% 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_346__3253458_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:13:57.202

% 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__Set__Oset_I_062_It__Com__Opname_Mt__Com__Opname_J_J, type,
    set_pname_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__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 (35)
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_Com_Ocom_OWhile, type,
    while : (state > $o) > com > com).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Com__Opname_Mt__Com__Opname_J, type,
    finite1291246666_pname : set_pname_pname > $o).
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_HOL_OThe_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    the_Ho1908033100_state : (hoare_958474565_state > $o) > hoare_958474565_state).
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_Lattices_Osup__class_Osup_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    sup_su737426425_state : set_Ho840737317_state > set_Ho840737317_state > set_Ho840737317_state).
thf(sy_c_Map_Odom_001t__Com__Opname_001t__Com__Ocom, type,
    dom_pname_com : (pname > option_com) > set_pname).
thf(sy_c_Natural_Oevalc, type,
    evalc : com > state > 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__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_001_062_It__Com__Opname_Mt__Com__Opname_J, type,
    collect_pname_pname : ((pname > pname) > $o) > set_pname_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_Oimage_001t__Com__Opname_001t__Com__Opname, type,
    image_pname_pname : (pname > pname) > set_pname > set_pname).
thf(sy_c_Set_Oimage_001t__Com__Opname_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    image_2144627828_state : (pname > hoare_958474565_state) > set_pname > set_Ho840737317_state).
thf(sy_c_Set_Oimage_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    image_1849792670_state : (hoare_958474565_state > hoare_958474565_state) > set_Ho840737317_state > 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 (124)
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_single__stateE, axiom,
    ((hoare_405891322gleton => (![T : state]: (~ ((![S : state]: (S = T)))))))). % single_stateE
thf(fact_5_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_6_derivs__insertD, axiom,
    ((![G : set_Ho840737317_state, T2 : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G @ (insert776267541_state @ T2 @ Ts)) => ((hoare_604442164_state @ G @ (insert776267541_state @ T2 @ bot_bo105666705_state)) & (hoare_604442164_state @ G @ Ts)))))). % derivs_insertD
thf(fact_7_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_8_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_9_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_10_empty, axiom,
    ((![G : set_Ho840737317_state]: (hoare_604442164_state @ G @ bot_bo105666705_state)))). % empty
thf(fact_11_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_12_hoare__derivs_Oinsert, axiom,
    ((![G : set_Ho840737317_state, T2 : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G @ (insert776267541_state @ T2 @ bot_bo105666705_state)) => ((hoare_604442164_state @ G @ Ts) => (hoare_604442164_state @ G @ (insert776267541_state @ T2 @ Ts))))))). % hoare_derivs.insert
thf(fact_13_state__not__singleton__def, axiom,
    ((hoare_405891322gleton = (?[S3 : state]: (?[T3 : state]: (~ ((S3 = T3)))))))). % state_not_singleton_def
thf(fact_14_singletonI, axiom,
    ((![A : hoare_958474565_state]: (member109514606_state @ A @ (insert776267541_state @ A @ bot_bo105666705_state))))). % singletonI
