2 : state]: ((P5 @ Z2 @ S) => (Q3 @ Z2 @ S5))) => (Q @ Z @ S5))))))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state)))))). % conseq
thf(fact_137_these__empty__eq, axiom,
    ((![B4 : set_option_com]: (((these_com @ B4) = bot_bot_set_com) = (((B4 = bot_bo697186986on_com)) | ((B4 = (insert_option_com @ none_com @ bot_bo697186986on_com)))))))). % these_empty_eq
thf(fact_138_these__empty__eq, axiom,
    ((![B4 : set_op1790085995_state]: (((these_1282564994_state @ B4) = bot_bo105666705_state) = (((B4 = bot_bo472264663_state)) | ((B4 = (insert677809883_state @ none_H2001963430_state @ bot_bo472264663_state)))))))). % these_empty_eq
thf(fact_139_these__not__empty__eq, axiom,
    ((![B4 : set_option_com]: ((~ (((these_com @ B4) = bot_bot_set_com))) = (((~ ((B4 = bot_bo697186986on_com)))) & ((~ ((B4 = (insert_option_com @ none_com @ bot_bo697186986on_com)))))))))). % these_not_empty_eq
thf(fact_140_these__not__empty__eq, axiom,
    ((![B4 : set_op1790085995_state]: ((~ (((these_1282564994_state @ B4) = bot_bo105666705_state))) = (((~ ((B4 = bot_bo472264663_state)))) & ((~ ((B4 = (insert677809883_state @ none_H2001963430_state @ bot_bo472264663_state)))))))))). % these_not_empty_eq
thf(fact_141_Comp, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o, C : com, Q : state > state > $o, D : com, R : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ 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 @ P @ (semi @ C @ D) @ R) @ bot_bo105666705_state))))))). % Comp
thf(fact_142_hoare__derivs_OSkip, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o]: (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ skip @ P) @ bot_bo105666705_state))))). % hoare_derivs.Skip
thf(fact_143_is__singletonE, axiom,
    ((![A2 : set_Ho840737317_state]: ((is_sin536631353_state @ A2) => (~ ((![X : hoare_958474565_state]: (~ ((A2 = (insert776267541_state @ X @ bot_bo105666705_state))))))))))). % is_singletonE
thf(fact_144_is__singleton__def, axiom,
    ((is_sin536631353_state = (^[A6 : set_Ho840737317_state]: (?[X2 : hoare_958474565_state]: (A6 = (insert776267541_state @ X2 @ bot_bo105666705_state))))))). % is_singleton_def
thf(fact_145_MGF__complete, axiom,
    ((![C : com, P : 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 @ P @ C @ Q) @ bot_bo105666705_state)) => (hoare_604442164_state @ bot_bo105666705_state @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))))))). % MGF_complete
thf(fact_146_Set_Ois__empty__def, axiom,
    ((is_emp266830871_state = (^[A6 : set_Ho840737317_state]: (A6 = bot_bo105666705_state))))). % Set.is_empty_def
thf(fact_147_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_148_Collect__empty__eq__bot, axiom,
    ((![P : hoare_958474565_state > $o]: (((collec305460656_state @ P) = bot_bo105666705_state) = (P = bot_bo1428770700tate_o))))). % Collect_empty_eq_bot
thf(fact_149_bot__empty__eq, axiom,
    ((bot_bot_pname_o = (^[X2 : pname]: (member_pname @ X2 @ bot_bot_set_pname))))). % bot_empty_eq
thf(fact_150_bot__empty__eq, axiom,
    ((bot_bo1428770700tate_o = (^[X2 : hoare_958474565_state]: (member109514606_state @ X2 @ bot_bo105666705_state))))). % bot_empty_eq