thf(fact_15_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_16_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_17_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_18_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_19_empty__iff, axiom,
    ((![C : hoare_958474565_state]: (~ ((member109514606_state @ C @ bot_bo105666705_state)))))). % empty_iff
thf(fact_20_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_21_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_22_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_23_evalc__elim__cases_I1_J, axiom,
    ((![S4 : state, T2 : state]: ((evalc @ skip @ S4 @ T2) => (T2 = S4))))). % evalc_elim_cases(1)
thf(fact_24_evalc_OSkip, axiom,
    ((![S4 : state]: (evalc @ skip @ S4 @ S4)))). % evalc.Skip
thf(fact_25_bot__set__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ bot_bo1428770700tate_o)))). % bot_set_def
thf(fact_26_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_27_equals0I, axiom,
    ((![A2 : set_Ho840737317_state]: ((![Y4 : hoare_958474565_state]: (~ ((member109514606_state @ Y4 @ A2)))) => (A2 = bot_bo105666705_state))))). % equals0I
thf(fact_28_equals0D, axiom,
    ((![A2 : set_Ho840737317_state, A : hoare_958474565_state]: ((A2 = bot_bo105666705_state) => (~ ((member109514606_state @ A @ A2))))))). % equals0D
thf(fact_29_emptyE, axiom,
    ((![A : hoare_958474565_state]: (~ ((member109514606_state @ A @ bot_bo105666705_state)))))). % emptyE
thf(fact_30_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_31_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_32_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_33_insert__absorb, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ A2) => ((insert776267541_state @ A @ A2) = A2))))). % insert_absorb
thf(fact_34_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_35_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_36_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_37_insertI1, axiom,
    ((![A : hoare_958474565_state, B : set_Ho840737317_state]: (member109514606_state @ A @ (insert776267541_state @ A @ B))))). % insertI1
thf(fact_38_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_39_com__det, axiom,
    ((![C : com, S4 : state, T2 : state, U : state]: ((evalc @ C @ S4 @ T2) => ((evalc @ C @ S4 @ U) => (U = T2)))))). % com_det
thf(fact_40_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_41_insert__not__empty, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: (~ (((insert776267541_state @ A @ A2) = bot_bo105666705_state)))))). % insert_not_empty
thf(fact_42_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_43_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_44_singletonD, axiom,
    ((![B2 : hoare_958474565_state, A : hoare_958474565_state]: ((member109514606_state @ B2 @ (insert776267541_state @ A @ bot_bo105666705_state)) => (B2 = A))))). % singletonD
thf(fact_45_com_Odistinct_I11_J, axiom,
    ((![X7 : pname]: (~ ((skip = (body2 @ X7))))))). % com.distinct(11)
thf(fact_46_the__elem__eq, axiom,
    ((![X : hoare_958474565_state]: ((the_el1300254266_state @ (insert776267541_state @ X @ bot_bo105666705_state)) = X)))). % the_elem_eq
thf(fact_47_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_48_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_49_is__singletonI, axiom,
    ((![X : hoare_958474565_state]: (is_sin536631353_state @ (insert776267541_state @ X @ bot_bo105666705_state))))). % is_singletonI
thf(fact_50_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_51_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_52_Set_Ois__empty__def, axiom,
    ((is_emp266830871_state = (^[A3 : set_Ho840737317_state]: (A3 = bot_bo105666705_state))))). % Set.is_empty_def
thf(fact_53_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_54_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_55_com_Odistinct_I41_J, axiom,
    ((![X41 : com, X42 : com, X7 : pname]: (~ (((semi @ X41 @ X42) = (body2 @ X7))))))). % com.distinct(41)
thf(fact_56_evalc__elim__cases_I4_J, axiom,
    ((![C1 : com, C22 : com, S4 : state, T2 : state]: ((evalc @ (semi @ C1 @ C22) @ S4 @ T2) => (~ ((![S1 : state]: ((evalc @ C1 @ S4 @ S1) => (~ ((evalc @ C22 @ S1 @ T2))))))))))). % evalc_elim_cases(4)
thf(fact_57_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_58_com_Odistinct_I5_J, axiom,
    ((![X41 : com, X42 : com]: (~ ((skip = (semi @ X41 @ X42))))))). % com.distinct(5)
thf(fact_59_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_60_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_61_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_62_evalc__elim__cases_I6_J, axiom,
    ((![P2 : pname, S4 : state, S12 : state]: ((evalc @ (body2 @ P2) @ S4 @ S12) => (evalc @ (the_com @ (body @ P2)) @ S4 @ S12))))). % evalc_elim_cases(6)
thf(fact_63_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_64_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_65_bot__empty__eq, axiom,
    ((bot_bo1428770700tate_o = (^[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ bot_bo105666705_state))))). % bot_empty_eq
thf(fact_66_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_67_finite__dom__body, axiom,
    ((finite_finite_pname @ (dom_pname_com @ body)))). % finite_dom_body
thf(fact_68_MGT__alternD, axiom,
    ((![G : set_Ho840737317_state, C : com]: (hoare_405891322gleton => ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ (^[Z3 : state]: (^[S02 : state]: (![S13 : state]: (((evalc @ C @ S02 @ S13)) => ((Z3 = S13)))))) @ C @ (^[Y5 : state]: (^[Z4 : state]: (Y5 = Z4)))) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_Mirabelle_MGT @ C) @ bot_bo105666705_state))))))). % MGT_alternD
thf(fact_69_singleton__conv, axiom,
    ((![A : hoare_958474565_state]: ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (X4 = A))) = (insert776267541_state @ A @ bot_bo105666705_state))))). % singleton_conv
thf(fact_70_singleton__conv2, axiom,
    ((![A : hoare_958474565_state]: ((collec305460656_state @ ((^[Y5 : hoare_958474565_state]: (^[Z4 : hoare_958474565_state]: (Y5 = Z4))) @ A)) = (insert776267541_state @ A @ bot_bo105666705_state))))). % singleton_conv2
thf(fact_71_empty__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ (^[X4 : hoare_958474565_state]: $false))))). % empty_def
thf(fact_72_Collect__conv__if, axiom,
    ((![P2 : hoare_958474565_state > $o, A : hoare_958474565_state]: (((P2 @ A) => ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (((X4 = A)) & ((P2 @ X4))))) = (insert776267541_state @ A @ bot_bo105666705_state))) & ((~ ((P2 @ A))) => ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (((X4 = A)) & ((P2 @ X4))))) = bot_bo105666705_state)))))). % Collect_conv_if
thf(fact_73_Collect__conv__if2, axiom,
    ((![P2 : hoare_958474565_state > $o, A : hoare_958474565_state]: (((P2 @ A) => ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (((A = X4)) & ((P2 @ X4))))) = (insert776267541_state @ A @ bot_bo105666705_state))) & ((~ ((P2 @ A))) => ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (((A = X4)) & ((P2 @ X4))))) = bot_bo105666705_state)))))). % Collect_conv_if2