thf(fact_151_option_Osimps_I15_J, axiom,
    ((![X22 : com]: ((set_option_com2 @ (some_com @ X22)) = (insert_com @ X22 @ bot_bot_set_com))))). % option.simps(15)
thf(fact_152_option_Osimps_I15_J, axiom,
    ((![X22 : hoare_958474565_state]: ((set_op1542142163_state @ (some_H1327573290_state @ X22)) = (insert776267541_state @ X22 @ bot_bo105666705_state))))). % option.simps(15)
thf(fact_153_finite__map__freshness, axiom,
    ((![F : pname > option_com]: ((finite_finite_pname @ (dom_pname_com @ F)) => ((~ ((finite_finite_pname @ top_top_set_pname))) => (?[X : pname]: ((F @ X) = none_com))))))). % finite_map_freshness
thf(fact_154_UNIV__I, axiom,
    ((![X3 : pname]: (member_pname @ X3 @ top_top_set_pname)))). % UNIV_I
thf(fact_155_finite__option__UNIV, axiom,
    (((finite1292484209_pname @ top_to1034942016_pname) = (finite_finite_pname @ top_top_set_pname)))). % finite_option_UNIV
thf(fact_156_finite__Plus__UNIV__iff, axiom,
    (((finite1993869834_pname @ top_to380724057_pname) = (((finite_finite_pname @ top_top_set_pname)) & ((finite_finite_pname @ top_top_set_pname)))))). % finite_Plus_UNIV_iff
thf(fact_157_elem__set, axiom,
    ((![X3 : pname, Xo : option_pname]: ((member_pname @ X3 @ (set_option_pname2 @ Xo)) = (Xo = (some_pname @ X3)))))). % elem_set
thf(fact_158_elem__set, axiom,
    ((![X3 : com, Xo : option_com]: ((member_com @ X3 @ (set_option_com2 @ Xo)) = (Xo = (some_com @ X3)))))). % elem_set
thf(fact_159_set__empty__eq, axiom,
    ((![Xo : option_com]: (((set_option_com2 @ Xo) = bot_bot_set_com) = (Xo = none_com))))). % set_empty_eq
thf(fact_160_set__empty__eq, axiom,
    ((![Xo : option1481361675_state]: (((set_op1542142163_state @ Xo) = bot_bo105666705_state) = (Xo = none_H2001963430_state))))). % set_empty_eq
thf(fact_161_option_Oset__sel, axiom,
    ((![A : option_pname]: ((~ ((A = none_pname))) => (member_pname @ (the_pname @ A) @ (set_option_pname2 @ A)))))). % option.set_sel
thf(fact_162_option_Oset__sel, axiom,
    ((![A : option_com]: ((~ ((A = none_com))) => (member_com @ (the_com @ A) @ (set_option_com2 @ A)))))). % option.set_sel
thf(fact_163_empty__not__UNIV, axiom,
    ((~ ((bot_bo105666705_state = top_to519946101_state))))). % empty_not_UNIV
thf(fact_164_Finite__Set_Ofinite__set, axiom,
    (((finite505202775_pname @ top_to1806476070_pname) = (finite_finite_pname @ top_top_set_pname)))). % Finite_Set.finite_set
thf(fact_165_finite__prod, axiom,
    (((finite1490780830_pname @ top_to2104644005_pname) = (((finite_finite_pname @ top_top_set_pname)) & ((finite_finite_pname @ top_top_set_pname)))))). % finite_prod
thf(fact_166_finite__Prod__UNIV, axiom,
    (((finite_finite_pname @ top_top_set_pname) => ((finite_finite_pname @ top_top_set_pname) => (finite1490780830_pname @ top_to2104644005_pname))))). % finite_Prod_UNIV

% Conjectures (5)
thf(conj_0, hypothesis,
    (hoare_405891322gleton)).
thf(conj_1, hypothesis,
    (wT_bodies)).
thf(conj_2, hypothesis,
    ((wt @ c))).
thf(conj_3, hypothesis,
    ((member_pname @ pn @ (dom_pname_com @ body)))).
thf(conj_4, conjecture,
    ((wt @ (the_com @ (body @ pn))))).