thf(fact_74_insert__Collect, axiom,
    ((![A : hoare_958474565_state, P2 : hoare_958474565_state > $o]: ((insert776267541_state @ A @ (collec305460656_state @ P2)) = (collec305460656_state @ (^[U2 : hoare_958474565_state]: (((~ ((U2 = A)))) => ((P2 @ U2))))))))). % insert_Collect
thf(fact_75_insert__compr, axiom,
    ((insert776267541_state = (^[A4 : hoare_958474565_state]: (^[B4 : set_Ho840737317_state]: (collec305460656_state @ (^[X4 : hoare_958474565_state]: (((X4 = A4)) | ((member109514606_state @ X4 @ B4)))))))))). % insert_compr
thf(fact_76_MGT__def, axiom,
    ((hoare_Mirabelle_MGT = (^[C3 : com]: (hoare_1659279548_state @ (^[Y5 : state]: (^[Z4 : state]: (Y5 = Z4))) @ C3 @ (evalc @ C3)))))). % MGT_def
thf(fact_77__Cconstant_C, axiom,
    ((![C4 : $o, G : set_Ho840737317_state, P2 : state > state > $o, C : com, Q : state > state > $o]: ((C4 => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ C @ Q) @ bot_bo105666705_state))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ (^[Z3 : state]: (^[S3 : state]: (((P2 @ Z3 @ S3)) & (C4)))) @ C @ Q) @ bot_bo105666705_state)))))). % "constant"
thf(fact_78_escape, axiom,
    ((![P2 : state > state > $o, G : set_Ho840737317_state, C : com, Q : state > state > $o]: ((![Z : state, S : state]: ((P2 @ Z @ S) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ (^[Za : state]: (^[S5 : state]: (S5 = S))) @ C @ (^[Z5 : state]: (Q @ Z))) @ bot_bo105666705_state)))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ C @ Q) @ bot_bo105666705_state)))))). % escape
thf(fact_79_MGT__alternI, axiom,
    ((![G : set_Ho840737317_state, C : com]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_Mirabelle_MGT @ C) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ (^[Z3 : state]: (^[S02 : state]: (![S13 : state]: (((evalc @ C @ S02 @ S13)) => ((Z3 = S13)))))) @ C @ (^[Y5 : state]: (^[Z4 : state]: (Y5 = Z4)))) @ bot_bo105666705_state)))))). % MGT_alternI
thf(fact_80_finite__insert, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((finite1986656878_state @ (insert776267541_state @ A @ A2)) = (finite1986656878_state @ A2))))). % finite_insert
thf(fact_81_finite__insert, axiom,
    ((![A : pname, A2 : set_pname]: ((finite_finite_pname @ (insert_pname @ A @ A2)) = (finite_finite_pname @ A2))))). % finite_insert
thf(fact_82_finite__Collect__conjI, axiom,
    ((![P2 : pname > $o, Q : pname > $o]: (((finite_finite_pname @ (collect_pname @ P2)) | (finite_finite_pname @ (collect_pname @ Q))) => (finite_finite_pname @ (collect_pname @ (^[X4 : pname]: (((P2 @ X4)) & ((Q @ X4)))))))))). % finite_Collect_conjI
thf(fact_83_finite__Collect__disjI, axiom,
    ((![P2 : pname > $o, Q : pname > $o]: ((finite_finite_pname @ (collect_pname @ (^[X4 : pname]: (((P2 @ X4)) | ((Q @ X4)))))) = (((finite_finite_pname @ (collect_pname @ P2))) & ((finite_finite_pname @ (collect_pname @ Q)))))))). % finite_Collect_disjI
thf(fact_84_infinite__finite__induct, axiom,
    ((![P2 : set_pname > $o, A2 : set_pname]: ((![A5 : set_pname]: ((~ ((finite_finite_pname @ A5))) => (P2 @ A5))) => ((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_85_infinite__finite__induct, axiom,
    ((![P2 : set_Ho840737317_state > $o, A2 : set_Ho840737317_state]: ((![A5 : set_Ho840737317_state]: ((~ ((finite1986656878_state @ A5))) => (P2 @ A5))) => ((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_86_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_pname, B : set_pname, R : pname > pname > $o]: ((~ ((finite_finite_pname @ A2))) => ((finite_finite_pname @ B) => ((![X5 : pname]: ((member_pname @ X5 @ A2) => (?[Xa : pname]: ((member_pname @ Xa @ B) & (R @ X5 @ Xa))))) => (?[X5 : pname]: ((member_pname @ X5 @ B) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A2)) & ((R @ A4 @ X5)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_87_not__finite__existsD, axiom,
    ((![P2 : pname > $o]: ((~ ((finite_finite_pname @ (collect_pname @ P2)))) => (?[X_1 : pname]: (P2 @ X_1)))))). % not_finite_existsD
thf(fact_88_finite_OemptyI, axiom,
    ((finite_finite_pname @ bot_bot_set_pname))). % finite.emptyI
thf(fact_89_finite_OemptyI, axiom,
    ((finite1986656878_state @ bot_bo105666705_state))). % finite.emptyI
thf(fact_90_infinite__imp__nonempty, axiom,
    ((![S6 : set_pname]: ((~ ((finite_finite_pname @ S6))) => (~ ((S6 = bot_bot_set_pname))))))). % infinite_imp_nonempty
thf(fact_91_infinite__imp__nonempty, axiom,
    ((![S6 : set_Ho840737317_state]: ((~ ((finite1986656878_state @ S6))) => (~ ((S6 = bot_bo105666705_state))))))). % infinite_imp_nonempty
thf(fact_92_finite_OinsertI, axiom,
    ((![A2 : set_Ho840737317_state, A : hoare_958474565_state]: ((finite1986656878_state @ A2) => (finite1986656878_state @ (insert776267541_state @ A @ A2)))))). % finite.insertI
thf(fact_93_finite_OinsertI, axiom,
    ((![A2 : set_pname, A : pname]: ((finite_finite_pname @ A2) => (finite_finite_pname @ (insert_pname @ A @ A2)))))). % finite.insertI
thf(fact_94_finite__set__of__finite__funs, axiom,
    ((![A2 : set_pname, B : set_pname, D : pname]: ((finite_finite_pname @ A2) => ((finite_finite_pname @ B) => (finite1291246666_pname @ (collect_pname_pname @ (^[F2 : pname > pname]: (![X4 : pname]: (((((member_pname @ X4 @ A2)) => ((member_pname @ (F2 @ X4) @ B)))) & ((((~ ((member_pname @ X4 @ A2)))) => (((F2 @ X4) = D)))))))))))))). % finite_set_of_finite_funs
thf(fact_95_finite_Ocases, axiom,
    ((![A : set_pname]: ((finite_finite_pname @ A) => ((~ ((A = bot_bot_set_pname))) => (~ ((![A5 : set_pname]: ((?[A6 : pname]: (A = (insert_pname @ A6 @ A5))) => (~ ((finite_finite_pname @ A5)))))))))))). % finite.cases
thf(fact_96_finite_Ocases, axiom,
    ((![A : set_Ho840737317_state]: ((finite1986656878_state @ A) => ((~ ((A = bot_bo105666705_state))) => (~ ((![A5 : set_Ho840737317_state]: ((?[A6 : hoare_958474565_state]: (A = (insert776267541_state @ A6 @ A5))) => (~ ((finite1986656878_state @ A5)))))))))))). % finite.cases
thf(fact_97_finite_Osimps, axiom,
    ((finite_finite_pname = (^[A4 : set_pname]: (((A4 = bot_bot_set_pname)) | ((?[A3 : set_pname]: (?[B5 : pname]: (((A4 = (insert_pname @ B5 @ A3))) & ((finite_finite_pname @ A3))))))))))). % finite.simps
thf(fact_98_finite_Osimps, axiom,
    ((finite1986656878_state = (^[A4 : set_Ho840737317_state]: (((A4 = bot_bo105666705_state)) | ((?[A3 : set_Ho840737317_state]: (?[B5 : hoare_958474565_state]: (((A4 = (insert776267541_state @ B5 @ A3))) & ((finite1986656878_state @ A3))))))))))). % finite.simps
thf(fact_99_finite__induct, axiom,
    ((![F3 : set_pname, P2 : set_pname > $o]: ((finite_finite_pname @ F3) => ((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 @ F3))))))). % finite_induct
thf(fact_100_finite__induct, axiom,
    ((![F3 : set_Ho840737317_state, P2 : set_Ho840737317_state > $o]: ((finite1986656878_state @ F3) => ((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 @ F3))))))). % finite_induct
thf(fact_101_finite_Oinducts, axiom,
    ((![X : set_pname, P2 : set_pname > $o]: ((finite_finite_pname @ X) => ((P2 @ bot_bot_set_pname) => ((![A5 : set_pname, A6 : pname]: ((finite_finite_pname @ A5) => ((P2 @ A5) => (P2 @ (insert_pname @ A6 @ A5))))) => (P2 @ X))))))). % finite.inducts
thf(fact_102_finite_Oinducts, axiom,
    ((![X : set_Ho840737317_state, P2 : set_Ho840737317_state > $o]: ((finite1986656878_state @ X) => ((P2 @ bot_bo105666705_state) => ((![A5 : set_Ho840737317_state, A6 : hoare_958474565_state]: ((finite1986656878_state @ A5) => ((P2 @ A5) => (P2 @ (insert776267541_state @ A6 @ A5))))) => (P2 @ X))))))). % finite.inducts
thf(fact_103_finite__ne__induct, axiom,
    ((![F3 : set_pname, P2 : set_pname > $o]: ((finite_finite_pname @ F3) => ((~ ((F3 = 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 @ F3)))))))). % finite_ne_induct
thf(fact_104_finite__ne__induct, axiom,
    ((![F3 : set_Ho840737317_state, P2 : set_Ho840737317_state > $o]: ((finite1986656878_state @ F3) => ((~ ((F3 = 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 @ F3)))))))). % finite_ne_induct
thf(fact_105_LoopF, axiom,
    ((![G : set_Ho840737317_state, P2 : state > state > $o, B2 : state > $o, C : com]: (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ (^[Z3 : state]: (^[S3 : state]: (((P2 @ Z3 @ S3)) & ((~ ((B2 @ S3))))))) @ (while @ B2 @ C) @ P2) @ bot_bo105666705_state))))). % LoopF
thf(fact_106_finite__pointwise, axiom,
    ((![U3 : set_pname, G : set_Ho840737317_state, P : pname > state > state > $o, C0 : pname > com, Q2 : pname > state > state > $o, P2 : pname > state > state > $o, Q : pname > state > state > $o]: ((finite_finite_pname @ U3) => ((![P4 : pname]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ (P @ P4) @ (C0 @ P4) @ (Q2 @ P4)) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ (P2 @ P4) @ (C0 @ P4) @ (Q @ P4)) @ bot_bo105666705_state)))) => ((hoare_604442164_state @ G @ (image_2144627828_state @ (^[P5 : pname]: (hoare_1659279548_state @ (P @ P5) @ (C0 @ P5) @ (Q2 @ P5))) @ U3)) => (hoare_604442164_state @ G @ (image_2144627828_state @ (^[P5 : pname]: (hoare_1659279548_state @ (P2 @ P5) @ (C0 @ P5) @ (Q @ P5))) @ U3)))))))). % finite_pointwise
thf(fact_107_the__elem__def, axiom,
    ((the_el1300254266_state = (^[X6 : set_Ho840737317_state]: (the_Ho1908033100_state @ (^[X4 : hoare_958474565_state]: (X6 = (insert776267541_state @ X4 @ bot_bo105666705_state)))))))). % the_elem_def
thf(fact_108_com_Oinject_I5_J, axiom,
    ((![X61 : state > $o, X62 : com, Y61 : state > $o, Y62 : com]: (((while @ X61 @ X62) = (while @ Y61 @ Y62)) = (((X61 = Y61)) & ((X62 = Y62))))))). % com.inject(5)
thf(fact_109_image__empty, axiom,
    ((![F4 : hoare_958474565_state > hoare_958474565_state]: ((image_1849792670_state @ F4 @ bot_bo105666705_state) = bot_bo105666705_state)))). % image_empty
thf(fact_110_empty__is__image, axiom,
    ((![F4 : hoare_958474565_state > hoare_958474565_state, A2 : set_Ho840737317_state]: ((bot_bo105666705_state = (image_1849792670_state @ F4 @ A2)) = (A2 = bot_bo105666705_state))))). % empty_is_image
thf(fact_111_image__is__empty, axiom,
    ((![F4 : hoare_958474565_state > hoare_958474565_state, A2 : set_Ho840737317_state]: (((image_1849792670_state @ F4 @ A2) = bot_bo105666705_state) = (A2 = bot_bo105666705_state))))). % image_is_empty
thf(fact_112_finite__imageI, axiom,
    ((![F3 : set_pname, H : pname > pname]: ((finite_finite_pname @ F3) => (finite_finite_pname @ (image_pname_pname @ H @ F3)))))). % finite_imageI
thf(fact_113_image__insert, axiom,
    ((![F4 : hoare_958474565_state > hoare_958474565_state, A : hoare_958474565_state, B : set_Ho840737317_state]: ((image_1849792670_state @ F4 @ (insert776267541_state @ A @ B)) = (insert776267541_state @ (F4 @ A) @ (image_1849792670_state @ F4 @ B)))))). % image_insert
thf(fact_114_pigeonhole__infinite, axiom,
    ((![A2 : set_pname, F4 : pname > pname]: ((~ ((finite_finite_pname @ A2))) => ((finite_finite_pname @ (image_pname_pname @ F4 @ A2)) => (?[X5 : pname]: ((member_pname @ X5 @ A2) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A2)) & (((F4 @ A4) = (F4 @ X5)))))))))))))))). % pigeonhole_infinite
thf(fact_115_com_Odistinct_I51_J, axiom,
    ((![X61 : state > $o, X62 : com, X7 : pname]: (~ (((while @ X61 @ X62) = (body2 @ X7))))))). % com.distinct(51)
thf(fact_116_evalc_OWhileTrue, axiom,
    ((![B2 : state > $o, S0 : state, C : com, S12 : state, S22 : state]: ((B2 @ S0) => ((evalc @ C @ S0 @ S12) => ((evalc @ (while @ B2 @ C) @ S12 @ S22) => (evalc @ (while @ B2 @ C) @ S0 @ S22))))))). % evalc.WhileTrue
thf(fact_117_evalc_OWhileFalse, axiom,
    ((![B2 : state > $o, S4 : state, C : com]: ((~ ((B2 @ S4))) => (evalc @ (while @ B2 @ C) @ S4 @ S4))))). % evalc.WhileFalse
thf(fact_118_evalc__WHILE__case, axiom,
    ((![B2 : state > $o, C : com, S4 : state, T2 : state]: ((evalc @ (while @ B2 @ C) @ S4 @ T2) => (((T2 = S4) => (B2 @ S4)) => (~ (((B2 @ S4) => (![S1 : state]: ((evalc @ C @ S4 @ S1) => (~ ((evalc @ (while @ B2 @ C) @ S1 @ T2))))))))))))). % evalc_WHILE_case
thf(fact_119_com_Odistinct_I39_J, axiom,
    ((![X41 : com, X42 : com, X61 : state > $o, X62 : com]: (~ (((semi @ X41 @ X42) = (while @ X61 @ X62))))))). % com.distinct(39)
thf(fact_120_com_Odistinct_I9_J, axiom,
    ((![X61 : state > $o, X62 : com]: (~ ((skip = (while @ X61 @ X62))))))). % com.distinct(9)
thf(fact_121_image__constant__conv, axiom,
    ((![A2 : set_Ho840737317_state, C : hoare_958474565_state]: (((A2 = bot_bo105666705_state) => ((image_1849792670_state @ (^[X4 : hoare_958474565_state]: C) @ A2) = bot_bo105666705_state)) & ((~ ((A2 = bot_bo105666705_state))) => ((image_1849792670_state @ (^[X4 : hoare_958474565_state]: C) @ A2) = (insert776267541_state @ C @ bot_bo105666705_state))))))). % image_constant_conv
thf(fact_122_Body1, axiom,
    ((![G : set_Ho840737317_state, P2 : pname > state > state > $o, Q : pname > state > state > $o, Procs : set_pname, Pn : pname]: ((hoare_604442164_state @ (sup_su737426425_state @ G @ (image_2144627828_state @ (^[P5 : pname]: (hoare_1659279548_state @ (P2 @ P5) @ (body2 @ P5) @ (Q @ P5))) @ Procs)) @ (image_2144627828_state @ (^[P5 : pname]: (hoare_1659279548_state @ (P2 @ P5) @ (the_com @ (body @ P5)) @ (Q @ P5))) @ Procs)) => ((member_pname @ Pn @ Procs) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ (P2 @ Pn) @ (body2 @ Pn) @ (Q @ Pn)) @ bot_bo105666705_state))))))). % Body1
thf(fact_123_Un__empty, axiom,
    ((![A2 : set_Ho840737317_state, B : set_Ho840737317_state]: (((sup_su737426425_state @ A2 @ B) = bot_bo105666705_state) = (((A2 = bot_bo105666705_state)) & ((B = bot_bo105666705_state))))))). % Un_empty

% Conjectures (3)
thf(conj_0, hypothesis,
    (hoare_405891322gleton)).
thf(conj_1, hypothesis,
    ((![X8 : pname]: ((member_pname @ X8 @ (dom_pname_com @ body)) => (hoare_604442164_state @ g @ (insert776267541_state @ (hoare_1659279548_state @ (^[Y5 : state]: (^[Z4 : state]: (Y5 = Z4))) @ (body2 @ X8) @ (evalc @ (body2 @ X8))) @ bot_bo105666705_state)))))).
thf(conj_2, conjecture,
    ((hoare_604442164_state @ g @ (insert776267541_state @ (hoare_1659279548_state @ (^[Y5 : state]: (^[Z4 : state]: (Y5 = Z4))) @ skip @ (evalc @ skip)) @ bot_bo105666705_state)))).
